Discriminant divisors for conic bundles

1. Introduction

Conic bundles played a crucial role in the classification of Fano threefolds [MM81], [MM83]. Especially, Mori–Mukai establish the following formula

(1.0.1)

\Delta_{f}\equiv-f_{*}K_{X/T}^{2},

where $f:X\to T$ is a conic bundle from a smooth projective threefold to a smooth projective surface over an algebraically closed field of characteristic zero. Originally, $\Delta_{f}$ was defined as the reduced closed subscheme of $T$ that parametrises the singular fibres. Although this naive definition does not behave nicely in characteristic two, [ABBB21] introduced a well-behaved definition. The main objective of this article is to establish the above formula (1.0.1) in arbitrary characteristic by using the definition of [ABBB21].

Theorem 1.1 (Remark 5.11, Theorem 5.15).

Let $k$ be an algebraically closed field. Let $f:X\to T$ be a generically smooth conic bundle, where $X$ and $T$ are smooth projective varieties over $k$ . Then the numerical equivalence

\Delta_{f}\equiv-f_{*}(K_{X/T}^{2})

holds, where $K_{X/T}:=K_{X}-f^{*}K_{T}$ .

This theorem will be applied in forthcoming articles on the classification of Fano threefolds in positive characteristic [Tan-Fano1], [Tan-Fano2], [AT], [Tan-Fano4].

1.1. Overview of proofs and contents

In Section 3, we shall introduce a discriminant scheme $\Delta_{f}$ for a conic bundle $f:X\to T$ , which is a closed subscheme of $T$ . The definition is designed to satisfy the following two axiomatic properties (I) and (II).

(I)

Given a point $t\in T$ , $X_{t}$ is not smooth if and only if $t\in\Delta_{f}$ .
(II)

An equality $\Delta_{f^{\prime}}=\beta^{-1}\Delta_{f}$ of closed subschemes holds for every carterian diagram of noetherian schemes

$\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T,\end{CD}$

where $f$ and $f^{\prime}$ are conic bundles.

The actual definition of $\Delta_{f}$ is given as follows. We first consider the case when the base scheme $T$ is affine and $X$ is a conic on $\mathbb{P}^{2}_{T}$ . We have $T={\operatorname{Spec}}\,A$ , there is a closed embedding $X\subset\mathbb{P}^{2}_{A}$ , and

X={\operatorname{Proj}}\,A[x,y,z]/(Q),\quad Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy,\quad a,b,c,\alpha,\beta,\gamma\in A.

In this case, $\Delta_{f}$ is explicitly defined as follows:

(1.1.1)

\Delta_{f}={\operatorname{Spec}}\,(A/\delta(Q)A),\quad\text{where}\quad\delta(Q):=4abc+\alpha\beta\gamma-a\alpha^{2}-b\beta^{2}-c\gamma^{2}\in A.

We shall check that the definition $\Delta_{f}$ is independent of the choice of the closed embedding $X\subset\mathbb{P}^{2}_{A}$ and so on. The proof is done by using the fact that two embeddings are related by a linear transform. As mentioned already, this definition is based on [ABBB21]. Although [ABBB21] does not check the well-definedness, the equation (1.1.1) is extracted from [ABBB21], which would be the most essential part to reach the definition explained above. Similarly, we also introduce a closed subscheme $\Sigma_{f}$ of $T$ such that, in addition to the property corresponding to (II), $\Sigma_{f}$ consists of the points $t$ of $T$ such that $X_{t}$ is not geometrically reduced.

In Section 4, we study the relation between $\Delta_{f}$ and the singularities of $X$ . We here only treat the case when the base scheme $T$ is regular. We shall prove that the following are equivalent (Theorem 4.4):

•

$\Delta_{f}$ is regular.
•

$X$ is regular and any fibre of $f$ is geometrically reduced.

In particular, if both $X$ and $T$ are regular, then the singular locus of $\Delta_{f}$ is set-theoretically equal to $\Sigma_{f}$ .

In Section 5, we prove the formula (Theorem 1.1):

(1.1.2)

\Delta_{f}\equiv-f_{*}K_{X/T}^{2}.

Note that the argument in characteristic zero [MM83, Proposition 6.8] does not work (cf. Remark 7.6). We here overview some of the ideas of the proof. As a toy case, let us consider the case when $f:X\to T$ coincides with ${\operatorname{Univ}}_{\mathbb{P}^{2}_{k}/k}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}^{2}_{k}/k}^{\theta}$ , where

•

$k$ is an algebraically closed field,
•

$\theta$ is the Hilbert polynomial of conics,
•

${\operatorname{Hilb}}_{\mathbb{P}^{2}_{k}/k}^{\theta}$ denotes the Hilbert scheme parametrising all the conics on $\mathbb{P}^{2}$ , and
•

${\operatorname{Univ}}_{\mathbb{P}^{2}_{k}/k}^{\theta}$ is its universal family.

In particular, we have ${\operatorname{Hilb}}_{\mathbb{P}^{2}_{k}/k}^{\theta}\simeq\mathbb{P}^{5}_{k}$ . In order to check the numerical equivalence (1.1.2), it is enough to find a line $L$ on ${\operatorname{Hilb}}_{\mathbb{P}^{2}_{k}/k}^{\theta}\simeq\mathbb{P}^{5}_{k}$ such that $\Delta_{f}\cdot L=-(f_{*}K_{X/T}^{2})\cdot L$ . This is carried out by taking a general line $L$ such that $\Delta_{f}\cap L$ is smooth. For the general case, the problem is reduced, by standard argument, to the case when $T$ is a smooth projective curve. We embed $T$ to the relative Hilbert scheme ${\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ for $E:=f_{*}\omega_{X/T}^{-1}$ , which is a locally free sheaf on $T$ of rank $3$ . Then the problem is further reduced to the case when $f$ coincides with a relative version ${\operatorname{Univ}}_{\mathbb{P}_{T}(E)/T}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ of ${\operatorname{Univ}}_{\mathbb{P}^{2}_{k}/k}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}^{2}_{k}/k}^{\theta}$ . By $\rho({\operatorname{Hilb}}_{\mathbb{P}^{2}_{k}/k}^{\theta})=\rho(\mathbb{P}^{5})=1$ , it was enough to find a line $L$ satisfying $\Delta_{f}\cdot L=-(f_{*}K_{X/T}^{2})\cdot L$ in the above case. For our case, we can check that $\rho({\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta})=2$ . Then the proof is carried out by finding suitable two curves $C_{1}$ and $C_{2}$ such that $\Delta_{f}\cdot C_{1}=-(f_{*}K_{X/T}^{2})\cdot C_{1}$ and $\Delta_{f}\cdot C_{2}=-(f_{*}K_{X/T}^{2})\cdot C_{2}$ . For more details, see Theorem 5.15.

In order to justify the above argument, we shall introduce an invertible sheaf $\Delta_{f}^{{\operatorname{bdl}}}$ , called the discriminant bundle, which is a variant of the discriminant scheme $\Delta_{f}$ (Subsection 5.3). If $f:X\to T$ is a generically smooth conic bundle such that $T$ is a noetherian integral scheme, then $\Delta_{f}$ is an effective Cartier divisor on $T$ . However, $\Delta_{f}$ is useless when $f:X\to T$ is a wild conic bundle, i.e., no fibre is smooth (more explicitly, we have $\Delta_{f}=T$ ). On the other hand, the discriminant bundle $\Delta^{{\operatorname{bdl}}}_{f}$ is an invertible sheaf on $T$ even if $f$ is a wild conic bundle. Furthermore, the discriminant bundles $\Delta^{{\operatorname{bdl}}}_{f}$ satisfy the following properties (Theorem 5.13).

(A)

If $f:X\to T$ is a generically smooth conic bundle such that $T$ is a noetherian integral scheme, then $\Delta^{{\operatorname{bdl}}}_{f}\simeq\mathcal{O}_{T}(\Delta_{f})$ .
(B)

An isomorphism $\Delta_{f^{\prime}}^{{\operatorname{bdl}}}\simeq\beta^{*}\Delta_{f}^{{\operatorname{bdl}}}$ of invertible sheaves holds for every carterian diagram

$\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T,\end{CD}$

where $T$ and $T^{\prime}$ are smooth varieties over a field.

The definition of $\Delta^{{\operatorname{bdl}}}_{f}$ is given as follows. For a conic bundle $f:X\to T$ , we have the following cartesian diagram for the Hilbert polynomial $\theta$ of conics:

\begin{CD}X@>{\varphi}>{}>\widetilde{X}:={\operatorname{Univ}}^{\theta}_{\mathbb{P}_{T}(f_{*}\omega_{X/T}^{-1})/T}\\ @V{}V{f}V@V{}V{\widetilde{f}}V\\ T@>{\psi}>{}>\widetilde{T}:={\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T}(f_{*}\omega_{X/T}^{-1})/T}.\end{CD}

Then $\Delta^{{\operatorname{bdl}}}_{f}$ is defined by $\Delta^{{\operatorname{bdl}}}_{f}:=\psi^{*}\mathcal{O}_{\widetilde{T}}(\Delta_{\widetilde{f}})$ . It is clear that (A) holds. The property (B) can be checked as follows. By standard argument, we may assume that $\beta:T^{\prime}\to T$ is either smooth or a closed immersion such that $\beta(T^{\prime})$ is an effective Cartier divisor. For each case, it is not so hard to check (B).

In Section 6, we study $\Delta_{f}$ for surface conic bundles. We restrict ourselves to treating the case when $T$ is a smooth curve and $X$ is a surface having at worst canonical singularities. Since we are interested in the relation between $\Delta_{f}$ and the singularities of $X$ , let us assume that there is a closed point $0\in T$ such that $X_{0}$ is the unique singular fibre. We shall first prove that

\deg\Delta_{f}=m-1,

where $m$ is the number of the irreducible components of the central fibre $Y_{0}$ for the minimal resolution $\varphi:Y\to X$ of $X$ . Given such a conic bundle $f:X\to T$ , we shall prove that the singularity of $X$ is determined by $\deg\Delta_{f}$ and whether the $X_{0}$ is reduced (Theorem 6.3). We then exhibit several examples.

In Section 7, we observe some phenomena which occur only in characteristic two. For example, the following hold in characteristic $\neq 2$ (Proposition 7.2).

(1)

If $f:X\to T$ is a conic bundle of smooth varieties, then $\Delta_{f}$ is a reduced divisor.
(2)

If $f:X\to T$ is a conic bundle of smooth varieties with $\dim T=2$ , then $\Delta_{f}$ is normal crossing.

We shall see that both the properties fail in characteristic two (Example 7.3). Furthermore, (2) does not hold even after replacing $\Delta_{f}$ by $(\Delta_{f})_{{\operatorname{red}}}$ . Roughly speaking, the singularities of $\Delta_{f}$ and $(\Delta_{f})_{{\operatorname{red}}}$ can be arbitrarily bad even if $f:X\to T$ is a generically smooth conic bundle from a smooth threefold $X$ to a smooth surface $T$ .

Remark 1.2.

As explained above, we shall establish some foundational results on discriminant divisors $\Delta_{f}$ for conic bundles $f:X\to T$ , where $X$ and $T$ are noetherian schemes. Most results in Section 3 and Section 4 are essentially contained in [Bea77, Chapitre I] and [Sar82, Section 1] for the case when $X$ and $T$ are smooth varieties over an algebraically closed field of characteristic $\neq 2$ . As far as the author knows, the contents in Section 3 and Section 4 are new even when $X$ and $T$ are smooth varieties over an algebraically closed field of characteristic two.

Acknowledgements: The author thanks the referee for reading the manuscript carefully and for suggesting several improvements. The author was funded by JSPS KAKENHI Grant numbers JP22H01112 and JP23K03028.

2. Preliminaries

2.1. Notation

(1)

We say that $X$ is a variety (over a field $k$ ) if $X$ is an integral scheme that is separated and of finite type over $k$ . We say that $X$ is a curve (resp. surface, resp. threefold) if $X$ is a variety of dimension one (resp. two, resp. three).
(2)

We say that a noetherian scheme $X$ is excellent if all the stalks $\mathcal{O}_{X,x}$ are excellent.
(3)

Given a morphism $f:X\to S$ of schemes and a point $s\in S$ , $X_{s}$ denotes the fibre of $f$ over $s$ , i.e., $X_{s}:=X\times_{S}{\operatorname{Spec}}\,\kappa(s)$ , where $\kappa(s)$ denotes the residue field of $S$ at $s$ . Unless otherwise specified, we consider $\kappa(s)$ as the base field of $X_{s}$ . For example, we say that $X_{s}$ is smooth (resp. geometrically reduced) if $X_{s}$ is smooth (resp. geometrically reduced) over $\kappa(s)$ . When $S$ is an integral scheme, the base field of the generic fibre is the function field $K(S)$ of $S$ .
(4)

For the definition of the relative dualising sheaf $\omega_{X/T}$ , we refer to Remark 2.6.
(5)

For the definition of conic bundles, see Subsection 2.2.
(6)

For the definition and basic properties on strictly henselian local rings, we refer to [Fu15, Subsection 2.8].
(7)

For a ring $A$ , ${\operatorname{GL}}_{n}(A)$ denotes the group consisting of invertible matrices of size $n\times n$ . Given a matrix $M$ of size $n\times n$ , it is well known that $M$ is invertible if and only if $\det M\in A^{\times}$ .

(8)

Given $f\in A[x,y,z]$ and $M\in{\operatorname{GL}}_{3}(A)$ , we define $f^{M}\in A[x,y,z]$ as follows. For $f=f(x,y,z)$ , we set

f^{M}(x,y,z):=f((M(x,y,z)^{T})^{T}),

where $(-)^{T}$ denotes the transposed matrix. More explicitly, for

M\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}m_{11}&m_{12}&m_{13}\\ m_{21}&m_{22}&m_{23}\\ m_{31}&m_{32}&m_{33}\\ \end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}m_{11}x+m_{12}y+m_{13}z\\ m_{21}x+m_{22}y+m_{23}z\\ m_{31}x+m_{32}y+m_{33}z\\ \end{pmatrix},

we have

f^{M}(x,y,z)=f(m_{11}x+m_{12}y+m_{13}z,m_{21}x+m_{22}y+m_{23}z,m_{31}x+m_{32}y+m_{33}z).

(9)

Set $\theta:=2m+1\in\mathbb{Z}[m]$ , which is the Hilbert polynomial of an arbitrary conic on $\mathbb{P}^{2}_{k}$ for a field $k$ .
(10)

Given a scheme $S$ and $S$ -schemes $X$ and $Y$ , we say that $X$ is $S$ -isomorphic to $Y$ if there exists an isomorphism $\theta:X\to Y$ of schemes such that both $\theta:X\to Y$ and $\theta^{-1}:Y\to X$ are $S$ -morphisms. When $S={\operatorname{Spec}}\,A$ for a ring $A$ , we say that $X$ is $A$ -isomorphic to $Y$ if $X$ is ${\operatorname{Spec}}\,A$ -isomorphic to $Y$ .

2.1.1. Singularities of minimal model program

We will freely use the standard notation in birational geometry, for which we refer to [Kol13] and [KM98]. Let $X$ be an integral normal excellent scheme admitting a dualising complex. We say that $X$ is canonical or has at worst canonical singularities if

(1)

$K_{X}$ is $\mathbb{Q}$ -Cartier, and
(2)

all the coefficients $a_{i}$ are $\geq 0$ for every proper birational morphism $f:Y\to X$ from an integral normal excellent scheme $Y$ and

$K_{Y}=f^{*}K_{X}+\sum_{i=1}^{r}a_{i}E_{i},$

where $E_{1},...,E_{r}$ are all the $f$ -exceptional prime divisors.

Under assuming (1), if there exists a proper birational morphism $g:Z\to X$ from an integral regular excellent scheme $Z$ , then (2) is known to be equivalent to (2)’ below [KM98, Lemma 2.30].

(2)’

all the coefficients $b_{i}$ are $\geq 0$ for some proper birational morphism $g:Z\to X$ from an integral regular excellent scheme $Z$ and

$K_{Z}=g^{*}K_{X}+\sum_{j=1}^{s}b_{j}F_{j},$

where $F_{1},...,F_{s}$ are all the $g$ -exceptional prime divisors.

2.1.2. Linear algebra over local rings

Lemma 2.1.

Let $A$ be a local ring. If $M\in{\operatorname{GL}}_{n}(A)$ , then we can write $M=M_{1}\cdots M_{r}$ , where each $M_{i}$ is an elementary matrix. Recall, e.g., that if $n=3$ , then the list of elementary matrices are as follows:

	$\displaystyle\begin{pmatrix}\lambda&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&0\\ 0&\lambda&0\\ 0&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&\lambda\\ \end{pmatrix}\qquad(\lambda\in A^{\times})$
	$\displaystyle\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&0\\ 0&0&1\\ 0&1&0\\ \end{pmatrix}\qquad\begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\\ \end{pmatrix}$
	$\displaystyle\begin{pmatrix}1&\mu&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&\mu\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&0\\ 0&1&\mu\\ 0&0&1\\ \end{pmatrix}$
	$\displaystyle\begin{pmatrix}1&0&0\\ \mu&1&0\\ 0&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&0\\ 0&1&0\\ \mu&0&1\\ \end{pmatrix}\qquad\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&\mu&1\\ \end{pmatrix}\qquad(\mu\in A).$

Proof.

If $A$ is a field, then the assertion follows from linear algebra, i.e., $M$ becomes the identity matrix $E$ after applying elementary transformations finitely many times.

The same argument works even when $A$ is a local ring. For example, we can find an entry $m_{ij}$ of $M$ which is a unit of $A$ , where $m_{ij}$ denotes the $(i,j)$ -entry of $M$ . Then, applying suitable elementary transformations, we may assume that $m_{11}=1$ . ∎

2.2. Definitions of conic bundles

In this subsection, we first introduce the definition of conic bundles (Definition 2.3) and conics over a noetherian ring (Definition 2.8). We also summarise some basic properties, which should be well known to experts.

2.2.1. Conic bundles

Definition 2.2.

Let $\kappa$ be a field.

(1)

We say that $C$ is a conic on $\mathbb{P}^{2}_{\kappa}$ if the equality

C={\operatorname{Proj}}\,\kappa[x,y,z]/(ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta za+\gamma xy)\subset\mathbb{P}^{2}_{\kappa}={\operatorname{Proj}}\,\kappa[x,y,z]

holds for some $(a,b,c,\alpha,\beta,\gamma)\in\kappa^{6}\setminus\{(0,0,0,0,0,0)\}$ .

(2)

We say that $C$ is a conic over $\kappa$ if $C$ is $\kappa$ -isomorphic to a conic on $\mathbb{P}^{2}_{\kappa}$ .

Definition 2.3.

We say that $f:X\to T$ is a conic bundle if $f:X\to T$ is a flat proper morphism of noetherian schemes such that $X_{t}$ is a conic over $\kappa(t)$ for any point $t\in T$ .

Remark 2.4.

If $f:X\to T$ is a conic bundle and $T^{\prime}\to T$ is a morphism of noetherian schemes, then also the base change $X\times_{T}T^{\prime}\to T^{\prime}$ is a conic bundle.

Lemma 2.5.

Let $f:X\to T$ be a conic bundle. Then $f_{*}\mathcal{O}_{X}=\mathcal{O}_{T}$ and $R^{i}f_{*}\mathcal{O}_{X}=0$ for every $i>0$ .

Proof.

We have $H^{i}(X_{t},\mathcal{O}_{X_{t}})=0$ for every $i>0$ . By [Har77, Ch. III, Theorem 12.11], it holds that $R^{i}f_{*}\mathcal{O}_{X}=0$ for every $i>0$ and $f_{*}\mathcal{O}_{X}$ is an invertible sheaf on $T$ . Then $f^{\sharp}:\mathcal{O}_{T}\to f_{*}\mathcal{O}_{X}$ is an isomorphism, because the induced ring homomorphism

f^{\sharp}\otimes k(t):\mathcal{O}_{T}\otimes k(t)\to(f_{*}\mathcal{O}_{X})\otimes k(t)

is a nonzero $k(t)$ -linear map of one-dimensional $k(t)$ -vector spaces for every point $t\in T$ . ∎

Since a conic bundle $f:X\to T$ is a flat proper morphism of noetherian schemes whose fibres are Cohen–Macaulay of pure dimension one, there exists a dualising sheaf $\omega_{X/T}$ in the sense of [Con00, page 157].

Remark 2.6.

We here summarise some basic properties on $\omega_{X/T}$ for later usage.

(1)

Let $f:X\to T$ be a conic bundle of noetherian schemes. Since every fibre $X_{t}$ is Gorenstein of pure dimension one, we have $\omega_{X/T}=\mathcal{H}^{-1}(f^{!}\mathcal{O}_{Y})$ and $\omega_{X/T}$ is an invertible sheaf on $X$ .
(2)

Let

$\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}$

be a carterian diagram of noetherian schemes, where $f$ (and hence $f^{\prime}$ ) is a conic bundle. We then obtain

$\omega_{X^{\prime}/T^{\prime}}\simeq\alpha^{*}\omega_{X/T}.$

In particular, $\omega_{X/T}|_{X_{t}}\simeq\omega_{X_{t}}$ for any point $t\in T$ and the fibre $X_{t}$ over $t$ .
(3)

Let $f:X\to T$ be a conic bundle, where $X$ and $T$ are Gorenstein normal varieties over an algebraically closed field. Then $\omega_{X/T}\simeq\mathcal{O}_{X}(K_{X}-f^{*}K_{T})$ .

Proposition 2.7 (cf. [Bea77]*Proposition 1.2).

Let $f:X\to T$ be a conic bundle. Then the following hold.

(1)

$R^{i}f_{*}\omega_{X/T}^{-1}=0$ for every $i>0$ .
(2)

$f_{*}\omega_{X/T}^{-1}:=f_{*}(\omega_{X/T}^{-1})$ is a locally free sheaf of rank $3$ .
(3)

$\omega_{X/T}^{-1}$ is very ample over $T$ , and hence it defines a closed immersion $\iota:X\hookrightarrow\mathbb{P}(f_{*}\omega_{X/T}^{-1})$ over $T$ .

Proof.

Let us show (1) and (2). Fix a point $t\in T$ . By [Har77, Ch. III, Corollary 12.9], it is enough to show that $\dim_{\kappa(t)}H^{0}(X_{t},\omega^{-1}_{X/T}|_{X_{t}})=3$ and $H^{i}(X_{t},\omega^{-1}_{X/T})=0$ for every $i>0$ . This follows from $\omega^{-1}_{X/T}|_{X_{t}}\simeq\omega^{-1}_{X_{t}}$ and the fact that $X_{t}$ is a conic on $\mathbb{P}^{2}_{\kappa(t)}$ . Thus (1) and (2) hold.

Let us show (3). First, we prove that $\omega_{X/T}^{-1}$ is $f$ -free, i.e., $f^{*}f_{*}\omega_{X/T}^{-1}\to\omega_{X/T}^{-1}$ is surjective. Fix a closed point $x\in X$ and set $t:=f(x)$ . Then $x\in X_{t}$ , and there exists a section $\sigma\in H^{0}(X_{t},\omega_{X/T}^{-1}|_{X_{t}})$ such that $\sigma|_{\{x\}}\neq 0$ . We have two maps

(2.7.1)

f_{*}\omega_{X}^{-1}\to(f_{*}\omega_{X/T}^{-1})\otimes k(t)\xrightarrow{\simeq}H^{0}(X_{t},\omega_{X/T}^{-1}|_{X_{t}}).

The first map is the projection, and the second map is obtained by [Har77, Ch. III, Corollary 12.9]. Let $T^{\prime}$ be an affine open subset containing $t$ , and set ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}X^{\prime}:=f^{-1}(T^{\prime})}$ . Then, by the above (2.7.1), we obtain

(2.7.2)

H^{0}(X^{\prime},\omega^{-1}_{X/T}|_{X^{\prime}})\to H^{0}(X^{\prime},\omega^{-1}_{X/T}|_{X^{\prime}})\otimes k(t)\xrightarrow{\simeq}H^{0}(X_{t},\omega_{X/T}^{-1}|_{X_{t}}).

Since the first map is surjective and the second map is an isomorphism, (2.7.2) is surjective. Hence we can take $\tau\in H^{0}(X^{\prime},\omega^{-1}_{X/T}|_{X^{\prime}})$ such that $\tau|_{X_{t}}=\sigma$ . Thus $\omega_{X/T}^{-1}$ is $f$ -free. Let $\iota:X\to\mathbb{P}(f_{*}\omega_{X/T}^{-1})$ be the induced morphism.

Next, we show that $\iota:X\to\mathbb{P}(f_{*}\omega_{X/T}^{-1})$ is injective. Pick distinct points $x,x^{\prime}\in X$ such that $\iota(x)=\iota(x^{\prime})=:p$ . Set $t:=f(x)=f(x^{\prime})$ . Then $x,x^{\prime}\in X_{t}$ , and hence we can take a $\sigma\in H^{0}(X_{t},\omega_{X}^{-1}|_{X_{t}})$ such that $\sigma|_{\{x\}}=0$ and $\sigma|_{\{x^{\prime}\}}\neq 0$ . By the surjection (2.7.2), there exists $\tau\in H^{0}(X^{\prime},\omega_{X/T}^{-1}|_{X^{\prime}})$ such that $\tau|_{X_{t}}=\sigma$ . This implies $\iota(x)\neq\iota(x^{\prime})$ , which is absurd.

Finally, we show that $\iota:X\to\mathbb{P}(f_{*}\omega_{X/T}^{-1})=:{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}$ is a closed immersion. Now, we have a commutative diagram

where $P_{t}$ is a fibre of $\pi\colon P\to T$ . Since $X_{t}$ is a conic over $\kappa(t)$ , we can embed this into $\mathbb{P}^{2}_{\kappa(t)}$ , and this closed immersion is induced by $|-K_{X_{t}}|$ . By construction of $\iota$ , also $\iota|_{X_{t}}$ is a closed immersion. Then we can check that $\iota$ is a closed immersion by applying Nakayama’s lemma. Thus (3) holds. ∎

2.2.2. Conics over rings

Definition 2.8.

Let $A$ be a noetherian ring.

(1)

We say that $X$ is a conic on $\mathbb{P}^{2}_{A}$ if the equation

X={\operatorname{Proj}}\,A[x,y,z]/(ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta za+\gamma xy)\subset\mathbb{P}^{2}_{\kappa}={\operatorname{Proj}}\,A[x,y,z]

holds for some $(a,b,c,\alpha,\beta,\gamma)\in A^{6}$ and the induced morphism $X\to{\operatorname{Spec}}\,A$ is a conic bundle.

(2)

We say that $X$ is a conic over $A$ if $X$ is $A$ -isomorphic to a conic on $\mathbb{P}^{2}_{A}$ .

Proposition 2.9.

Let $A$ be a noetherian ring and take

Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z].

For $X:={\operatorname{Proj}}\,A[x,y,z]/(Q)$ , the following are equivalent.

(1)

$X$ is a conic on $\mathbb{P}^{2}_{A}$ .
(2)

The induced morphism $f:X\to{\operatorname{Spec}}\,A$ is a conic bundle.
(3)

The induced morphism $f:X\to{\operatorname{Spec}}\,A$ is flat and any fibre of $f$ is one-dimensional.

Furthermore, if $(A,\mathfrak{m})$ is a noetherian local ring, then each of (1)–(3) is equivalent to (4).

(4)

At least one of $a,b,c,\alpha,\beta,\gamma$ is not contained in $\mathfrak{m}$ .

Proof.

The implications $(1)\Rightarrow(2)\Rightarrow(3)$ are clear. Assume (3). Then $f:X\to{\operatorname{Spec}}\,A$ is a proper flat morphism. For any $t\in{\operatorname{Spec}}\,A$ , its fibre $f^{-1}(t)$ is a conic over $k(t)$ , as it is one-dimensional. Thus (1) holds.

Assume that $(A,\mathfrak{m})$ is a local ring. Then it is obvious that (3) implies (4). Assume (4). We first reduce the problem to the case when $a\not\in\mathfrak{m}$ . By symmetry, we may assume that $\gamma\not\in\mathfrak{m}$ and $a,b,c\in\mathfrak{m}$ . Applying the linear transform $(x,y,z)\mapsto(x,y+x,z)$ , the problem is reduced to the case when $a\not\in\mathfrak{m}$ .

Then the affine open subset $D_{+}(z)$ of $X$ can be written as

{\operatorname{Spec}}\,A[x,y]/(ax^{2}+\varphi(y)x+\psi(y)),\qquad\varphi(y),\psi(y)\in A[y].

We get an $A$ -module isomorphism $A[x,y]/(ax^{2}+\varphi(y)x+\psi(y))\simeq A[y]\oplus xA[y]$ , which is a free $A$ -module. Therefore, $f:X\to{\operatorname{Spec}}\,A$ is flat. As the fibre $f^{-1}(\mathfrak{m})$ is one-dimensional, any fibre of $f$ is one-dimensional. Thus (3) holds. ∎

2.3. Local description of conic bundles

Given a strictly henselian noetherian local ring $A$ and a conic $X={\operatorname{Proj}}\,A[x,y,z]/(Q)$ on $\mathbb{P}^{2}_{A}$ , the purpose of this subsection is to simplify the defining equation $Q$ via $A$ -linear transformations. In Subsection 2.3.1 and Subsection 2.3.2, we treat the case when the residue field of $A$ is of characteristic $\neq 2$ and $=2$ , respectively.

Lemma 2.10.

Let $(A,\mathfrak{m},\kappa)$ be a noetherian local ring. Let

X:={\rm Proj}\,A[x,y,z]/(Q).

be a conic on $\mathbb{P}^{2}_{A}$ with $Q\in A[x,y,z]$ . Then there exists $M\in{\operatorname{GL}}_{3}(A)$ such that

Q^{M}=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy

for some $a,b,c\in A^{\times}$ and $\alpha,\beta,\gamma\in A$ .

Proof.

By the same argument as in the second paragraph of the proof of Proposition 2.9, we may assume that $a\in A^{\times}$ . Applying a linear transform $(x,y,z)\mapsto{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(x+dx+ex,y,z)}$ for some $d,e\in\{0,1\}$ , we obtain $a\not\in\mathfrak{m},b\not\in\mathfrak{m}$ , and $c\not\in\mathfrak{m}$ . ∎

2.3.1. Local description in characteristic $\neq 2$

Proposition 2.11.

Let $\kappa$ be a field of characteristic $\neq 2$ . Let

X:={\rm Proj}\,\kappa[x,y,z]/(Q).

be a conic on $\mathbb{P}^{2}_{\kappa}$ with $Q\in\kappa[x,y,z]$ . Take the $3\times 3$ symmetric matrix $S_{Q}$ such that $Q=(x\,y\,z)S_{Q}(x\,y\,z)^{T}$ . Then the following hold.

(1)

There exists $M\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M}=ax^{2}+by^{2}+cz^{2}$

for some $a,b,c\in\kappa$ .
(2)
The following are equivalent.
1. (a)
  
  $X$ is smooth.
2. (b)
  
  ${\rm rank}(S_{Q})=3$ .
(3)
The following are equivalent.
1. (a)
  
  $X$ is reduced.
2. (b)
  
  $X$ is geometrically reduced.
3. (c)
  
  ${\rm rank}(S_{Q})=2$ .
(4)
The following are equivalent.
1. (a)
  
  $X$ is not reduced.
2. (b)
  
  $X$ is not geometrically reduced.
3. (c)
  
  ${\rm rank}(S_{Q})=1$ .

Proof.

All the assertions follow from linear algebra. ∎

Proposition 2.12.

Let $(A,\mathfrak{m},\kappa)$ be a noetherian local ring such that $\kappa$ is of characteristic $\neq 2$ . Let

X:={\rm Proj}\,A[x,y,z]/(Q).

be a conic on $\mathbb{P}^{2}_{A}$ with $Q\in A[x,y,z]$ . Let $0\in{\operatorname{Spec}}\,A$ be the closed point. Then the following hold.

(1)

There exists $M\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M}=ax^{2}+by^{2}+cz^{2}+\alpha yz.$

for some $a\in A^{\times}$ and $b,c,\alpha\in A$ . Furthermore, $X_{0}$ is not reduced if and only if $b,c,\alpha\in\mathfrak{m}$ .
(2)

If $X_{0}$ is reduced, then there exists $M^{\prime}\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M^{\prime}}=a^{\prime}x^{2}+b^{\prime}y^{2}+c^{\prime}z^{2}$

for some $a^{\prime},b^{\prime}\in A^{\times}$ and $c^{\prime}\in A$ . Furthermore, $X_{0}$ is not smooth if and only if $c^{\prime}\in\mathfrak{m}$ .

Proof.

Note that $2\in A^{\times}$ . Indeed, there exists $u\in A$ such that $2u\in 1+\mathfrak{m}$ , which implies that $2\in A^{\times}$ .

Let us show (1). By Proposition 2.9, we can write

Q=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy,

where $a\in A^{\times}$ and $b,c,\alpha,\beta,\gamma\in A$ . Completing the square

ax^{2}+\beta zx+\gamma xy=a\left(x+\frac{\beta}{2a}z+\frac{\gamma}{2a}y\right)^{2}-\left(\frac{\beta}{2a}z+\frac{\gamma}{2a}y\right)^{2},

we may assume that $\beta=\gamma=0$ :

Q=ax^{2}+by^{2}+cz^{2}+\alpha yz.

By $a\in A^{\times}$ and Proposition 2.11, $X_{0}$ is not reduced if and only if $b,c,\alpha\in\mathfrak{m}$ . Thus (1) holds.

Let us show (2). By (1), we may assume that

Q=ax^{2}+by^{2}+cz^{2}+\alpha yz

for some $a\in A^{\times}$ and $b,c,\alpha\in A$ . Since $X_{0}$ is reduced, one of $b,c,\alpha$ is contained in $A^{\times}$ . We may assume that $b\in A^{\times}$ by applying $(x,y,z)\mapsto(x,y,z+y)$ for the case when $b,c\in\mathfrak{m}$ and $\alpha\in A^{\times}$ . Then we are done by completing the square again:

by^{2}+cz^{2}+\alpha yz=b\left(y+\frac{\alpha}{2b}z\right)^{2}+\left(c-\frac{\alpha^{2}}{4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b^{2}}}\right)z^{2}.

Thus (2) holds. ∎

Corollary 2.13.

Let $(A,\mathfrak{m},\kappa)$ be a strictly henselian noetherian local ring such that $\kappa$ is of characteristic $\neq 2$ . Let $0\in{\operatorname{Spec}}\,A$ be the closed point. Let

X:={\rm Proj}\,A[x,y,z]/(Q).

be a conic on $\mathbb{P}^{2}_{A}$ with $Q\in A[x,y,z]$ . Then the following hold.

(1)

There exists $M\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M}=x^{2}+by^{2}+cz^{2}+\alpha yz.$

for some $b,c,\alpha\in A$ . Furthermore, $X_{0}$ is not reduced if and only if $b,c,\alpha\in\mathfrak{m}$ .
(2)

If $X_{0}$ is reduced, then there exists $M^{\prime}\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M^{\prime}}=x^{2}+y^{2}+c^{\prime}z^{2}$

for some $c^{\prime}\in A$ . Furthermore, $X_{0}$ is not smooth over $\kappa$ if and only if $c^{\prime}\in\mathfrak{m}$ .
(3)

If $X_{0}$ is smooth over $\kappa$ , then there exists $M^{\prime\prime}\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M^{\prime\prime}}=x^{2}+y^{2}+z^{2}.$

Proof.

Since $A$ is a strictly henselian local ring, there exists $\sqrt{d}\in A$ for any element $d\in A^{\times}$ . Then the assertions follow from Proposition 2.12. ∎

2.3.2. Local description in characteristic $2$

Proposition 2.14.

Let $(A,\mathfrak{m},\kappa)$ be a noetherian local ring such that $\kappa$ is of characteristic two. Let

X:={\operatorname{Proj}}\,A[x,y,z]/(Q)

be a conic on $\mathbb{P}^{2}_{A}$ with $Q\in A[x,y,z]$ . Let $0\in{\operatorname{Spec}}\,A$ be the closed point. Then the following hold.

(1)

If $X_{0}$ is geometrically reduced, then there exists $M\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M}=ax^{2}+by^{2}+cz^{2}+yz+2\beta zx+2\gamma xy$

for some $a,c,\beta,\gamma\in A$ and $b\in A^{\times}$ .
(2)

If $A$ is a strictly henselian local ring and $X_{0}$ is geometrically reduced, then there exists $M^{\prime}\in{\operatorname{GL}}_{3}(A)$ such that

$Q^{M^{\prime}}=a^{\prime}x^{2}+yz$

for some $a^{\prime}\in A$ . In this case, $X_{0}$ is smooth if and only if $a^{\prime}\in A^{\times}$ .

Proof.

Let us show (1). We can write

Q=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy

for some $(a,b,c,\alpha,\beta,\gamma)\in A^{6}\setminus\{(0,...,0)\}$ . We set $\overline{\alpha}:=\alpha\mod\mathfrak{m},\overline{\beta}:=\beta\mod\mathfrak{m},...$ etc. As $X_{0}$ is geometrically reduced, we have $(\overline{\alpha},\overline{\beta},\overline{\gamma})\neq(0,0,0)$ . By symmetry, we may assume that $\overline{\alpha}\neq 0$ , i.e., $\alpha\in A^{\times}$ . Applying $(x,y,z)\mapsto(x,\alpha^{-1}y,z)$ , the problem is reduced to the case when $\alpha=1$ :

Q(x,y,z)=ax^{2}+by^{2}+cz^{2}+yz+\beta zx+\gamma xy.

We have

yz+\beta zx+\gamma xy=(y+\beta x)(z+\gamma x)-\beta\gamma x^{2}.

Therefore, by applying $(x,y+\beta x,z+\gamma x)\mapsto(x,y,z)$ , we may assume that $\alpha=1,\beta=2\beta^{\prime},\gamma=2\gamma^{\prime}$ for some $\beta^{\prime},\gamma^{\prime}\in A$ :

Q(x,y,z)=ax^{2}+by^{2}+cz^{2}+yz+2\beta^{\prime}zx+2\gamma^{\prime}xy.

If $b\in A^{\times}$ or $c\in A^{\times}$ , then we may assume $b\in A^{\times}$ by switching $y$ and $z$ if necessary. If $b\in\mathfrak{m}$ and $c\in\mathfrak{m}$ . then we apply $(x,y,z)\mapsto(x,y,y+z)$ :

	$\displaystyle Q(x,y,y+z)$	$\displaystyle=$	$\displaystyle ax^{2}+by^{2}+c(y+z)^{2}+y(y+z)+2\beta^{\prime}(y+z)x+2\gamma^{\prime}xy$
		$\displaystyle=$	$\displaystyle ax^{2}+(b+c+1)y^{2}+cz^{2}+(1+2c)yz+2\beta^{\prime}zx+2(\beta^{\prime}+\gamma^{\prime})xy.$

By applying $(1+2c)y\mapsto y$ , we are done, because $1+b+c\in 1+\mathfrak{m}\subset A^{\times}$ . Thus (1) holds.

Let us show (2). By (1), we may assume that

Q(x,y,z)=ax^{2}+by^{2}+cz^{2}+yz+2\beta zx+2\gamma xy

for some $a,c,\beta,\gamma\in A$ and $b\in A^{\times}$ . Since $\kappa$ is separably closed, we have

\overline{b}Y^{2}+Y+\overline{c}=\overline{b}(Y+\overline{\delta})(Y+\overline{\epsilon})

for some $\overline{\delta},\overline{\epsilon}\in\kappa$ with $\overline{\delta}\neq\overline{\epsilon}$ . By Hensel’s lemma, it holds that

bY^{2}+Y+c=b(Y+\delta)(Y+\epsilon)\qquad\text{in}\qquad A[Y]

for some lifts $\delta,\epsilon\in A$ of $\overline{\delta},\overline{\epsilon}\in\kappa$ . Hence we have $by^{2}+yz+cz^{2}=b(y+\delta z)(y+\epsilon z)$ and

Q(x,y,z)=ax^{2}+by^{2}+cz^{2}+yz+2\beta zx+2\gamma xy=ax^{2}+b(y+\delta z)(y+\epsilon z)+2\beta zx+2\gamma xy.

Consider the matrix

\begin{pmatrix}1&\delta\\ 1&\epsilon\end{pmatrix}.

This is an invertible matrix, because its determinant $\epsilon-\delta$ is contained in $A^{\times}$ (indeed, its reduction $\overline{\epsilon}-\overline{\delta}\in\kappa$ is nonzero). Applying $(x,y+\delta z,y+\epsilon z)\mapsto(x,y,z)$ , the problem is reduced to the case when $Q(x,y,z)=ax^{2}+\alpha yz+2\beta zx+2\gamma xy$ for some $a,\beta,\gamma\in A$ and $\alpha\in A^{\times}$ . By using $(x,y,z)\mapsto(x,y,\alpha^{-1}z)$ , we may assume that $\alpha=1$ : $Q(x,y,z)=ax^{2}+yz+2\beta zx+2\gamma xy$ . For $a^{\prime}:=a-4\beta\gamma$ , $y^{\prime}:={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y+2\beta x}$ and $z^{\prime}:=z+2\gamma x$ , we obtain

Q(x,y,z)=ax^{2}+yz+2\beta zx+2\gamma xy=(a-4\beta\gamma)x^{2}+(y+2\beta x)(z+2\gamma x)=a^{\prime}x^{2}+y^{\prime}z^{\prime}.

Then it is clear that $X_{0}$ is smooth if and only if $a^{\prime}\in A^{\times}$ . Thus (2) holds. ∎

Proposition 2.15.

Let $\kappa$ be a separably closed field of characteristic two. Let

C:={\operatorname{Proj}}\,\kappa[x,y,z]/(Q)

be a conic on $\mathbb{P}^{2}_{\kappa}$ . Then the following hold.

(1)
There exists $M\in{\operatorname{GL}}_{3}(\kappa)$ such that one of the following holds.
1. (a)
  
  $Q^{M}=ax^{2}+by^{2}+cz^{2}$ for some $a,b,c\in\kappa$ . In this case, $C$ is not geometrically reduced.
2. (b)
  
  $Q^{M}=ax^{2}+yz$ for some $a\in\kappa$ . In this case, $C$ is geometrically reduced. Furthermore, $C$ is smooth if and only if $a\neq 0$ .
(2)
If $\kappa$ is algebraically closed, then there exists $M^{\prime}\in{\operatorname{GL}}_{3}(\kappa)$ such that one of the following holds.
1. (a)
  
  $Q^{M^{\prime}}=x^{2}$ . In this case, $C$ is not reduced.
2. (b)
  
  $Q^{M^{\prime}}=yz$ . In this case, $C$ is not smooth but is reduced.
3. (c)
  
  $Q^{M^{\prime}}=x^{2}+yz$ . In this case, $C$ is smooth.

Proof.

Let us show (1). If $X$ is geometrically reduced, then the assertion follows from Proposition 2.14(2). We may assume that $C$ is not geometrically reduced. Then $\alpha=\beta=\gamma=0$ . Thus (1) holds. The assertion (2) immediately follows from (1). ∎

2.4. Generic smoothness

Definition 2.16.

Let $f:X\to T$ be a conic bundle such that $T$ is a noetherian integral scheme. We say that $f$ is generically smooth if the generic fibre of $f$ is smooth. We say that $f$ is wild if $X$ is normal and $f$ is not generically smooth.

For a conic bundle $f:X\to T$ such that $T$ is a noetherian integral scheme, $f$ is generically smooth if and only if there exists a non-empty open subset $T^{\prime}$ of $T$ such that $f^{-1}(t)$ is smooth for every point $t\in T^{\prime}$ [EGAIV3, Théorème 12.2.4].

Lemma 2.17.

Let $f:X\to T$ be a conic bundle such that $T$ is a noetherian integral scheme. If $X$ is normal and the function field $K(T)$ of $T$ is of characteristic $\neq 2$ , then $f$ is generically smooth.

Proof.

Taking the base change by ${\operatorname{Spec}}\,K(T)\to T$ , we may assume that $T={\operatorname{Spec}}\,k$ for a field $k$ . Then $X$ is a conic on $\mathbb{P}^{2}_{k}$ and $X$ is a regular scheme. Taking the base change to the separable closure of $k$ , we may assume that $k$ is separably closed. By Proposition 2.11(3) or Corollary 2.13(2), we can write $X\simeq\{x^{2}+y^{2}+cz^{2}=0\}\subset\mathbb{P}^{2}_{k}$ for some $c\in k$ . If $c=0$ , then $X$ is not regular, as the $k$ -rational point $[0:0:1]\in\mathbb{P}^{2}_{k}$ is a non-regular point. Hence $c\neq 0$ . In this case, $X$ is smooth over $T={\operatorname{Spec}}\,k$ . ∎

Example 2.18.

We work over an algebraically closed field of characteristic two. Set

X:=\{t_{0}x_{0}^{2}+t_{1}x_{1}^{2}+t_{2}x_{2}^{2}=0\}\subset\mathbb{P}^{2}_{x_{0},x_{1},x_{2}}\times\mathbb{P}^{2}_{t_{0},t_{1},t_{2}}.

Then the induced morphism $f:X\to T:=\mathbb{P}^{2}_{t_{0},t_{1},t_{2}}={\operatorname{Proj}}\,k[t_{0},t_{1},t_{2}]$ is a wild conic bundle, because we have $a_{0}x_{0}^{2}+a_{1}x_{1}^{2}+a_{2}x_{2}^{2}=(\sqrt{a_{0}}x_{0}+\sqrt{a_{1}}x_{1}+\sqrt{a_{2}}x_{2})^{2}$ for all $a_{0},a_{1},a_{2}\in k$ .

3. Discriminant divisors

Let $f:X\to T$ be a conic bundle. The purpose of this section is to introduce two closed subschemes $\Delta_{f}$ and $\Sigma_{f}$ of $T$ which satisfy the following properties (I) and (II).

(I)

Given a point $t\in T$ , $X_{t}$ is not smooth (resp. not geometrically reduced) if and only if $t\in\Delta_{f}$ (resp. $t\in\Sigma_{f}$ ) (Theorem 3.13).
(II)

Equalities $\Delta_{f^{\prime}}=\beta^{-1}\Delta_{f}$ and $\Sigma_{f^{\prime}}=\beta^{-1}\Sigma_{f}$ of closed subschemes hold for a carterian diagram of noetherian schemes

$\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T,\end{CD}$

where $f$ and $f^{\prime}$ are conic bundles (Remark 3.5, Remark 3.11).

In Subsection 3.1 (resp. Subsection 3.2), we give the definition of $\Delta_{f}$ (resp. $\Sigma_{f}$ ). The main technical difficulty is their well-definedness. The property (I) will be established in Subsection 3.3. In Subsection 3.4, we introduce a simpler version $\Sigma^{\prime}_{f}$ of $\Sigma_{f}$ which can be defined only in characteristic two.

3.1. Definition of discriminant divisors

In this subsection, we introduce a closed subscheme $\Delta_{f}$ of $T$ , called the discriminant scheme of $f$ , associated with a conic bundle $f:X\to T$ . We first define $\Delta_{f}$ for the case when $T$ is affine, say $T={\operatorname{Spec}}\,A$ , and a closed embedding

X={\operatorname{Proj}}\,A[x,y,z]/(Q)\subset\mathbb{P}^{2}_{A}

is fixed (Definition 3.1). In this case, we set $\Delta_{f}:={\operatorname{Spec}}\,(A/\delta(Q)A)$ , where $\delta(Q)\in A$ is determined by $Q$ . Then we shall check that $\Delta_{f}={\operatorname{Spec}}\,(A/\delta(Q)A)$ is independent of the choice of the embedding $X\subset\mathbb{P}^{2}_{A}$ (Proposition 3.3), which enables us to define $\Delta_{f}$ for the general case, i.e., we may glue $\{\Delta_{f_{i}}\}_{i\in I}$ together for a suitable affine open cover $T=\bigcup_{i\in I}T_{i}$ and the induced morphisms $f_{i}:f^{-1}(T_{i})\to T_{i}$ (Definition 3.4).

Definition 3.1 (cf.[ABBB21]*Section 2.b).

Let $A$ be a noetherian ring. For

Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z],

we set

\delta(Q):=4abc+\alpha\beta\gamma-a\alpha^{2}-b\beta^{2}-c\gamma^{2}\in A.

Remark 3.2.

Note that if $2\in A^{\times}$ , then

\delta(Q)=\frac{1}{2}\det\begin{pmatrix}2a&\gamma&\beta\\ \gamma&2b&\alpha\\ \beta&\alpha&2c\end{pmatrix}.

Proposition 3.3.

Let $A$ be a noetherian ring. For $M\in{\operatorname{GL}}_{3}(A)$ and

Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z],

the equality

\delta(Q)A=\delta(Q^{M})A

of ideals hold.

Proof.

We first reduce the problem to the case when $A$ is a noetherian local ring. For $J:=\delta(Q)A\cap\delta(Q^{M})A$ , we have the inclusion

\varphi:J=\delta(Q)A\cap\delta(Q^{M})A\hookrightarrow\delta(Q)A,

which is an $A$ -module homomorphism. For a prime ideal $\mathfrak{p}\in{\operatorname{Spec}}\,A$ , we get the following inclusion [AM69, Corollary 3.4, ii)]:

\varphi_{\mathfrak{p}}:J_{\mathfrak{p}}=\delta(Q)A_{\mathfrak{p}}\cap\delta(Q^{M})A_{\mathfrak{p}}\hookrightarrow\delta(Q)A_{\mathfrak{p}}.

As we are assuming that $\varphi_{\mathfrak{p}}$ is an isomorphism for every $\mathfrak{p}\in{\operatorname{Spec}}\,A$ , also $\varphi$ is an isomorphism, i.e., $\delta(Q)A\subset\delta(Q^{M})A$ . By symmetry, we obtain the opposite inclusion $\delta(Q)A\supset\delta(Q^{M})A$ , which implies $\delta(Q)A=\delta(Q^{M})A$ . Thus the problem is reduced to the case when $A$ is a noetherian local ring.

Since $A$ is a local ring, we have $M=M_{1}\cdots M_{r}$ for some elementary matrices $M_{1},...,M_{r}$ (Lemma 2.1). By symmetry, we may assume that one of (1)–(3) holds.

(1)

$M=\begin{pmatrix}\lambda&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ for some $\lambda\in A^{\times}$ .
(2)

$M=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\\ \end{pmatrix}$ .
(3)

$M=\begin{pmatrix}1&\mu&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ for some $\mu\in A$ .

Set $Q(x,y,z):=Q$ .

(1) In this case, we have

Q^{M}(x,y,z)=Q(\lambda x,y,z)=(\lambda^{2}a)x^{2}+by^{2}+cz^{2}+\alpha yz+(\lambda\beta)zx+(\lambda\gamma)xy

and

\delta(Q^{M})=4(\lambda^{2}a)bc+\alpha(\lambda\beta)(\lambda\gamma)-(\lambda^{2}a)\alpha^{2}-b(\lambda\beta)^{2}-c(\lambda\gamma)^{2}=\lambda^{2}\delta(Q),

as required.

(2) In this case, we obtain $Q^{M}(x,y,z)=Q(y,x,z)$ , which implies $\delta(Q^{M})=\delta(Q)$ by symmetry (Definition 3.1).

(3) In this case, we have

$\displaystyle Q^{M}(x,y,z)$	$\displaystyle=$	$\displaystyle Q(x+\mu y,y,z)$
	$\displaystyle=$	$\displaystyle a(x+\mu y)^{2}+by^{2}+cz^{2}+\alpha yz+\beta z(x+\mu y)+\gamma(x+\mu y)y$
	$\displaystyle=$	$\displaystyle ax^{2}+(a\mu^{2}+b+\gamma\mu)y^{2}+cz^{2}+(\alpha+\beta\mu)yz+\beta zx+(2a\mu+\gamma)xy.$

Therefore, the assertion holds by the following computation:

$\displaystyle\delta(Q^{M})$	$\displaystyle=$	$\displaystyle 4a(a\mu^{2}+b+\gamma\mu)c+(\alpha+\beta\mu)\cdot\beta\cdot(2a\mu+\gamma)$
	$\displaystyle-$	$\displaystyle a(\alpha+\beta\mu)^{2}-(a\mu^{2}+b+\gamma\mu)\beta^{2}-c(2a\mu+\gamma)^{2}$
	$\displaystyle=$	$\displaystyle(4a^{2}c\mu^{2}+4ac\gamma\mu+4abc)$
	$\displaystyle+$	$\displaystyle(2a\beta^{2}\mu^{2}+(2a\alpha\beta+\beta^{2}\gamma)\mu+a\beta\gamma)$
	$\displaystyle+$	$\displaystyle(-a\beta^{2}\mu^{2}-2a\alpha\beta\mu-a\alpha^{2})$
	$\displaystyle+$	$\displaystyle(-a\beta^{2}\mu^{2}-\beta^{2}\gamma\mu-b\beta^{2})$
	$\displaystyle+$	$\displaystyle(-4a^{2}c\mu^{2}-4ac\gamma\mu-c\gamma^{2})$
	$\displaystyle=$	$\displaystyle 4abc+\alpha\beta\gamma-a\alpha^{2}-b\beta^{2}-c\gamma^{2}$
	$\displaystyle=$	$\displaystyle\delta(Q).$

∎

Definition 3.4.

Let $f:X\to T$ be a conic bundle. We define the closed subscheme $\Delta_{f}$ of $T$ , called the discriminant locus or discriminant scheme of $f$ , as follows.

(1)

If $T={\operatorname{Spec}}\,A$ and $X={\operatorname{Proj}}\,A[x,y,z]/(Q)$ , then we set

$\Delta_{f}:={\operatorname{Spec}}\,(A/\delta(Q)A).$

By Proposition 3.3, this definition is independent of the choice of $Q$ .

(2)

Fix an affine open cover $T=\bigcup_{i\in I}T_{i}$ such that $f^{-1}(T_{i})$ is $T_{i}$ -isomorphic to a conic on $\mathbb{P}^{2}_{T_{i}}$ for every $i\in I$ (the existence of such an open cover is guaranteed by Proposition 2.7). We can write

X_{i}:=f^{-1}(T_{i})={\operatorname{Proj}}\,A_{i}[x,y,z]/(Q_{i}),\qquad Q_{i}\in A_{i}[x,y,z],\qquad A_{i}:=\Gamma(T_{i},\mathcal{O}_{T}).

By (1), we have a closed subscheme $\Delta_{f_{i}}$ of $T_{i}$ for every $i\in I$ . Again by (1), we obtain $\Delta_{f_{i}}|_{T_{i}\cap T_{j}}=\Delta_{f_{j}}|_{T_{i}\cap T_{j}}$ , so that there exists a closed subscheme $\Delta_{f}$ on $T$ such that $\Delta_{f}|_{T_{i}}=\Delta_{f_{i}}$ for every $i\in I$ . It follows from (1) that $\Delta_{f}$ does not depend on the choice of the affine open cover $T=\bigcup_{i\in I}T_{i}$ .

We also call $\Delta_{f}$ the discriminant divisor of $f$ when $\Delta_{f}$ is an effective Cartier divisor on $T$ .

Remark 3.5.

Let

\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}

be a cartesian diagram of noetherian schemes, where $f$ is a conic bundle. In particular, also $f^{\prime}$ is a conic bundle. In this case, the equality

\beta^{-1}(\Delta_{f})=\Delta_{f^{\prime}}

holds by Definition 3.4, where $\beta^{-1}(\Delta_{f})$ is the scheme-theoretic inverse image of $\Delta_{f}$ .

3.2. Locus of non-reduced fibres

In this subsection, we introduce a closed subscheme $\Sigma_{f}$ of $T$ associated with a conic bundle $f:X\to T$ . The outline is similar to that of Subsection 3.2, whilst its proof is more involved, because the defining ideal $\sigma(Q)$ is no longer principal. The proof is carried out by reducing the problem to the case of characteristic zero.

Definition 3.6.

Let $A$ be a noetherian ring. For

Q(x,y,z):=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z],

we set

\sigma(Q):=(4ab-\gamma^{2},4bc-\alpha^{2},4ca-\beta^{2},2a\alpha-\beta\gamma,2b\beta-\gamma\alpha,2c\gamma-\alpha\beta)\subset A,

which is an ideal of $A$ .

Remark 3.7.

Note that $\sigma(Q)$ is the ideal of $A$ generated by the $2\times 2$ minors of

\begin{pmatrix}2a&\gamma&\beta\\ \gamma&2b&\alpha\\ \beta&\alpha&2c\end{pmatrix}.

The following lemma is the key to prove that $\sigma(Q)$ does not depend on the choice of the closed embedding into $\mathbb{P}^{2}_{A}$ .

Lemma 3.8.

Let $A$ be a noetherian ring and fix $\mu\in A$ . Set

Q(x,y,z):=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z]

and

M:=\begin{pmatrix}1&0&\mu\\ 0&1&0\\ 0&0&1\end{pmatrix}.

Then the equality

(3.8.1)

\sigma(Q)=\sigma(Q^{M}).

of ideals of $A$ holds.

Proof.

We introduce the following notation.

•

$\sigma_{11}(Q):=4bc-\alpha^{2}$ .
•

$\sigma_{22}(Q):=4ca-\beta^{2}$ .
•

$\sigma_{33}(Q):=4ab-\gamma^{2}$ .
•

$\sigma_{12}(Q):=2c\gamma-\alpha\beta$ .
•

$\sigma_{23}(Q):=2a\alpha-\beta\gamma$ .
•

$\sigma_{31}(Q):=2b\beta-\gamma\alpha$ .

Step 1.

The equality (3.8.1) holds if the following equalities (1)–(6) hold.

(1)

$\sigma_{11}(Q^{M})=\sigma_{11}(Q)-2\mu\sigma_{31}(Q)+\mu^{2}\sigma_{33}(Q)$ .
(2)

$\sigma_{22}(Q^{M})=\sigma_{22}(Q)$ .
(3)

$\sigma_{33}(Q^{M})=\sigma_{33}(Q)$ .
(4)

$\sigma_{12}(Q^{M})=\sigma_{12}(Q)-\mu\sigma_{23}(Q)$ .
(5)

$\sigma_{23}(Q^{M})=\sigma_{23}(Q)$ .
(6)

$\sigma_{31}(Q^{M})=\sigma_{31}(Q)+\mu\sigma_{33}(Q)$ .

Proof of Step 1.

Note that $\{\sigma_{ij}(Q)\}_{i,j}$ and $\{\sigma_{ij}(Q^{M})\}_{i,j}$ are generators of $\sigma(Q)$ and $\sigma(Q^{M})$ , respectively:

	$\displaystyle\sigma(Q)$	$\displaystyle=$	$\displaystyle(\sigma_{11}(Q),\sigma_{22}(Q),\sigma_{33}(Q),\sigma_{12}(Q),\sigma_{23}(Q),\sigma_{31}(Q))$
	$\displaystyle\sigma(Q^{M})$	$\displaystyle=$	$\displaystyle(\sigma_{11}(Q^{M}),\sigma_{22}(Q^{M}),\sigma_{33}(Q^{M}),\sigma_{12}(Q^{M}),\sigma_{23}(Q^{M}),\sigma_{31}(Q^{M})).$

Therefore, (1)–(6) imply the equality (3.8.1). This completes the proof of Step 1. ∎

Step 2.

If $A$ is a field of characteristic zero, then the equalities (1)–(6) of Step 1 hold.

Proof of Step 2.

For the symmetric matrix

S:=\begin{pmatrix}2a&\gamma&\beta\\ \gamma&2b&\alpha\\ \beta&\alpha&2c\end{pmatrix},

we have

Q(x,y,z)=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy=\frac{1}{2}\begin{pmatrix}x&y&z\end{pmatrix}S\begin{pmatrix}x\\ y\\ z\end{pmatrix}.

Furthermore, the following equalities (i)–(vi) hold if each $S_{ij}$ denotes the $2\times 2$ -matrix obtained from $S$ by excluding the $i$ -th row and the $j$ -th column.

(i)

$\sigma_{11}(Q)=4bc-\alpha^{2}=\det S_{11}$ .
(ii)

$\sigma_{22}(Q)=4ca-\beta^{2}=\det S_{22}$ .
(iii)

$\sigma_{33}(Q)=4ab-\gamma^{2}=\det S_{33}$ .
(iv)

$\sigma_{12}(Q)=2c\gamma-\alpha\beta=\det S_{12}=\det S_{21}$ .
(v)

$\sigma_{23}(Q)=2a\alpha-\beta\gamma=\det S_{23}=\det S_{32}$ .
(vi)

$\sigma_{31}(Q)=2b\beta-\gamma\alpha=-\det S_{31}=-\det S_{13}$ .

It holds that

Q^{M}(x,y,z)=\frac{1}{2}\begin{pmatrix}x&y&z\end{pmatrix}(M^{T}SM)\begin{pmatrix}x\\ y\\ z\end{pmatrix},

and

M^{T}SM=\begin{pmatrix}2a&\gamma&\beta+\mu\cdot(2a)\\ \gamma&2b&\alpha+\mu\gamma\\ \beta+\mu\cdot(2a)&\alpha+\mu\gamma&2c+2\mu\beta+\mu^{2}\cdot(2a)\end{pmatrix}

where $M^{T}$ denotes the transposed matrix of $M$ .

(1)

$\displaystyle\sigma_{11}(Q^{M})$	$\displaystyle=$	$\displaystyle\det(M^{T}SM)_{11}$
	$\displaystyle=$	$\displaystyle\det\begin{pmatrix}2b&\alpha+\mu\gamma\\ \alpha+\mu\gamma&2c+2\mu\beta+\mu^{2}\cdot(2a)\end{pmatrix}$
	$\displaystyle=$	$\displaystyle\det\begin{pmatrix}2b&\alpha+\mu\gamma\\ \alpha&2c+\mu\beta\end{pmatrix}+\mu\det\begin{pmatrix}2b&\alpha+\mu\gamma\\ \gamma&\beta+\mu\cdot(2a)\end{pmatrix}$
	$\displaystyle=$	$\displaystyle\det\begin{pmatrix}2b&\alpha\\ \alpha&2c\end{pmatrix}+\mu\det\begin{pmatrix}2b&\gamma\\ \alpha&\beta\end{pmatrix}+\mu\det\begin{pmatrix}2b&\alpha\\ \gamma&\beta\end{pmatrix}+\mu^{2}\det\begin{pmatrix}2b&\gamma\\ \gamma&2a\end{pmatrix}$
	$\displaystyle=$	$\displaystyle\det S_{11}-\mu\det S_{13}-\mu\det S_{31}+\mu^{2}\det S_{33}$
	$\displaystyle=$	$\displaystyle\sigma_{11}(Q)-2\mu\sigma_{31}(Q)+\mu^{2}\sigma_{33}(Q).$

(2)

\sigma_{22}(Q^{M})=\det(M^{T}SM)_{22}=\det\begin{pmatrix}2a&\beta+\mu\cdot(2a)\\ \beta+\mu\cdot(2a)&2c+2\mu\beta+\mu^{2}\cdot(2a)\end{pmatrix}

=\det\begin{pmatrix}2a&\beta+\mu\cdot(2a)\\ \beta&2c+\mu\beta\end{pmatrix}=\det\begin{pmatrix}2a&\beta\\ \beta&2c\end{pmatrix}=\det S_{22}=\sigma_{22}(Q).

(3) $\sigma_{33}(Q^{M})=\det(M^{T}SM)_{33}=\det S_{33}=\sigma_{33}(Q)$ .

(4)

$\displaystyle\sigma_{12}(Q^{M})$	$\displaystyle=$	$\displaystyle\det(M^{T}SM)_{12}$
	$\displaystyle=$	$\displaystyle\det\begin{pmatrix}\gamma&\alpha+\mu\gamma\\ \beta+\mu\cdot(2a)&2c+2\mu\beta+\mu^{2}\cdot(2a)\end{pmatrix}$
	$\displaystyle=$	$\displaystyle\det\begin{pmatrix}\gamma&\alpha\\ \beta+\mu\cdot(2a)&2c+\mu\beta\end{pmatrix}$
	$\displaystyle=$	$\displaystyle\det\begin{pmatrix}\gamma&\alpha\\ \beta&2c\end{pmatrix}+\mu\det\begin{pmatrix}\gamma&\alpha\\ 2a&\beta\end{pmatrix}$
	$\displaystyle=$	$\displaystyle\det S_{12}-\mu\det S_{32}$
	$\displaystyle=$	$\displaystyle\sigma_{12}(Q)-\mu\sigma_{23}(Q).$

(5)

\sigma_{23}(Q^{M})=\det(MSM^{T})_{23}=\det\begin{pmatrix}2a&\gamma\\ \beta+\mu\cdot(2a)&\alpha+\mu\gamma\end{pmatrix}=\det\begin{pmatrix}2a&\gamma\\ \beta&\alpha\end{pmatrix}

=\det S_{23}=\sigma_{23}(Q).

(6)

\sigma_{31}(Q^{M})=-\det(M^{T}SM)_{13}=-\det\begin{pmatrix}\gamma&2b\\ \beta+\mu\cdot(2a)&\alpha+\mu\gamma\end{pmatrix}

=-\det\begin{pmatrix}\gamma&2b\\ \beta&\alpha\end{pmatrix}-\mu\det\begin{pmatrix}\gamma&2b\\ 2a&\gamma\end{pmatrix}=-\det S_{13}+\mu\det S_{33}=\sigma_{31}(Q)+\mu\sigma_{33}(Q).

This completes the proof of Step 2. ∎

Step 3.

The equalities (1)–(6) of Step 1 hold without any additional assumptions.

Proof of Step 3.

Let $A^{\prime}:=\mathbb{Z}[a^{\prime},b^{\prime},c^{\prime},\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\mu^{\prime}]$ be the polynomial ring over $\mathbb{Z}$ with seven variables $a^{\prime},b^{\prime},c^{\prime},\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\mu^{\prime}$ . Let

\varphi:A^{\prime}=\mathbb{Z}[a^{\prime},b^{\prime},c^{\prime},\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\mu^{\prime}]\to A

be the ring homomorphism such that

\varphi(a^{\prime})=a,\quad\varphi(b^{\prime})=b,\quad\varphi(c^{\prime})=c,\quad\varphi(\alpha^{\prime})=\alpha,\quad\varphi(\beta^{\prime})=\beta,\quad\varphi(\gamma^{\prime})=\gamma,\quad\varphi(\mu^{\prime})=\mu.

Take the embedding $\psi:A^{\prime}\hookrightarrow A^{\prime\prime}:={\operatorname{Frac}}\,A^{\prime}$ into its total ring of fractions $A^{\prime\prime}={\operatorname{Frac}}\,A^{\prime}$ . Set

Q^{\prime}(x,y,z):=a^{\prime}x^{2}+b^{\prime}y^{2}+c^{\prime}z^{2}+\alpha^{\prime}yz+\beta^{\prime}zx+\gamma^{\prime}xy\in A^{\prime}[x,y,z]\subset A^{\prime\prime}[x,y,z].

and

M^{\prime}:=\begin{pmatrix}1&0&\mu^{\prime}\\ 0&1&0\\ 0&0&1\end{pmatrix}.

Let us only show that (1) of Step 1 holds, as the other ones (2)–(6) are similar. Applying Step 2 to $A^{\prime\prime}$ , we obtain an equality

\sigma_{11}(Q^{\prime M^{\prime}})=\sigma_{11}(Q^{\prime})-2\mu^{\prime}\sigma_{31}(Q^{\prime})+\mu^{\prime 2}\sigma_{33}(Q^{\prime})\quad{\rm in}\quad A^{\prime\prime}.

Since both sides are elements of $A^{\prime}$ , it follows from $A^{\prime}\subset A^{\prime\prime}$ that

\sigma_{11}(Q^{\prime M^{\prime}})=\sigma_{11}(Q^{\prime})-2\mu^{\prime}\sigma_{31}(Q^{\prime})+\mu^{\prime 2}\sigma_{33}(Q^{\prime})\quad{\rm in}\quad A^{\prime}.

Take the image by $\varphi:A^{\prime}\to A$ :

\varphi(\sigma_{11}(Q^{\prime M^{\prime}}))=\varphi(\sigma_{11}(Q^{\prime}))-2\mu\varphi(\sigma_{31}(Q^{\prime}))+\mu^{2}\varphi(\sigma_{33}(Q^{\prime})).

We can check that $\varphi(\sigma_{11}(Q^{\prime M^{\prime}}))=\sigma_{11}(Q^{M})$ and $\varphi(\sigma_{ij}(Q^{\prime}))=\sigma_{ij}(Q)$ for all $i,j$ . This completes the proof of Step 3. ∎

Step 1 and Step 3 complete the proof of Lemma 3.8. ∎

Proposition 3.9.

Let $A$ be a noetherian ring. For $M\in{\operatorname{GL}}_{3}(A)$ and

Q(x,y,z):=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z],

the equality

\sigma(Q)=\sigma(Q^{M})

of ideals hold.

Proof.

By the same argument as in the proof of Proposition 3.3, we may assume that $A$ is a noetherian local ring and one of (1)-(3) holds.

(1)

$M=\begin{pmatrix}\lambda&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ for some $\lambda\in A^{\times}$ .
(2)

$M=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\\ \end{pmatrix}$ .
(3)

$M=\begin{pmatrix}1&0&\mu\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ for some $\mu\in A$ .

(1) In this case, we have

Q^{M}(x,y,z)=Q(\lambda x,y,z)=(\lambda^{2}a)x^{2}+by^{2}+cz^{2}+\alpha yz+(\lambda\beta)zx+(\lambda\gamma)xy

and

\sigma(Q^{M})=(4\lambda^{2}ab-(\lambda\gamma)^{2},4bc-\alpha^{2},4c(\lambda^{2}a)-(\lambda\beta)^{2},

2(\lambda^{2}a)\alpha-(\lambda\beta)(\lambda\gamma),2b(\lambda\beta)-(\lambda\gamma)\alpha,2c(\lambda\gamma)-\alpha(\lambda\beta))=\sigma(Q),

as required.

(2) In this case, we have $\sigma(Q^{M})=\sigma(Q)$ by symmetry.

(3) In this case, the equality $\sigma(Q^{M})=\sigma(Q)$ follows from Lemma 3.8. ∎

Definition 3.10.

Let $f:X\to T$ be a conic bundle. We define the closed subscheme $\Sigma_{f}$ of $T$ as follows.

(1)

If $T={\operatorname{Spec}}\,A$ and $X={\operatorname{Proj}}\,A[x,y,z]/(Q)$ , then we set

$\Sigma_{f}:={\operatorname{Spec}}\,(A/\sigma(Q)).$

By Proposition 3.9, this definition is independent of the choice of $Q$ .

(2)

Fix an affine open cover $T=\bigcup_{i\in I}T_{i}$ such that $f^{-1}(T_{i})$ is $T_{i}$ -isomorphic to a conic on $\mathbb{P}^{2}_{T_{i}}$ for every $i\in I$ (the existence of such an open cover is guaranteed by Proposition 2.7). We can write

X_{i}:=f^{-1}(T_{i})={\operatorname{Proj}}\,A_{i}[x,y,z]/(Q_{i}),\qquad Q_{i}\in A_{i}[x,y,z],\qquad A_{i}:=\Gamma(T_{i},\mathcal{O}_{T}).

By (1), we have a closed subscheme $\Sigma_{f_{i}}$ of $T_{i}$ for every $i\in I$ . Again by (1), we obtain $\Sigma_{f_{i}}|_{T_{i}\cap T_{j}}=\Sigma_{f_{j}}|_{T_{i}\cap T_{j}}$ , so that there exists a closed subscheme $\Sigma_{f}$ on $T$ such that $\Sigma_{f}|_{T_{i}}=\Sigma_{f_{i}}$ for every $i\in I$ . It follows from (1) that $\Sigma_{f}$ does not depend on the choice of the affine open cover $T=\bigcup_{i\in I}T_{i}$ .

Remark 3.11.

Let

\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}

be a cartesian diagram of noetherian schemes, where $f$ is a conic bundle. In particular, also $f^{\prime}$ is a conic bundle. In this case, the equation

\beta^{-1}(\Sigma_{f})=\Sigma_{f^{\prime}}

holds by Definition 3.10, where $\beta^{-1}(\Sigma_{f})$ is the scheme-theoretic inverse image of $\Sigma_{f}$ .

3.3. Singular fibres

The purpose of this subsection is to prove Theorem 3.13, which states that $\Delta_{f}$ (resp. $\Sigma_{f}$ ) is set-theoretically equal to the locus parametrising non-smooth fibres (resp. geometrically non-reduced fibres). Since the problem is reduced to the case when the base scheme is ${\operatorname{Spec}}\,\kappa$ for a field $\kappa$ , we start with the following.

Proposition 3.12.

Let $\kappa$ be a field. Let

Q:=Q(x,y,z):=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in\kappa[x,y,z]\setminus\{0\}

and set

C:={\operatorname{Proj}}\,\kappa[x,y,z]/(Q),

which is a conic on $\mathbb{P}^{2}_{\kappa}$ . Then the following hold.

(1)

$C$ is not smooth over $\kappa$ if and only if $\delta(Q)=0$ .
(2)

$C$ is not geometrically reduced if and only if $\sigma(Q)=0$ .

Proof.

By Proposition 3.3 and Proposition 3.9, we may perform coordinate changes.

We first treat the case when ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}$ is of characteristic $\neq 2$ . By Proposition 2.11, we may assume that $\alpha=\beta=\gamma=0$ :

Q=ax^{2}+by^{2}+cz^{2}.

In this case, the following hold (Definition 3.1, Definition 3.6):

\delta(Q)=4abc,\qquad\sigma(Q)=(ab,bc,ca).

Thus (1) and (2) hold by Proposition 2.11.

Hence we may assume that ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}$ is an algebraically closed field of characteristic two. By Proposition 2.15(2), the problem is reduced to the case when $Q$ is equal to one of $x^{2},x^{2}+yz,yz$ . For each case, (1) and (2) hold by Definition 3.1 and Definition 3.6, respectively. ∎

Theorem 3.13.

Let $f:X\to T$ be a conic bundle and let $t$ be a point of $T$ . Then the following hold.

(1)

$X_{t}$ is not smooth if and only if $t\in\Delta_{f}$ .
(2)

$X_{t}$ is not geometrically reduced if and only if $t\in\Sigma_{f}$ .

Proof.

By Remark 3.5, we may assume that $T={\operatorname{Spec}}\,\kappa$ , where $\kappa$ is a field. Then the assertion follows from Proposition 3.12. ∎

Theorem 3.13 immediately deduces the set-theoretic inclusion $\Sigma_{f}\subset\Delta_{f}$ . As the following proposition shows, this inclusion holds even as closed subschemes.

Proposition 3.14.

Let $f:X\to T$ be a conic bundle. Then the inclusion $\Sigma_{f}\subset\Delta_{f}$ of closed subschemes holds, i.e., the inclusion $I_{\Sigma_{f}}\supset I_{\Delta_{f}}$ holds for the corresponding ideal sheaves.

Proof.

By Proposition 2.7, we may assume that $T={\operatorname{Spec}}\,A$ and

X={\operatorname{Proj}}\,A[x,y,z]/(Q)\subset\mathbb{P}^{2}_{A}

for

Q=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z].

It suffices to show that $\delta(Q)\in\sigma(Q)$ . Recall that

•

$\delta(Q)=4abc+\alpha\beta\gamma-a\alpha^{2}-b\beta^{2}-c\gamma^{2}$ and
•

$\sigma(Q)=(4ab-\gamma^{2},4bc-\alpha^{2},4ca-\beta^{2},2a\alpha-\beta\gamma,2b\beta-\gamma\alpha,2c\gamma-\alpha\beta)$ .

It holds that

4abc-a\alpha^{2},\quad 4abc-b\beta^{2},\quad 4abc-c\gamma^{2}\in\sigma(Q),

and hence

12abc-a\alpha^{2}-b\beta^{2}-c\gamma^{2}\in\sigma(Q).

By $8abc-2a\alpha^{2}=2a(4bc-\alpha^{2})\in\sigma(Q)$ and $2a\alpha^{2}-\alpha\beta\gamma=\alpha(2a\alpha-\beta\gamma)\in\sigma(Q)$ , we get

8abc-\alpha\beta\gamma\in\sigma(Q),

which implies

\delta(Q)=(12abc-a\alpha^{2}-b\beta^{2}-c\gamma^{2})-(8abc-\alpha\beta\gamma)\in\sigma(Q).

∎

3.4. Case of characteristic two

In this subsection, we introduce a simpler version $\Sigma^{\prime}_{f}$ of $\Sigma_{f}$ which can be defined only in characteristic two. Let us start by recalling the defining equations of $\Delta_{f}$ and $\Sigma_{f}$ .

3.15.

Let $A$ be a noetherian $\mathbb{F}_{2}$ -algebra. For

Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta za+\gamma xy\in A[x,y,z]

we have the following (Definition 3.1, Definition 3.6).

(1)

$\delta(Q)=\alpha\beta\gamma+a\alpha^{2}+b\beta^{2}+c\gamma^{2}$ .
(2)

$\sigma(Q)=(\alpha^{2},\beta^{2},\gamma^{2},\alpha\beta,\beta\gamma,\gamma\alpha)$ .

We now introduce the ring-theoretic counterpart $\sigma^{\prime}$ to $\Sigma^{\prime}_{f}$ .

Definition 3.16.

Let $A$ be a noetherian $\mathbb{F}_{2}$ -algebra. For

Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta za+\gamma xy\in A[x,y,z],

we define

\sigma^{\prime}(Q)=(\alpha,\beta,\gamma),

which is an ideal of $A$ .

Proposition 3.17.

Let $A$ be a noetherian $\mathbb{F}_{2}$ -algebra. For $M\in{\operatorname{GL}}_{3}(A)$ and

Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z],

the equality

\sigma^{\prime}(Q)=\sigma^{\prime}(Q^{M})

of ideals hold.

Proof.

By the same argument as in the proof of Proposition 3.3, we may assume that $A$ is a noetherian local ring and one of the following holds.

(1)

$M=\begin{pmatrix}\lambda&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ for some $\lambda\in A^{\times}$ .
(2)

$M=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\\ \end{pmatrix}$ .
(3)

$M=\begin{pmatrix}1&0&\mu\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ for some $\mu\in A$ .

We only treat the case (3), i.e., we assume that $M=\begin{pmatrix}1&0&\mu\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}$ . Then there exist $c^{\prime}\in A$ such that the following holds:

$\displaystyle Q^{M}(x,y,z)$	$\displaystyle=$	$\displaystyle Q(x+\mu z,y,z)$
	$\displaystyle=$	$\displaystyle a(x+\mu z)^{2}+by^{2}+cz^{2}+\alpha yz+\beta z(x+\mu z)+\gamma(x+\mu z)y$
	$\displaystyle=$	$\displaystyle ax^{2}+by^{2}+c^{\prime}z^{2}+(\alpha+\mu\gamma)yz+\beta zx+\gamma xy.$

Hence we obtain $\sigma^{\prime}(Q^{M})=(\alpha+\mu\gamma,\beta,\gamma)=(\alpha,\beta,\gamma)=\sigma(Q)$ . ∎

Definition 3.18.

Let $f:X\to T$ be a conic bundle. We define the closed subscheme $\Sigma^{\prime}_{f}$ of $T$ as follows.

(1)

If $T={\operatorname{Spec}}\,A$ and $X={\operatorname{Proj}}\,A[x,y,z]/(Q)$ , then we set

$\Sigma^{\prime}_{f}:={\operatorname{Spec}}\,(A/\sigma^{\prime}(Q)).$

By Proposition 3.17, this definition is independent of the choice of $Q$ .

(2)

Fix an affine open cover $T=\bigcup_{i\in I}T_{i}$ such that $f^{-1}(T_{i})$ is $T_{i}$ -isomorphic to a conic on $\mathbb{P}^{2}_{T_{i}}$ for every $i\in I$ (the existence of such an open cover is guaranteed by Proposition 2.7). We can write

X_{i}:=f^{-1}(T_{i})={\operatorname{Proj}}\,A_{i}[x,y,z]/(Q_{i}),\qquad Q_{i}\in A_{i}[x,y,z],\qquad A_{i}:=\Gamma(T_{i},\mathcal{O}_{T}).

By (1), we have a closed subscheme $\Sigma^{\prime}_{f_{i}}$ of $T_{i}$ for every $i\in I$ . Again by (1), we obtain $\Sigma^{\prime}_{f_{i}}|_{T_{i}\cap T_{j}}=\Sigma^{\prime}_{f_{j}}|_{T_{i}\cap T_{j}}$ , so that there exists a closed subscheme $\Sigma^{\prime}_{f}$ on $T$ such that $\Sigma^{\prime}_{f}|_{T_{i}}=\Sigma^{\prime}_{f_{i}}$ for every $i\in I$ . It follows from (1) that $\Sigma^{\prime}_{f}$ does not depend on the choice of the affine open cover $T=\bigcup_{i\in I}T_{i}$ .

Remark 3.19.

Let

\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}

be a cartesian diagram of schemes, where $T$ and $T^{\prime}$ are noetherian schemes and $f$ is a conic bundle. In particular, also $f^{\prime}$ is a conic bundle. In this case, the equation

\beta^{-1}(\Sigma^{\prime}_{f})=\Sigma^{\prime}_{f^{\prime}}

holds by Definition 3.18, where $\beta^{-1}(\Sigma^{\prime}_{f})$ is the scheme-theoretic inverse image of $\Sigma^{\prime}_{f}$ .

Remark 3.20.

Given a conic bundle $f:X\to T$ , we have an inclusion $\Sigma_{f}\supset\Sigma^{\prime}_{f}$ of closed subschemes of $T$ and the following holds ((3.15), Definition 3.16):

(\Sigma_{f})_{{\operatorname{red}}}=(\Sigma^{\prime}_{f})_{{\operatorname{red}}}.

4. Singularities of ambient spaces

Given a conic bundle $f:X\to T$ , it is natural to seek a relation between $\Delta_{f}$ and the singularities of $X$ . The main result of this section is to prove that the following are equivalent under the assumption that $T$ is regular (Theorem 4.4).

(1)

$\Delta_{f}$ is regular.
(2)

$X$ is regular and every fibre of $f$ is geometrically reduced.

In particular, if $f:X\to T$ is a conic bundle between regular noetherian schemes, then the equality $(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}=\Sigma_{f}$ holds, where $(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}$ denotes the non-regular locus of $\Delta_{f}$ (Theorem 4.3). These results are known when $f:X\to T$ is a conic bundle of smooth varieties over an algebraically closed field of characteristic $\neq 2$ [Sar82, Proposition 1.8].

Proposition 4.1.

Let $f:X\to T$ be a generically smooth conic bundle, where $T$ is a regular noetherian integral scheme. Fix a point $0\in\Delta_{f}$ . If $\Delta_{f}$ is regular at $0$ , then the fibre $X_{0}$ is geometrically reduced.

Proof.

Assume that $X_{0}$ is not geometrically reduced. It suffices to show that $\Delta_{f}$ is not regular at $0$ . By taking the strict henselisation of the local ring $\mathcal{O}_{T,0}$ , we may assume that $T={\operatorname{Spec}}\,A$ , $(A,\mathfrak{m},\kappa)$ is a strictly henselian regular local ring, and $0$ is the closed point of $T$ . We can write

X={\operatorname{Proj}}\,A[x,y,z]/(Q)\qquad\text{for}\qquad Q=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy\in A[x,y,z].

By Definition 3.1 and Definition 3.4, we get

\Delta_{f}={\operatorname{Spec}}\,(A/\delta(Q)A)\qquad\text{and}\qquad\delta(Q)=4abc+\alpha\beta\gamma-a\alpha^{2}-b\beta^{2}-c\gamma^{2}\in A.

If $\kappa$ is of characteristic two, then Theorem 3.13 and Remark 3.20 imply

2,\alpha,\beta,\gamma\in\mathfrak{m}.

In particular, we get $\delta(Q)\in\mathfrak{m}^{2}$ , and hence $\Delta_{f}$ is not regular at $0$ .

Then the problem is reduced to the case when $\kappa$ is of characteristic $\neq 2$ . By Corollary 2.13(1), we may assume that

X={\operatorname{Proj}}\,A[x,y,z]/(x^{2}+by^{2}+cz^{2}+\alpha yz),

where $b,c,\alpha\in\mathfrak{m}$ . By $\delta(Q)=4bc-\alpha^{2}\in\mathfrak{m}^{2}$ , $\Delta_{f}$ is not regular at $0$ . ∎

Proposition 4.2.

Let $f:X\to T$ be a generically smooth conic bundle, where $X$ and $T$ are regular noetherian integral schemes. Fix a point $0\in\Delta_{f}$ . Then the following are equivalent.

(1)

$X_{0}$ is geometrically reduced.
(2)

$\Delta_{f}$ is regular at $0$ .

Proof.

The implication (2) $\Rightarrow$ (1) has been settled in Proposition 4.1. Let us show the opposite one: (1) $\Rightarrow$ (2). By taking the strict henselisation of the local ring $\mathcal{O}_{T,0}$ , we may assume that $T={\operatorname{Spec}}\,A$ , $(A,\mathfrak{m},\kappa)$ is a strictly henselian regular local ring, and $0$ is the closed point of $T$ .

Step 1.

The implication (1) $\Rightarrow$ (2) holds if $\kappa$ is of characteristic $\neq 2$ .

Proof of Step 1.

Assume (1). By Corollary 2.13(2), we may assume that

X={\operatorname{Proj}}\,A[x,y,z]/(x^{2}+y^{2}+cz^{2})

for some $c\in A$ . Then the following hold (Definition 3.1, Theorem 3.13):

\delta(Q)=4c\qquad{\rm and}\qquad\sigma(Q)=A.

It suffices to show that $c\not\in\mathfrak{m}^{2}$ . Suppose $c\in\mathfrak{m}^{2}$ . Let $\mathfrak{n}$ be the closed point of $X$ lying over $0\in T$ corresponding to $[0:0:1]\in{\operatorname{Proj}}\,\kappa[x,y,z]=\mathbb{P}^{2}_{\kappa}$ . For $\widetilde{x}:=x/z$ and $\widetilde{y}:=y/z$ , we have the induced injective ring homomorphism

A\hookrightarrow B:=A[\widetilde{x},\widetilde{y}]/(\widetilde{x}^{2}+\widetilde{y}^{2}+c),

where ${\operatorname{Spec}}\,B$ is an open neighbourhood of $\mathfrak{n}\in X$ . Via this injection, we consider $A$ as a subring of $B$ . We have $\mathfrak{m}\subset\mathfrak{n}$ , which implies $c\in\mathfrak{m}^{2}\subset\mathfrak{n}^{2}$ . By $\widetilde{x}\in\mathfrak{n}$ and $\widetilde{y}\in\mathfrak{n}$ , we obtain $\widetilde{x}^{2}+\widetilde{y}^{2}+c\in\mathfrak{n}^{2}$ , and hence $B$ is not regular at $\mathfrak{n}$ , which is a contradiction. Thus (1) implies (2). This completes the proof of Step 1. ∎

Step 2.

The implication (1) $\Rightarrow$ (2) holds if $\kappa$ is of characteristic two.

Proof of Step 2.

Assume (1), i.e., $X_{0}$ is geometrically reduced. It follows from $0\in\Delta_{f}$ that $X_{0}$ is not smooth. By Proposition 2.14(2), we can write

X={\operatorname{Proj}}\,A[x,y,z]/(ax^{2}+yz)

for some $a\in\mathfrak{m}$ . Then we have

\delta(Q)=-a\qquad\text{and}\qquad\Delta_{f}={\operatorname{Spec}}\,(A/\delta(Q)A).

By

D_{+}(x)={\operatorname{Spec}}\,A[y,z]/(a+yz)\subset X,

we have $a\not\in\mathfrak{m}^{2}$ , as otherwise $D_{+}(x)$ would not be regular. Hence $\Delta_{f}=\{a=0\}$ is regular at $0$ . Thus (2) holds. This completes the proof of Step 2. ∎

Step 1 and Step 2 complete the proof of Proposition 4.2. ∎

Theorem 4.3.

Let $f:X\to T$ be a generically smooth conic bundle, where $X$ and $T$ are regular noetherian integral schemes. Set

(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}:=\{t\in\Delta_{f}\,|\,\mathcal{O}_{\Delta_{f},t}\text{ is not a regular local ring}\}.

Then the set-theoretic equality

(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}=\Sigma_{f}

holds.

Proof.

By Theorem 3.13 or Proposition 3.14, both sides are subsets of $\Delta_{f}$ .

Let us show the inclusion $(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}\supset\Sigma_{f}$ . Pick a point $t\in\Delta_{f}\setminus(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}$ , i.e., $t$ is a point of $\Delta_{f}$ at which $\Delta_{f}$ is regular. By Proposition 4.1, the fibre $X_{t}$ over $t$ is geometrically reduced. It follows from Theorem 3.13 that $t\not\in\Sigma_{f}$ . This completes the proof of the inclusion $(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}\supset\Sigma_{f}$ .

Let us prove the opposite inclusion $(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}\subset\Sigma_{f}$ . Pick a point $t\in\Delta_{f}\setminus\Sigma_{f}$ . Then $X_{t}$ is not smooth but geometrically reduced (Theorem 3.13). By Proposition 4.2, $\Delta_{f}$ is regular at $t$ , i.e., $t\not\in(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}$ . Therefore, we get $(\Delta_{f})_{{\rm non}\text{-}{\rm reg}}\subset\Sigma_{f}$ . ∎

Theorem 4.4.

Let $f:X\to T$ be a generically smooth conic bundle, where $T$ is a regular noetherian integral scheme. Then the following are equivalent.

(1)

$\Delta_{f}$ is regular.
(2)

$X$ is regular and any fibre of $f$ is geometrically reduced.

Proof.

By Proposition 4.2, (2) implies (1). Assume (1), i.e., $\Delta_{f}$ is regular. By Proposition 4.1, any fibre of $f$ is geometrically reduced. Fix a point ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}\in X$ and set $0:=f({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P})$ . Hence it is enough to show that $X$ is regular at ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}$ . Taking the strict henselisation of the local ring $\mathcal{O}_{T,0}$ , we may assume that $T={\operatorname{Spec}}\,A$ , $(A,\mathfrak{m},\kappa)$ is a strictly henselian regular local ring, and $0$ is the closed point of $T$ .

Since $X_{0}$ is geometrically reduced, we can write

X={\operatorname{Proj}}\,A[x,y,z]/(ax^{2}+yz)

for some $a\in A$ (Corollary 2.13(2), Proposition 2.14(2)). Then we obtain $\delta(Q)=-a$ for $Q:=ax^{2}+yz$ (Definition 3.1). As $\Delta_{f}={\operatorname{Spec}}\,(A/\delta(Q)A)$ is regular, we have $a\not\in\mathfrak{m}^{2}$ . It suffices to show that $A[y,z]/(a+yz)$ is regular. We may assume that $a\in\mathfrak{m}$ . Since $(A/\mathfrak{m})[y,z]/(yz)=\kappa[y,z]/(yz)$ is smooth outside the origin $(0,0)$ , we may assume that ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}=(0,0)$ . Then the prime ideal ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}$ is the image of the maximal ideal $\mathfrak{n}:=\mathfrak{m}A[y,z]+(y,z)A[y,z]$ of $A[y,z]$ by the natural surjection $A[y,z]\to A[y,z]/(a+yz)$ . By $a\not\in\mathfrak{m}^{2}$ , we obtain $a+yz\not\in\mathfrak{n}^{2}$ , and hence $A[y,z]/(a+yz)$ is regular at ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}$ . Thus $X$ is regular. ∎

Proposition 4.5.

Let $k$ be an algebraically closed field. Let $f:X\to T$ be a conic bundle, where $T$ is a smooth variety over $k$ . Assume that

(1)

$f$ is generically smooth, and
(2)

any fibre of $f$ is geometrically reduced.

Then $X$ is canonical in the sense of [Kol13, Definition 2.8]. If $k$ is characteristic $p>0$ , then $X$ is strongly $F$ -regular.

In the proof below, we use the following fact: if $X^{\prime}\to X$ is an étale surjective morphism, then $X$ is canonical if and only if so is $X^{\prime}$ [Kol13, Proposition 2.15].

Proof.

Let us prove the assertion under the assumption that $k$ is of characteristic $p>0$ . We prove the assertion by induction on $\dim T$ .

We first treat the case when $\dim T=1$ . By Corollary 2.13(2) and Proposition 2.14(2), we may assume, after taking suitable étale cover of $X$ , that

X={\operatorname{Proj}}\,A[x,y,z]/(xy+cz^{2}),\qquad T={\operatorname{Spec}}\,A,\qquad c\in A.

The singular locus of $X$ is contained in

D_{+}(z)={\operatorname{Spec}}\,A[x,y]/(xy+c).

This is an $A_{n}$ -singularity, and hence canonical and strongly $F$ -regular (note that these singularities are toric).

Assume that $\dim T\geq 2$ and that the assertion holds for the lower dimensional case. By (1), $\Delta_{f}$ is a nonzero effective Cartier divisor on $T$ . Fix a closed point $0\in T$ around which we will work. Take a general smooth prime divisor $T^{\prime}$ on $T$ passing through $0$ . Set $X^{\prime}:=X\times_{T}T^{\prime}$ . Then $f^{\prime}:X^{\prime}\to T^{\prime}$ is a conic bundle over $T^{\prime}$ . In particular, $X^{\prime}$ is strongly $F$ -regular by the induction hypothesis. Since $\dim(X^{\prime}\cap{\rm Sing}\,X)<\dim X^{\prime}-1$ (where $\dim\emptyset:=-\infty$ ), we have

(K_{X}+X^{\prime})|_{X^{\prime}}=K_{X^{\prime}},

i.e., the different ${\operatorname{Diff}}_{X^{\prime}}(0)$ is equal to zero (for the definition of ${\operatorname{Diff}}_{X^{\prime}}(0)$ , see [Kol13, Subsection 4.1]). Since $(X^{\prime},{\operatorname{Diff}}_{X^{\prime}}(0)=0)$ is strongly $F$ -regular, it follows from inversion of adjunction [Das15, Theorem 4.1] that $(X,X^{\prime})$ is purely $F$ -regular around $X^{\prime}$ . In particular, $X$ is strongly $F$ -regular and $(X,X^{\prime})$ is log canonical [HW02, Theorem 3.3]. Since $X^{\prime}$ is a nonzero effective Cartier divisor, $X$ is canonical. This completes the proof for the case when $k$ is of characteristic $p>0$ .

If $k$ is of characteristic zero, then the above argument works by using the inversion of adjunction for log canonicity [Kaw07, Theorem]. ∎

For a later use, we establish the following Jacobian criterion.

Proposition 4.6 (Jacobian criterion).

Let $k$ be a field. Set $A:=k[t_{1},...,t_{d}]$ and let

X:={\operatorname{Proj}}\,A[x,y,z]/(Q)

be a conic bundle over $A$ for $Q:=Q(t_{1},...,t_{d},x,y,z)\in A[x,y,z]$ . Fix a $k$ -rational point

w:=(t_{10},...,t_{d0})\times[x_{0}:y_{0}:z_{0}]\in X\subset\mathbb{P}^{2}_{A},

where $t_{10},...,t_{d0},x_{0},y_{0},z_{0}\in k$ . Then the following are equivalent.

(1)

$X$ is not smooth at $w$ .
(2)

The equation

$\partial_{x}Q(w)=\partial_{y}Q(w)=\partial_{z}Q(w)=\partial_{t_{1}}Q(w)=\cdots=\partial_{t_{d}}Q(w)=0,$

holds, where $\partial_{\bullet}$ denotes the partial differential with respect to $\bullet$ .

Proof.

Taking the base change to the algebraic closure, the problem is reduced to the case when $k$ is algebraically closed. If $k$ is of characteristic $\neq 2$ , then the assertion is well known. We may assume that $k$ is of characteristic two.

We have

Q=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy,\qquad a,b,c,\alpha,\beta,\gamma\in A=k[t_{1},...,t_{d}].

By symmetry, we may assume that $x_{0}=1$ . Set

Q^{\prime}:=Q^{\prime}(t_{1},...,t_{d},y,z):=Q(t_{1},...,t_{d},1,y,z)=a+by^{2}+cz^{2}+\alpha yz+\beta z+\gamma y,

which corresponds to the affine open subset $X^{\prime}:=D_{+}(x)$ of $X$ defined by $x\neq 0$ . Consider the following conditions.

(1)’

$X^{\prime}$ is not smooth at $w$ .
(2)’

$\partial_{y}Q^{\prime}(w)=\partial_{z}Q^{\prime}(w)=\partial_{t_{1}}Q^{\prime}(w)=\cdots=\partial_{t_{d}}Q^{\prime}(w)=0.$

It is clear that (1) $\Leftrightarrow$ (1)’ $\Leftrightarrow$ (2)’. Therefore it suffices to show that (2) $\Leftrightarrow$ (2)’. It is clear that (2) $\Rightarrow$ (2)’, because the partial differentials $\partial_{y},\partial_{z},\partial_{t_{1}},...,\partial_{t_{d}}$ commute with the substitution $x=1$ .

Therefore, it is enough to show that (2)’ $\Rightarrow$ (2). Assume (2)’. Then we have

\partial_{y}Q(w)=\partial_{z}Q(w)=\partial_{t_{1}}Q(w)=\cdots=\partial_{t_{d}}Q(w)=0.

It suffices to show $\partial_{x}Q(w)=0$ . By $\partial_{y}Q(w)=\partial_{z}Q(w)=0$ , we have

\alpha z_{0}+\gamma=0,\qquad\alpha y_{0}+\beta=0.

Therefore, we get

\partial_{x}Q(w)=\beta z_{0}+\gamma y_{0}=(\alpha y_{0})z_{0}+(\alpha z_{0})y_{0}=0,

as required. ∎

Remark 4.7.

Proposition 4.6 holds even when $A=k[[t_{1},...,t_{d}]]$ . In this case, the $k$ -rational point $w$ is lying over $\mathfrak{m}=(t_{1},...,t_{d})$ , and hence $t_{10}=\cdots=t_{d0}=0$ .

5. The Mori–Mukai formula

In this section, we prove the Mori–Mukai formula (Theorem 5.15 in Subsection 5.4):

(5.0.1)

\Delta_{f}\equiv-f_{*}(K_{X/T}^{2})

for an arbitrary generically smooth conic bundle $f:X\to T$ between smooth projective varieties. As explained in Subsection 1.1, the problem is reduced to the case when $f$ coincides with ${\operatorname{Univ}}_{\mathbb{P}_{T}(E)/T}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ , where $T$ is a smooth projective curve, $E$ is a locally free sheaf on $T$ of rank $3$ , and ${\operatorname{Univ}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ denotes the universal family associated with the relative Hilbert scheme ${\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ parametrising conics. We start by studying the absolute Hilbert scheme ${\operatorname{Univ}}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}^{2}_{\mathbb{Z}}/{\mathbb{Z}}}^{\theta}$ parametrising conics (Subsection 5.1), because the absolute case determines the local structure of the relative one ${\operatorname{Univ}}_{\mathbb{P}_{T}(E)/T}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ . More specifically, if $T=\bigcup_{i\in I}T_{i}$ is an open cover trivialising $E$ , then ${\operatorname{Univ}}_{\mathbb{P}_{T_{i}}(E|_{T_{i}})/T_{i}}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}_{T_{i}}(E|_{T_{i}})/T_{i}}^{\theta}$ coincides with ${\operatorname{Univ}}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}^{\theta}\times_{\mathbb{Z}}T_{i}\to{\operatorname{Hilb}}_{\mathbb{P}^{2}_{\mathbb{Z}}/{\mathbb{Z}}}^{\theta}\times_{\mathbb{Z}}T_{i}$ . We then summarise properties on the relative version ${\operatorname{Univ}}_{\mathbb{P}_{T}(E)/T}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}_{T}(E)/T}^{\theta}$ in Subsection 5.2, most of which are immediate conclusions from the absolute case.

In order to include generically non-smooth conic bundles, we shall introduce an invertible sheaf $\Delta_{f}^{{\operatorname{bdl}}}$ , which we shall call the discriminant bundle (Subsection 5.3), for an arbitrary conic bundle $f:X\to S$ with $S$ integral. When $f$ is generically smooth, it satisfies $\Delta_{f}^{{\operatorname{bdl}}}\simeq\mathcal{O}_{T}(\Delta_{f})$ (Remark 5.11) and the above Mori–Mukai formula (5.0.1) is generalised as follows:

(5.0.2)

\Delta_{f}^{{\operatorname{bdl}}}\equiv-f_{*}(K_{X/T}^{2}).

Moreover, the proof of (5.0.1) becomes simpler by using the notion of discriminant bundles, as otherwise we would need to be careful with the generically smooth condition.

5.1. The universal family

In this subsection, we study the conic bundle ${\operatorname{Univ}}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}^{2}_{\mathbb{Z}}/{\mathbb{Z}}}^{\theta}$ , where ${\operatorname{Hilb}}_{\mathbb{P}^{2}_{\mathbb{Z}}/{\mathbb{Z}}}^{\theta}$ denotes the Hilbert scheme of conics and ${\operatorname{Univ}}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}^{\theta}$ is its universal family. As explained above, this plays an essential role in the proof of the Mori–Mukai formula (5.0.1). It is also worth studying this conic bundle ${\operatorname{Univ}}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}^{\theta}\to{\operatorname{Hilb}}_{\mathbb{P}^{2}_{\mathbb{Z}}/{\mathbb{Z}}}^{\theta}$ as a concrete example. For foundations on Hilbert schemes, we refer to [FGI05].

Notation 5.1.

(1)

Set $\theta(m):=2m+1\in\mathbb{Z}[m]$ , which is nothing but the Hilbert polynomial of conics. In other words, if $\kappa$ is a field and $C$ is a conic on $\mathbb{P}^{2}_{\kappa}$ , then the following holds for any $m\in\mathbb{Z}$ :

$\theta(m)=\chi(C,\mathcal{O}_{\mathbb{P}^{2}}(m)|_{C})=2m+1.$

(2)

We have

{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\simeq\mathbb{P}^{5}_{\mathbb{Z}}={\operatorname{Proj}}\,\mathbb{Z}[a,b,c,\alpha,\beta,\gamma],

where $\mathbb{Z}[a,b,c,\alpha,\beta,\gamma]$ denotes the polynomial ring over $\mathbb{Z}$ with six variables $a,b,c,\alpha,\beta,\gamma$ . We identify ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}$ with $\mathbb{P}^{5}_{\mathbb{Z}}={\operatorname{Proj}}\,\mathbb{Z}[a,b,c,\alpha,\beta,\gamma]$ .

(3)

Let ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\subset{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\times_{\mathbb{Z}}\mathbb{P}^{2}_{\mathbb{Z}}$ be the universal closed subscheme:

{\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}=\{ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy=0\}\subset\mathbb{P}^{2}_{\mathbb{Z}}\times_{\mathbb{Z}}{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}

(4)

The induced composite morphism

f_{{\operatorname{univ}}}:{\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\hookrightarrow\mathbb{P}^{2}_{\mathbb{Z}}\times_{\mathbb{Z}}{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\xrightarrow{{\rm pr}_{2}}{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}

is a generically smooth conic bundle. Set $\Delta_{{\operatorname{univ}}}:=\Delta_{f_{{\operatorname{univ}}}}\subset{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}$ , which satisfies the following (Definition 3.1 and Definition 3.4):

\Delta_{{\operatorname{univ}}}=\{4abc+\alpha\beta\gamma-a\alpha^{2}-b\beta^{2}-c\gamma^{2}=0\}\subset{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}=\mathbb{P}^{5}_{\mathbb{Z}}={\operatorname{Proj}}\,\mathbb{Z}[a,b,c,\alpha,\beta,\gamma].

Similarly, we set $\Sigma_{{\operatorname{univ}}}:=\Sigma_{f_{{\operatorname{univ}}}}\subset{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}$ .

(5)

For a field $K$ and the morphism ${\operatorname{Spec}}\,K\to{\operatorname{Spec}}\,\mathbb{Z}$ , we set

\Delta_{{\operatorname{univ}},K}:=\Delta_{{\operatorname{univ}}}\times_{\mathbb{Z}}K\quad{\rm and}\quad\Sigma_{{\operatorname{univ}},K}:=\Sigma_{{\operatorname{univ}}}\times_{\mathbb{Z}}K.

Proposition 5.2.

We use Notation 5.1. Then ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}$ is smooth over $\mathbb{Z}$ . In particular, ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}},{\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Q}}/\mathbb{Q}},$ and ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{F}_{p}}/\mathbb{F}_{p}}$ are regular for any prime number $p$ .

Proof.

Both $f_{{\operatorname{univ}}}:{\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}$ and ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\to{\operatorname{Spec}}\,\mathbb{Z}$ are flat by ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\simeq\mathbb{P}^{5}_{\mathbb{Z}}$ . Hence also ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{Z}}/\mathbb{Z}}\to{\operatorname{Spec}}\,\mathbb{Z}$ is flat. Fix a prime number $p$ . It suffices to show that

{\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}/\overline{\mathbb{F}}_{p}}=\{Q:=ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy=0\}\subset\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}\times_{\overline{\mathbb{F}}_{p}}\mathbb{P}^{5}_{\overline{\mathbb{F}}_{p}}

is smooth, where $\overline{\mathbb{F}}_{p}$ denotes the algebraic closure of $\mathbb{F}_{p}$ . We shall apply the Jacobian criterion. Suppose that ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}/\overline{\mathbb{F}}_{p}}$ is not smooth at a closed point

w:=[x_{0}:y_{0}:z_{0}]\times[a_{0}:b_{0}:c_{0}:\alpha_{0}:\beta_{0}:\gamma_{0}]\in{\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}/\overline{\mathbb{F}}_{p}}\subset\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}\times_{\overline{\mathbb{F}}_{p}}\mathbb{P}^{5}_{\overline{\mathbb{F}}_{p}}

Then $[a_{0}:b_{0}:c_{0}:\alpha_{0}:\beta_{0}:\gamma_{0}]\in\mathbb{P}^{5}_{\overline{\mathbb{F}}_{p}}$ is its image. By symmetry, we may assume that $a_{0}\neq 0$ or $\alpha_{0}\neq 0$ .

Assume $\alpha_{0}\neq 0$ . By the Jacobian criterion (Proposition 4.6), we have $\partial_{a}Q(w)=\partial_{b}Q(w)=\partial_{c}Q(w)=0$ , which concludes that $x_{0}=y_{0}=z_{0}=0$ . This contradicts $[x_{0}:y_{0}:z_{0}]\in\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}$ .

Assume $a_{0}\neq 0$ . For $\mathbb{A}^{5}_{\overline{\mathbb{F}}_{p}}=D_{+}(a)$ and $A:=\overline{\mathbb{F}}_{p}[b,c,\alpha,\beta,\gamma]$ , we have

f_{{\operatorname{univ}}}^{-1}(D_{+}(a))\simeq\{x^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy=0\}\subset\mathbb{P}^{2}_{A}.

By Proposition 4.6, our singular point satisfies $\partial_{b}=\partial_{c}=0$ , which implies $[x_{0}:y_{0}:z_{0}]=[1:0:0]$ . However, such a point does not lie on ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\overline{\mathbb{F}}_{p}}/\overline{\mathbb{F}}_{p}}$ , because $[x_{0}:y_{0}:z_{0}]=[1:0:0]$ and $a_{0}\neq 0$ imply

a_{0}x_{0}^{2}+b_{0}y_{0}^{2}+c_{0}z_{0}^{2}+\alpha_{0}y_{0}z_{0}+\beta_{0}z_{0}x_{0}+\gamma_{0}x_{0}y_{0}=a_{0}x_{0}^{2}\neq 0.

∎

Theorem 5.3.

We use Notation 5.1. Then the following set-theoretic equality holds:

\Sigma_{{\operatorname{univ}}}=\{v\in\Delta_{{\operatorname{univ}}}\,|\,\Delta_{{\operatorname{univ}}}\to{\operatorname{Spec}}\,\mathbb{Z}\text{ is not smooth at }v\}.

Proof.

Set

\overline{\Sigma}_{{\operatorname{univ}}}:=\{v\in\Delta_{{\operatorname{univ}}}\,|\,\Delta_{{\operatorname{univ}}}\to{\operatorname{Spec}}\,\mathbb{Z}\text{ is not smooth at }v\},

which we equip with the reduced scheme structure. Fix a prime number $p$ . It is enough to show the set-theoretic equality $\Sigma_{{\operatorname{univ}}}\times_{\mathbb{Z}}\mathbb{F}_{p}=\overline{\Sigma}_{{\operatorname{univ}}}\times_{\mathbb{Z}}\mathbb{F}_{p}$ . Since ${\operatorname{Univ}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{F}_{p}}/\mathbb{F}_{p}}\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{\mathbb{F}_{p}}/\mathbb{F}_{p}}$ is a conic bundle of regular schemes (Proposition 5.2), the assertion follows from Theorem 3.13 and Proposition 4.2. ∎

Proposition 5.4.

We use Notation 5.1. Let $K$ be a field. Then $\Delta_{{\operatorname{univ}},K}$ is geometrically integral over $K$ and geometrically normal over $K$ . In particular, $\Delta_{{\operatorname{univ}},K}$ is a normal prime divisor on $\mathbb{P}^{5}_{K}$

Proof.

We may assume that $K$ is an algebraically closed field by taking the base change (Remark 3.5). Set $p:={\rm char}\,K$ , which is possibly zero. Recall that $\Delta_{{\operatorname{univ}},K}$ is an effective Cartier divisor on $\mathbb{P}^{5}_{K}$ . It suffices to show that $\Delta_{{\operatorname{univ}},K}$ is a normal prime divisor.

Step 1.

$\Delta_{{\operatorname{univ}},K}$ is smooth outside some closed subset of dimension two.

Proof of Step 1.

We first treat the case when $p\neq 2$ and $p\neq 3$ . Consider the hyperplane $\mathbb{P}^{4}_{K}$ defined by $\gamma=0$ :

\mathbb{P}^{4}_{K}=\{\gamma=0\}\subset\mathbb{P}^{5}_{K}.

We then have

\Delta_{K}\cap\mathbb{P}^{4}_{K}=\{\delta^{\prime}:=4abc-a\alpha^{2}-b\beta^{2}=0\}\subset\mathbb{P}^{4}_{K}={\operatorname{Proj}}\,K[a,b,c,\alpha,\beta].

Then the singular locus $(\Delta_{{\operatorname{univ}},K}\cap\mathbb{P}^{4}_{K})_{{\rm sing}}$ of $\Delta_{{\operatorname{univ}},K}\cap\mathbb{P}^{4}_{K}$ is given by

\partial_{a}\delta^{\prime}=4bc-\alpha^{2}=0,\qquad\partial_{b}\delta^{\prime}=4ac-\beta^{2}=0,\qquad\partial_{c}\delta^{\prime}=4ab=0,

\partial_{\alpha}\delta^{\prime}=-2a\alpha=0,\qquad\partial_{\beta}\delta^{\prime}=-2b\beta=0.

On the open subset $\{a\neq 0\}$ , these equations become $b=\alpha=4ac-\beta^{2}=0$ , which is one-dimensional. By symmetry, these equations define a one-dimensional locus also on the open subset $\{b\neq 0\}$ . Finally, on the remaining closed subset $\{a=b=0\}$ , the equations become $\{a=b=\alpha=\beta=0\}$ , which is zero-dimensional. To summarise, $\Delta_{{\operatorname{univ}},K}\cap\mathbb{P}^{4}_{K}$ is smooth outside a one-dimensional closed subset. Therefore, $\Delta_{{\operatorname{univ}},K}$ is smooth outside a two-dimensional closed subset when $p\neq 2$ and $p\neq 3$ . If $p=3$ , then we can not use the Jacobian criterion for homogeneous polynomials. However, we can still apply a similar argument to the above after taking the standard affine cover of $\mathbb{P}^{4}_{K}$ .

Assume $p=2$ . We then have

\Delta_{{\operatorname{univ}},K}=\{\delta:=\alpha\beta\gamma+a\alpha^{2}+b\beta^{2}+c\gamma^{2}=0\}\subset\mathbb{P}^{5}_{K}={\operatorname{Proj}}\,K[a,b,c,\alpha,\beta,\gamma].

We have

\partial_{a}\delta=\alpha^{2},\qquad\partial_{b}\delta=\beta^{2},\qquad\partial_{c}\delta=\gamma^{2},

\partial_{\alpha}\delta=\beta\gamma,\qquad\partial_{\beta}\delta=\alpha\gamma,\qquad\partial_{\gamma}\delta=\alpha\beta.

Therefore, the singular locus of $\Delta_{{\operatorname{univ}},K}$ is given by $\{\alpha=\beta=\gamma=0\}$ , which is two-dimensional. This completes the proof of Step 1. ∎

Step 2.

$\Delta_{{\operatorname{univ}},K}$ is a normal prime divisor on $\mathbb{P}^{5}_{K}$ .

Proof of Step 2.

Since $\Delta_{{\operatorname{univ}},K}$ is an effective Cartier divisor on $\mathbb{P}^{5}_{K}$ , $\Delta_{{\operatorname{univ}},K}$ is Cohen–Macaulay. By Step 1, $\Delta_{{\operatorname{univ}},K}$ is normal by Serre’s criterion. If $\Delta_{{\operatorname{univ}},K}$ is not irreducible, then $\Delta_{{\operatorname{univ}},K}$ would be non-normal, because two distinct prime divisors on $\mathbb{P}^{5}_{K}$ intersect along a three-dimensional non-empty closed subset. Therefore, $\Delta_{{\operatorname{univ}},K}$ is a normal prime divisor. This completes the proof of Step 2. ∎

Step 2 completes the proof of Proposition 5.4. ∎

Corollary 5.5.

$\Delta_{{\operatorname{univ}}}$ is flat over ${\operatorname{Spec}}\,\mathbb{Z}$ .

Proof.

The assertion follows from the fact that $\Delta_{{\operatorname{univ}}}\to{\operatorname{Spec}}\,\mathbb{Z}$ is a equi-dimensional morphism (Proposition 5.4) from a Cohen–Macaulay scheme to a regular scheme. ∎

5.2. Relative universal families

Notation 5.6.

(1)

Let $k$ be an algebraically closed field, let $C$ be a smooth projective curve over $k$ , and let $E$ be a locally free sheaf on $C$ of rank $3$ .
(2)

Set $\theta(m):=2m+1\in\mathbb{Z}[m]$ , which is the Hilbert polynomial of conics.
(3)

We have the Hilbert scheme ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ and let ${\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}\subset\mathbb{P}_{C}(E)\times_{C}{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ be the universal closed subscheme.

(4)

The induced morphism

f_{{\operatorname{univ}},C,E}:{\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}\hookrightarrow{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\times_{C}\mathbb{P}_{C}(E)\xrightarrow{{\rm pr}_{1}}{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}

is a generically smooth conic bundle. Set

\Delta_{{\operatorname{univ}},C,E}:=\Delta_{f_{{\operatorname{univ}},C,E}}\qquad{\rm and}\qquad\Sigma_{{\operatorname{univ}},C,E}:=\Sigma_{f_{{\operatorname{univ}},C,E}}.

Theorem 5.7.

We use Notation 5.6. Then the following hold.

(1)

The induced morphism $\pi:{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to C$ is a $\mathbb{P}^{5}$ -bundle, i.e., there exists an open cover $C=\bigcup_{i\in I}C_{i}$ such that the induced morphism ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\times_{C}C_{i}$ is isomorphic to $\mathbb{P}^{5}\times_{k}C_{i}$ for every $i\in I$ .
(2)

${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ is a $6$ -dimensional smooth projective variety.
(3)

${\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ is a $7$ -dimensional smooth projective variety.
(4)

$\Delta_{{\operatorname{univ}},C,E}$ is a reduced normal divisor on ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ .
(5)

For the singular locus $(\Delta_{{\operatorname{univ}},C,E})_{{\rm sing}}$ of $\Delta_{{\operatorname{univ}},C,E}$ , we have the set-theoretic equality

$(\Delta_{{\operatorname{univ}},C,E})_{{\rm sing}}=\Sigma_{{\operatorname{univ}},C,E}.$

Furthermore, it holds that $\dim\Sigma_{{\operatorname{univ}},C,E}\leq 3$ .

Proof.

Fix a non-empty open subset $C^{\prime}$ of $C$ such that $E^{\prime}:=E|_{C^{\prime}}\simeq\mathcal{O}_{C^{\prime}}^{\oplus 3}$ . Then we have the following diagrams in which all the squares are cartesian:

Then the assertion (1) holds by ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}^{2}_{k}/k}\simeq\mathbb{P}^{5}_{k}$ (cf. Notation 5.1).

Let us show (2) and (3). By (1) and Proposition 5.2, both ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to C$ and ${\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to C$ are smooth morphisms. Hence it is enough to show that ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ and ${\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ are connected. Since $C$ and all the fibres of $\pi:{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to C$ are connected, also ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ is connected. Similarly, ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}}$ is connected, because so are ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ and all the fibres of ${\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ . This completes the proof of (2) and (3).

The assertion (4) follows from Remark 3.5 and Proposition 5.4. Let us show (5). Since ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ and ${\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ are smooth varieties by (2) and (3), the set-theoretic equality $(\Delta_{{\operatorname{univ}},C,E})_{{\rm sing}}=\Sigma_{{\operatorname{univ}},C,E}$ follows from Theorem 4.3. Since $\Delta_{{\operatorname{univ}},C,E}$ is a reduced normal divisor on ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ , we obtain

\dim\Sigma_{{\operatorname{univ}},C,E}=\dim(\Delta_{{\operatorname{univ}},C,E})_{{\rm sing}}\leq\dim\Delta_{{\operatorname{univ}},C,E}-2\leq(\dim{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}-1)-2=3.

Thus (5) holds. ∎

Lemma 5.8.

We use Notation 5.6. Let $\Gamma$ and $L$ be curves on ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ such that $\pi(\Gamma)=C$ and $\pi(L)$ is a point. Then

N_{1}({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C})\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{Q}[\Gamma]+\mathbb{Q}[L],

where $N_{1}({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}):=Z_{1}({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C})/\equiv$ , $Z_{1}({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}):=\bigoplus_{B:\text{curve}}\mathbb{Z}B$ , $\equiv$ denotes the numerical equivalence, and $[\Gamma]$ and $[L]$ are the numerical equivalence classes.

Proof.

For an ample Cartier divisor $H$ on $C$ , we obtain $\pi^{*}H\cdot\Gamma>0$ and $\pi^{*}H\cdot L=0$ , which imply that $[\Gamma]$ and $[L]$ are linearly independent over $\mathbb{Q}$ . Hence it suffices to show that the following sequence is exact, because it implies $\rho({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C})=2$ :

0\to{\operatorname{Pic}}\,C\xrightarrow{\pi^{*}}{\operatorname{Pic}}\,({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C})\xrightarrow{\cdot L}\mathbb{Z}.

By $\pi_{*}\mathcal{O}_{{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}}=\mathcal{O}_{C}$ (Theorem 5.7(1)), $\pi^{*}:{\operatorname{Pic}}\,C\to{\operatorname{Pic}}\,({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C})$ is injective. Pick $M\in{\operatorname{Pic}}\,({\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C})$ satisfying $M\cdot L=0$ . By using Theorem 5.7(1), it is easy to see that $\pi_{*}M\in{\operatorname{Pic}}\,C$ and $\pi^{*}\pi_{*}M\to M$ is an isomorphism. ∎

5.3. Discriminant bundles

Recall that the discriminant scheme $\Delta_{f}$ for a conic bundle $f:X\to T$ satisfies the following properties.

(I)

For a point $t\in T$ , $X_{t}$ is not smooth if and only if $t\not\in\Delta_{f}$ (Theorem 3.13).
(II)

Let

$\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}$

be a cartesian diagram of noetherian schemes such that $f$ is a conic bundle. In particular, also $f^{\prime}$ is a conic bundle. Then the equality

$\Delta^{{\operatorname{bdl}}}_{f^{\prime}}=\beta^{-1}\Delta^{{\operatorname{bdl}}}_{f}$

of closed subschemes holds (Remark 3.5).

The purpose of this subsection is to introduce an invertible sheaf $\Delta_{f}^{{\operatorname{bdl}}}$ on $T$ that satisfies the following two properties.

(I)’

Let $f:X\to T$ be a generically smooth conic bundle such that $T$ is a noetherian integral scheme. Then $\Delta^{{\operatorname{bdl}}}_{f}\simeq\mathcal{O}_{T}(\Delta_{f})$ (Remark 5.11).
(II)’

Let

$\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}$

be a cartesian diagram, where $T$ and $T^{\prime}$ are notherian integral schemes, $f$ is a conic bundle, and $\beta$ is of finite type. In particular, also $f^{\prime}$ is a conic bundle. Then $\Delta^{{\operatorname{bdl}}}_{f^{\prime}}\simeq\beta^{*}\Delta^{{\operatorname{bdl}}}_{f}$ (Theorem 5.13).

Although our definition (Definition 5.9) might look unnatural at first sight, it is designed in order that $\Delta^{{\operatorname{bdl}}}_{f}$ satisfies (I)’ and (II)’. Indeed, it is easy to see that there is the unique way, if it exists, to define invertible sheaves $\Delta_{f}^{{\operatorname{bdl}}}$ satisfying (I)’and (II)’ when $f:X\to T$ is a conic bundle and $T$ is a regular noetherian integral scheme.

Definition 5.9.

Let $f:X\to T$ be a conic bundle, where $T$ is a noetherian integral scheme. We then have the following cartesian diagram:

\begin{CD}X@>{}>{}>{\operatorname{Univ}}^{\theta}_{\mathbb{P}(f_{*}\omega_{X/T}^{-1})/T}\\ @V{}V{f}V@V{}V{g}V\\ T@>{j}>{}>{\operatorname{Hilb}}^{\theta}_{\mathbb{P}(f_{*}\omega_{X/T}^{-1})/T},\end{CD}

where $\theta$ is the Hilbert polynomial of conics (Subsection 2.1(9)) and $j$ denotes the morphism induced by the closed immersion $X\hookrightarrow\mathbb{P}(f_{*}\omega_{X/T}^{-1})$ (Proposition 2.7(3)). We set

\Delta^{{\operatorname{bdl}}}_{f}:=j^{*}\mathcal{O}_{T}(\Delta_{g}),

which we call the discriminant bundle of $f$ . Note that $\Delta_{g}$ is an effective Cartier divisor on $T$ (Lemma 5.10) and $\mathcal{O}_{T}(\Delta_{g})$ denotes the invertible sheaf corresponding to $\Delta_{g}$ . In particular, $\Delta^{{\operatorname{bdl}}}_{f}$ is an invertible sheaf on $T$ .

Lemma 5.10.

We use the same notation as in Definition 5.9. Then $\Delta_{g}$ is an effective Cartier divisor.

Proof.

By the same argument as in Theorem 5.7(1)(2), we see that ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}(f_{*}\omega_{X/T}^{-1})/T}\to T$ is a $\mathbb{P}^{5}$ -bundle and ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}(f_{*}\omega_{X/T}^{-1})/T}$ is a noetherian integral scheme. Furthermore, ${\operatorname{Univ}}^{\theta}_{\mathbb{P}(f_{*}\omega_{X/T}^{-1})/T}\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}(f_{*}\omega_{X/T}^{-1})/T}$ is a generically smooth conic bundle. Thus $\Delta_{g}$ is an effective Cartier divisor. ∎

Remark 5.11.

Let $f:X\to T$ be a conic bundle, where $T$ is a noetherian integral scheme. If $f$ is generically smooth, then it follows from Definition 5.9 that $\Delta^{{\operatorname{bdl}}}_{f}\simeq\mathcal{O}_{T}(\Delta_{f})$ . On the other hand, if $f$ is not generically smooth, then $\Delta_{f}=T$ , which is no longer a Cartier divisor.

Lemma 5.12.

Let $\beta:T^{\prime}\to T$ be a morphism of noetherian schemes. Let $E$ be a locally free sheaf of rank $3$ on $T$ . Set $E^{\prime}:=\beta^{*}E$ . Then

{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T^{\prime}}(E^{\prime})/T^{\prime}}\simeq{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T}(E)/T}\times_{T}T^{\prime},

where $\theta:=2m+1\in\mathbb{Z}[m]$ , which is the Hilbert polynomial of conics.

Proof.

The assertion follows from $\mathbb{P}_{T^{\prime}}(E^{\prime})\simeq\mathbb{P}_{T}(E)\times_{T}T^{\prime}$ . ∎

Theorem 5.13.

Let

\begin{CD}X^{\prime}@>{\alpha}>{}>X\\ @V{}V{f^{\prime}}V@V{}V{f}V\\ T^{\prime}@>{\beta}>{}>T\end{CD}

be a cartesian diagram of noetherian schemes, where $T$ and $T^{\prime}$ are noetherian integral schemes and $f$ is a conic bundle. Assume that one of the following holds.

(A)

$\beta$ is flat.
(B)

$\beta$ is a closed immersion such that $\beta(T^{\prime})$ is an effective Cartier divisor on $T$ .
(C)

Both $T$ and $T^{\prime}$ are regular and $\beta$ is of finite type.

Then the following hold.

(1)

The induced homomorphism

$\theta:\beta^{*}f_{*}\omega_{X/T}^{-1}\to f^{\prime}_{*}\alpha^{*}\omega^{-1}_{X/T}$

is an isomorphism.
(2)

An isomorphism $\Delta^{{\operatorname{bdl}}}_{X^{\prime}/T^{\prime}}\simeq\beta^{*}\Delta^{{\operatorname{bdl}}}_{X/T}$ holds.

Proof.

By Remark 3.5 and Lemma 5.12, (1) implies (2). Hence it suffices to prove (1). We may assume that $T$ and $T^{\prime}$ are affine. If (A) holds, then the assertion (1) immediately follows from the flat base change theorem.

Assume (B). Recall that $\beta^{*}f_{*}\omega_{X/T}^{-1}$ is a locally free sheaf of rank $3$ (Proposition 2.7). By the same argument as in Proposition 2.7, also $f^{\prime}_{*}\alpha^{*}\omega^{-1}_{X/T}$ is a locally free sheaf of rank $3$ . Then it is enough to show that $\theta:\beta^{*}f_{*}\omega_{X/T}^{-1}\to f^{\prime}_{*}\alpha^{*}\omega^{-1}_{X/T}$ is surjective. Hence the problem is reduced to the surjectivity of

H^{0}(X,\omega_{X/T}^{-1})\to H^{0}(X^{\prime},\alpha^{*}\omega_{X/T}^{-1}).

We have an exact sequence:

0\to\omega^{-1}_{X/T}\otimes I_{X^{\prime}}\to\omega^{-1}_{X/T}\to\alpha_{*}\alpha^{*}\omega^{-1}_{X/T}\to 0.

By $I_{X^{\prime}}=\mathcal{O}_{X}(-X^{\prime})=f^{*}\mathcal{O}_{T}(-T^{\prime})$ , we obtain

R^{1}f_{*}(\omega^{-1}_{X/T}\otimes I_{X^{\prime}})\simeq R^{1}f_{*}(\omega^{-1}_{X/T}\otimes f^{*}\mathcal{O}_{T}(-T^{\prime}))\simeq R^{1}f_{*}(\omega^{-1}_{X/T})\otimes\mathcal{O}_{T}(-T^{\prime})\overset{{\rm(*)}}{=}0,

where ${\rm(*)}$ follows from Proposition 2.7. This completes the proof of (1) when (B) holds.

Assume (C). Since the problem is local on $T$ and $T^{\prime}$ , the problem is reduced to the case when $\beta:T^{\prime}\to T$ is factored as follows:

T=:T_{0}\hookrightarrow T_{1}\hookrightarrow\cdots\hookrightarrow T_{r}:=T^{\prime}\times\mathbb{A}^{n}\xrightarrow{{\rm pr}_{1}}T^{\prime},

where each $T_{i}$ is a regular affine noetherian integral scheme and each $T_{i}\hookrightarrow T_{i+1}$ is a closed immersion with $\dim T_{i}=\dim T_{i+1}-1$ [Mat86, Theorem 14.2 and Theorem 21.2]. Therefore, the problem is reduced to the case when (A) or (B) holds. This completes the proof of (1). ∎

5.4. Proof of the Mori–Mukai formula

Lemma 5.14.

We work over an algebraically closed field $k$ . Let $f:X\to T$ be a conic bundle, where $X$ is a smooth surface and $T$ is a smooth curve. Then $f$ is generically smooth and every singular fibre of $f$ is reduced and consists of exactly two $(-1)$ -curves.

Proof.

Let $X^{\prime}$ be a relatively minimal model of $f$ , so that we obtain the induced morphisms

f:X\xrightarrow{\varphi}X^{\prime}\xrightarrow{f^{\prime}}T.

Then the assertion follows from the fact that $f^{\prime}:X^{\prime}\to T$ is a $\mathbb{P}^{1}$ -bundle and $\varphi:X\to X^{\prime}$ is a sequence of blowups. ∎

We are ready to prove the Mori-Mukai formula in arbitrary characteristics, which generalises the case of characteristic zero [MM83, Proposition 6.2].

Theorem 5.15.

We work over an algebraically closed field $k$ . Let $f:X\to T$ be a conic bundle, where $T$ is a smooth projective variety. Then the equality

(5.15.1)

\Delta^{{\operatorname{bdl}}}_{f}\cdot C=-\omega_{X/T}\cdot\omega_{X/T}\cdot f^{-1}(C)

holds for every curve $C$ on $T$ . In particular, when also $X$ is smooth over $k$ , the numerical equivalence

\Delta^{{\operatorname{bdl}}}_{f}\equiv-f_{*}(K_{X/T}^{2})

holds for a Cartier divisor $K_{X/T}$ on $X$ satisfying $\omega_{X/T}\simeq\mathcal{O}_{X}(K_{X/T})$ .

Proof.

The in-particular part follows from the projection formula. Hence it suffices to show (5.15.1). In what follows, we write $\mathcal{L}^{2}:=\mathcal{L}\cdot\mathcal{L}$ (note that $\mathcal{L}^{2}$ differs from $\mathcal{L}^{\otimes 2}$ ).

Step 1.

Let $C$ be a projective curve on $T$ . Let $C^{N}\to C$ be the normalisation of $C$ and let $g:X\times_{T}C^{N}\to C^{N}$ be the base change of $f$ . Then the following hold.

(1)

$\Delta^{{\operatorname{bdl}}}_{f}\cdot C=\deg\Delta^{{\operatorname{bdl}}}_{g}$ .
(2)

$\omega_{X/T}^{2}\cdot f^{-1}(C)=\omega^{2}_{X\times_{T}C^{N}/C^{N}}$ .

Proof of Step 1.

We have the following cartesian diagram:

The assertion (1) holds by

\Delta^{{\operatorname{bdl}}}_{f}\cdot C=\deg(\beta^{*}\Delta^{{\operatorname{bdl}}}_{f})=\deg\Delta^{{\operatorname{bdl}}}_{g},

where the latter equality follows from $\beta^{*}\Delta^{{\operatorname{bdl}}}_{f}\simeq\Delta^{{\operatorname{bdl}}}_{g}$ (Theorem 5.13(2)). We obtain (2) by the following:

\omega_{X/T}^{2}\cdot f^{-1}(C)\overset{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\star)}}{=}(\omega_{X/T}|_{f^{-1}(C)})^{2}=(\alpha^{*}\omega_{X/T})^{2}\overset{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\star\star)}}{=}\omega_{X\times_{T}C^{N}/C^{N}}^{2},

where $(\star)$ holds by [Bad01, Lemma 1.10] and $(\star\star)$ follows from $\alpha^{*}\omega_{X/T}\simeq\omega_{X\times_{T}C^{N}/C^{N}}$ [Con00, Theorem 3.6.1]. For the definition and some properties of the intersection number $(\omega_{X/T}|_{f^{-1}(C)})^{2}$ , we refer to [Bad01, Section 1]. This completes the proof of Step 1. ∎

Step 2.

The assertion of Theorem 5.15 holds when $T$ is a smooth projective curve, $f$ is generically smooth, and $\Delta_{f}$ is reduced.

Proof of Step 2.

By Theorem 4.4, $X$ is smooth over $k$ and any fibre of $f$ is reduced, i.e., each singular fibre consists of two $(-1)$ -curves (Lemma 5.14). Let

f:X\xrightarrow{\mu}X^{\prime}\xrightarrow{f^{\prime}}T

be a relatively minimal model. Then any fibre of $f^{\prime}:X^{\prime}\to T$ is $\mathbb{P}^{1}$ . Set $e$ to be the number of the singular fibres of $f:X\to T$ . Then we have

K_{X}^{2}=K_{X^{\prime}}^{2}-e.

For the genus $g(T)$ of $T$ , it holds that

K_{X^{\prime}/T}^{2}=(K_{X^{\prime}}-f^{\prime*}K_{T})^{2}=K_{X^{\prime}}^{2}-2K_{X^{\prime}}\cdot f^{\prime*}K_{T}

=8(1-g(T))-2\cdot(-2)\cdot(2g(T)-2)=0.

By $\Delta_{f}^{{\operatorname{bdl}}}\simeq\mathcal{O}_{T}(\Delta_{f})$ (Remark 5.11), we obtain

\deg\Delta^{{\operatorname{bdl}}}_{f}=\deg\Delta_{f}=e=K_{X^{\prime}}^{2}-K_{X}^{2}=K_{X^{\prime}/T}^{2}-K_{X/T}^{2}=-K_{X/T}^{2}.

This completes the proof of Step 2. ∎

Step 3.

The assertion of Theorem 5.15 holds when $f:X\to T$ coincides with the induced morphism

f_{{\operatorname{univ}},C,E}:{\operatorname{Univ}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}

for some smooth projective curve $C$ and locally free sheaf $E$ on $C$ of rank $3$ .

Proof of Step 3.

We have the projection

\pi:{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to C.

Recall that there exists an open cover $C=\bigcup_{i\in I}C_{i}$ such that $\pi^{-1}(C_{i})\simeq\mathbb{P}^{5}\times C_{i}$ (Theorem 5.7). Take a general line $L$ contained in a fibre of $\pi$ . Fix a smooth projective curve $T^{\prime}$ obtained as a complete intersection of general hyperplane sections of $T={\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ . It follows from Lemma 5.8 that

N_{1}(T)\otimes_{\mathbb{Z}}\mathbb{Q}=\mathbb{Q}[T^{\prime}]+\mathbb{Q}[L].

Therefore, it is enough to show

\Delta^{{\operatorname{bdl}}}_{f}\cdot T^{\prime}=-\omega_{X/T}^{2}\cdot f^{-1}(T^{\prime})\qquad\text{and}\qquad\Delta^{{\operatorname{bdl}}}_{f}\cdot L=-\omega_{X/T}^{2}\cdot f^{-1}(L).

We now prove $\Delta^{{\operatorname{bdl}}}_{f}\cdot T^{\prime}=-\omega_{X/T}^{2}\cdot f^{-1}(T^{\prime})$ . Recall that $\Delta_{{\operatorname{univ}},C,E}$ is a reduced divisor on a $6$ -dimensional smooth projective variety $T={\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}$ (Theorem 5.7). Hence the scheme-theoretic intersection $T^{\prime}\cap\Delta_{{\operatorname{univ}},C,E}$ is smooth, because the curve $T^{\prime}$ is defined as a complete intersection of general hyperplane sections. For the base change $f^{\prime}:X\times_{T}T^{\prime}\to T^{\prime}$ of $f$ , we have $\Delta_{f^{\prime}}=T^{\prime}\cap\Delta_{f}=T^{\prime}\cap\Delta_{{\operatorname{univ}},C,E}$ (Remark 3.5), and hence $\Delta_{f^{\prime}}$ is smooth. Then

\Delta^{{\operatorname{bdl}}}_{f}\cdot T^{\prime}\overset{{\rm(i)}}{=}\deg\Delta_{f^{\prime}}\overset{{\rm(ii)}}{=}-\omega_{X\times_{T}T^{\prime}/T^{\prime}}^{2}\overset{{\rm(iii)}}{=}-\omega_{X/T}^{2}\cdot f^{-1}(T^{\prime}),

where (i) and (iii) hold by Step 1(1) and Step 1(2) respectively, and (ii) follows from Step 2. This completes the proof of $\Delta^{{\operatorname{bdl}}}_{f}\cdot T^{\prime}=-\omega_{X/T}^{2}\cdot f^{-1}(T^{\prime})$ .

It suffices to show $\Delta^{{\operatorname{bdl}}}_{f}\cdot L=-\omega_{X/T}^{2}\cdot f^{-1}(L)$ . By the same argument as in the previous paragraph, it is enough to prove that the scheme-theoretic intersection $L\cap\Delta_{{\operatorname{univ}},C,E}$ is smooth. Let $F(\simeq\mathbb{P}^{5}_{k})$ be the fibre of $\pi:{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{C}(E)/C}\to C$ containing $L$ . There is $i\in I$ such that $F\hookrightarrow\pi^{-1}(C_{i})$ . Consider the following composite isomorphism:

\iota:F\hookrightarrow\pi^{-1}(C_{i})\simeq\mathbb{P}^{5}\times C_{i}\xrightarrow{{\rm pr}_{1}}\mathbb{P}^{5}.

We have $\iota^{*}\Delta_{{\operatorname{univ}},k}=\Delta_{{\operatorname{univ}},C,E}|_{F}$ , because $\Delta_{{\operatorname{univ}},C,E}|_{F}$ is the pullback of $\Delta_{{\operatorname{univ}},C,E}|_{\pi^{-1}(C_{i})}$ . It follows from Proposition 5.4 that the scheme-theoretic intersection $M\cap\Delta_{{\operatorname{univ}},k}$ is smooth for a general line $M$ on $\mathbb{P}^{5}$ . As $L$ is chosen to be a general line on $F\simeq\mathbb{P}^{5}$ , also $L\cap\Delta_{{\operatorname{univ}},C,E}$ is smooth. This completes the proof of Step 3. ∎

Step 4.

The assertion of Theorem 5.15 holds when $T$ is a smooth projective curve.

Proof of Step 4.

Set $E:=f_{*}(\omega_{X/T}^{-1})$ . Then $E$ is a locally free sheaf of rank $3$ (Proposition 2.7(2)) and $f:X\hookrightarrow\mathbb{P}_{T}(E)\to T$ is a flat family of conics on $\mathbb{P}_{T}(E)$ relatively over $T$ . Therefore, we have the following cartesian diagram

Note that $\beta:T\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T}(E)/T}$ is a section of the projection ${\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T}(E)/T}\to T$ . In particular, $\beta:T\to{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T}(E)/T}$ is a closed immersion. We then obtain

\deg\Delta^{{\operatorname{bdl}}}_{f}\overset{{\rm(i)^{\prime}}}{=}\Delta^{{\operatorname{bdl}}}_{f_{{\operatorname{univ}},T,E}}\cdot\beta(T)\overset{{\rm(ii)^{\prime}}}{=}-\omega^{2}_{{\operatorname{Univ}}^{\theta}_{\mathbb{P}_{T}(E)/T}/{\operatorname{Hilb}}^{\theta}_{\mathbb{P}_{T}(E)/T}}\cdot f_{{\operatorname{univ}},T,E}^{-1}(\beta(T))\overset{{\rm(iii)^{\prime}}}{=}-\omega_{X/T}^{2},

where ${\rm(i)^{\prime}}$ and ${\rm(iii)^{\prime}}$ hold by Step 1(1) and Step 1(2) respectively, and ${\rm(ii)^{\prime}}$ follows from Step 3. This completes the proof of Step 4. ∎

Step 5.

Theorem 5.15 holds without any additional assumptions.

Proof of Step 5.

It suffices to show that $\Delta_{f}\cdot C=-f_{*}\omega_{X/T}^{2}\cdot C$ for any curve $C$ on $T$ . This follows from

\Delta^{{\operatorname{bdl}}}_{f}\cdot C\overset{{\rm(i)^{\prime\prime}}}{=}\deg\Delta^{{\operatorname{bdl}}}_{h}\overset{{\rm(ii)^{\prime\prime}}}{=}-\omega_{X\times_{T}C^{N}/C^{N}}^{2}\overset{{\rm(iii)^{\prime\prime}}}{=}-\omega_{X/T}^{2}\cdot f^{-1}(C),

where $h:X\times_{T}C^{N}\to C^{N}$ denotes the base change of $f:X\to T$ , ${\rm(i)^{\prime\prime}}$ and ${\rm(iii)^{\prime\prime}}$ hold by Step 1(1) and Step 1(2) respectively, and ${\rm(ii)^{\prime\prime}}$ follows from Step 4. This completes the proof of Step 5. ∎

Step 5 completes the proof of Theorem 5.15. ∎

6. Surfaces

In this section, we focus on surface conic bundles $f:X\to T$ . More precisely, we treat the case when $X$ has at worst canonical singularities, since there is nothing to do for the case when $X$ is smooth over $k$ (Lemma 5.14). The motivation is to seek a relation between $\Delta_{f}$ and the singularities of $X$ . For this purpose, we assume that $f:X\to T$ has the unique singular fibre $X_{0}$ , where $0\in T$ is a closed point. The primary goal of this section is to establish the following results (Theorem 6.3 in Subsection 6.2).

(1)

$\deg\Delta_{f}=m-1$ , where $m$ denotes the number of the irreducible components of the fibre $Y_{0}$ , where $Y\to X$ is the minimal resolution of $X$ .
(2)
If $X_{0}$ is reduced, then
- •
  
  $X$ has the unique singularity $x$ , and
- •
  
  $x$ is of type $A_{n}$ , where $n:=m-2=\deg\Delta_{f}-1$ .
(3)
If $X_{0}$ is not reduced, then one and only one of the following holds.
1. (a)
  
  $X$ has exactly two singularities $x$ and $x^{\prime}$ . Moreover, both $x$ and $x^{\prime}$ are of type $A_{1}$ . In this case, $\deg\Delta_{f}=2$ .
2. (b)
  
  $X$ has the unique singularity $x$ . Moreover, $x$ is of type $D_{n}$ with $n\geq 3$ , where $n=m-1=\deg\Delta_{f}$ .

For example, $X$ has the unique singularity of type $D_{4}$ if $\deg\Delta_{f}=4$ and $X_{0}$ is not reduced (note that each of these conditions can be checked, as far as $f:X\to T$ is explicitly given). In order to establish the above properties (1)–(3), we first classify the dual graphs of the exceptional loci of the minimal resolutions of surface conic bundles (Subsection 6.1). In Subsection 6.3, we shall summarise further classification results for surface conic bundles and exhibit several examples.

6.1. The classification of dual graphs

In this subsection, we classify the singularities of surface conic bundles for the case when the total space $X$ has at worst canonical singularities (Proposition 6.1, Proposition 6.2). The results in this subsection should be well known to experts. We include the proofs for the reader’s convenience. We refer to [KM98, §4.1, cf. 4.19] for some foundational results on canonical surface singularities.

Proposition 6.1.

We work over an algebraically closed field $k$ . Let $f:X\to T$ be a conic bundle, where $X$ is a canonical surface and $T$ is a smooth curve. Fix a closed point $0\in T$ . Assume that $X_{0}$ is reduced and $X_{0}$ is the unique singular fibre. Let $\varphi:Y\to X$ be the minimal resolution of $X$ . Then the dual graph of the singular fibre of the composition $Y\xrightarrow{\varphi}X\xrightarrow{f}T$ is as follows:

(-1)-(-2)-(-2)-\cdots-(-2)-(-1).

In particular, $X$ has the unique singularity $x$ and $x$ is of type $A_{n}$ for some $n\in\mathbb{Z}_{>0}$ .

Proof.

We run a $K_{Y}$ -MMP over $T$ :

\psi:Y=:Z_{m}\xrightarrow{\psi_{m-1}}Z_{m-1}\xrightarrow{\psi_{m-2}}\cdots\xrightarrow{\psi_{2}}Z_{2}\xrightarrow{\psi_{1}}Z_{1}=:Z,

i.e., each $\psi_{i}:Z_{i+1}\to Z_{i}$ is a contraction of a $(-1)$ -curve contained in the fibre over $0\in T$ . After possibly replacing $T$ by an open neighbourhood of $0\in T$ , we may assume that $Z_{1}=Z=\mathbb{P}^{1}\times T$ . Then the singular fibre $(Z_{2})_{0}$ consists of exactly two $(-1)$ -curves $E_{1}+E_{2}$ .

It is enough to prove the following three properties (1)-(3).

(1)

The blowup centre of $\psi_{i}:Z_{i+1}\to Z_{i}$ is a smooth point of the fibre $(Z_{i})_{0}$ for every $i$ .
(2)

The fibre $(Z_{i})_{0}$ over $0$ is reduced for every $i$ .
(3)

The blowup centre of $\psi_{i}:Z_{i+1}\to Z_{i}$ is disjoint from any $(-2)$ -curve contained in $(Z_{i})_{0}$ .

Indeed, if the singular fibre $(Z_{i})_{0}$ is a chain

(-1)-(-2)-(-2)-\cdots-(-2)-(-1),

which is reduced, then the next blowup centre lies on one of the leftmost and rightmost $(-1)$ -curves and avoids the $(-2)$ -curves.

Let us show (1). Suppose that $(Z_{i})_{0}$ is reduced and the blowup centre of $\psi_{i}:Z_{i+1}\to Z_{i}$ is a singular point of $(Z_{i})_{0}$ , i.e., it is an intersection of two irreducible components. Then $(Z_{i+1})_{0}$ is not reduced along the resulting $(-1)$ -curve $F_{i+1}$ . Then we can show, by induction on $j$ , that there is a $(-1)$ -curve $F_{j}$ on $Z_{j}$ such that $(Z_{j})_{0}$ is non-reduced along $F_{j}$ . This implies that $Y=Z_{m}$ contains a $(-1)$ -curve $F$ inside the singular fibre $Y_{0}$ such that $Y_{0}$ is not reduced along $F$ . Then $X_{0}$ is non-reduced along $\varphi(F)$ . This contradicts our assumption. Thus (1) holds. By induction on $i$ , the assertion (2) follows from (1).

Let us show (3). Note that the number of the irreducible components of $(Z_{i})_{0}$ is equal to $i$ . Let $\nu_{i}$ be the number of $(-2)$ -curves inside the fibre $(Z_{i})_{0}$ , e.g., we have $\nu_{1}=\nu_{2}=0$ and $\nu_{3}=1$ . For $i\geq 2$ , let us show that $\nu_{i}=i-2$ by induction on $i$ . Fix $i\geq 2$ and assume $\nu_{i}=i-2$ . By (1), we obtain

\nu_{i+1}\leq\nu_{i}+1.

Furthermore, the equality holds if and only if the blowup centre lies on a $(-1)$ -curve. We obtain

m-2\leq\nu_{m}\leq\nu_{2}+m-2=m-2,

where the first inequality follows from the fact that $X$ has two irreducible components and ${\operatorname{Ex}}(\varphi)$ consists of $(m-2)$ irreducible components. Therefore, we obtain $\nu_{i+1}=\nu_{i}+1$ . This completes the proof of $\nu_{i}=i-2$ for $i\geq 2$ . Then we get $\nu_{i+1}=\nu_{i}+1$ , i.e., the blowup centre of $\psi_{i}:Z_{i+1}\to Z_{i}$ lies on a $(-1)$ -curve. Thus (3) holds. ∎

Proposition 6.2.

We work over an algebraically closed field $k$ . Let $f:X\to T$ be a conic bundle, where $X$ is a canonical surface and $T$ is a smooth curve. Fix a closed point $0\in T$ . Assume that $X_{0}$ is non-reduced and $X_{0}$ is the unique singular fibre. Let $\varphi:Y\to X$ be the minimal resolution of $X$ . Then the dual graph of the singular fibre of the composition $Y\xrightarrow{\varphi}X\xrightarrow{f}T$ is either a chain

(-2)-(-1)-(-2)

or

D_{n}-(-1)

with $n\geq 3$ , where the latter case means that $(-1)$ -curve intersects only with the long tail.

Proof.

We run a $K_{Y}$ -MMP over $T$ :

\psi:Y=:Z_{m}\xrightarrow{\psi_{m-1}}Z_{m-1}\xrightarrow{\psi_{m-2}}\cdots\xrightarrow{\psi_{2}}Z_{2}\xrightarrow{\psi_{1}}Z_{1}=:Z,

i.e., each $\psi_{i}:Z_{i+1}\to Z_{i}$ is a contraction of a $(-1)$ -curve contained in the fibre over $0\in T$ . After possibly replacing $T$ by an open neighbourhood of $0\in T$ , we may assume that $Z_{1}=Z=\mathbb{P}^{1}\times T$ . It is clear that $(Z_{i})_{0}$ contains at least one $(-1)$ -curve for every $i\geq 2$ . Since $X_{0}$ is irreducible and ${\operatorname{Ex}}(\varphi:Y=Z_{m}\to X)$ consists of $(-2)$ -curves, the following holds:

(1)

We have the irreducible decomposition $((Z_{m})_{0})_{{\operatorname{red}}}=A_{1}\cup\cdots\cup A_{m-1}\cup B$ , where each $A_{i}$ is a $(-2)$ -curve and $B$ is a $(-1)$ -curve.

We see that

(2)

$(Z_{i})_{0}$ contains no prime divisor $C$ satisfying $C^{2}\leq-3$ ,

as otherwise its proper transform $C^{\prime}\subset(Z_{m})_{0}$ would satisfy $C^{\prime 2}\leq-3$ , which contradicts (1).

We now show that the blowup centre of $\psi_{2}:Z_{3}\to Z_{2}$ is the singular point $E_{1}\cap E_{2}$ for the irreducible decomposition $(Z_{2})_{0}=E_{1}\cup E_{2}$ . Otherwise, $(Z_{3})_{0}$ contains two $(-1)$ -curves $F$ and $F^{\prime}$ which are mutually disjoint. This property is stable under taking a blowup. Hence also $(Z_{m})_{0}$ has at least two $(-1)$ -curves, which contradicts (1). Therefore, the dual graph of $(Z_{3})_{0}$ is given by

(-2)-(-1)-(-2).

The blowup centre of the next blowup $Z_{4}\to Z_{3}$ is disjoint from the two $(-2)$ -curves by (2). Then $(Z_{4})_{0}$ consists of three $(-2)$ -curves and the unique $(-1)$ -curve. The blowup centre of $\psi_{4}:Z_{5}\to Z_{4}$ is disjoint from these $(-2)$ -curves. Repeating this procedure, we see that the resulting dual graph of $Y_{0}=(Z_{m})_{0}$ is as in the statement. ∎

6.2. Discriminants vs singularities

We are ready to prove a main theorem of this section.

Theorem 6.3.

We work over an algebraically closed field $k$ . Let $f:X\to T$ be a conic bundle, where $X$ is a canonical surface and $T$ is a smooth curve. Assume that there exists a closed point $0\in T$ such that $X_{0}$ is the unique singular fibre. Let $\varphi:Y\to X$ be the minimal resolution of $X$ and set $m$ to be the number of the irreducible components of the fibre $Y_{0}$ over $0\in T$ . Then the following hold.

(1)

$\deg\Delta_{f}=m-1$ .
(2)

Assume that $X_{0}$ is reduced. Then $X$ has the unique singularity $x$ and $x$ is of type $A_{n}$ , where $n=m-2=\deg\Delta_{f}-1$ .
(3)
Assume that $X_{0}$ is not reduced. Then one and only one of the following holds.
1. (a)
  
  $X$ has exactly two singularities $x$ and $x^{\prime}$ . Moreover, both $x$ and $x^{\prime}$ are of type $A_{1}$ . In this case, $\deg\Delta_{f}=2$ .
2. (b)
  
  $X$ has the unique singularity $x$ and $x$ is of type $D_{n}$ with $n\geq 3$ , where $n=m-1=\deg\Delta_{f}$ .

Proof.

Let us show (1). By compactifying $T$ suitably, we may assume that $T$ is a smooth projective curve. We run a $K_{Y}$ -MMP: $\psi:Y\to Z$ over $T$ , so that $f_{Z}:Z\to T$ is a $\mathbb{P}^{1}$ -bundle. Let $f_{Y}:Y\to T$ be the induced morphism. By $\deg\Delta_{f}=-K_{X/T}^{2}$ (Remark 5.11, Theorem 5.15), we obtain

\deg\Delta_{f}=-K_{X/T}^{2}=-(K_{X}-f^{*}K_{T})^{2}=-(K_{Y}-f_{Y}^{*}K_{T})^{2}

=-(K_{Z}-f_{Z}^{*}K_{T})^{2}+(m-1)=m-1.

Thus (1) holds. By (1), the assertions (2) and (3) follow from Proposition 6.1 and Proposition 6.2, respectively. ∎

6.3. Description and examples

In this subsection, we study generically smooth conic bundles over $k[[t]]$ . We shall classify such conic bundles for the case when either

•

${\rm char}\,k\neq 2$ (Proposition 6.4, Proposition 6.5), or
•

${\rm char}\,k=2$ and the fibre $X_{0}$ over the closed point is reduced (Proposition 6.6).

The remaining case is when ${\rm char}\,k=2$ and the fibre $X_{0}$ is non-reduced. Although we will not give the complete classification in this case, we shall check, by exhibiting several examples, that all the canonical (RDP) singularities of type $A$ and $D$ in Artin’s list [Art77] actually appear.

6.3.1. ${\rm char}\,k\neq 2$

In Proposition 6.4 (resp. Proposition 6.5), we treat the case when the singular fibre is reduced (resp. non-reduced).

Proposition 6.4.

Let $k$ be an algebraically closed field of characteristic $\neq 2$ . Set $A:=k[[t]]$ and let $0$ be the closed point of ${\operatorname{Spec}}\,A$ . Let $f:X\to{\operatorname{Spec}}\,A$ be a conic bundle, where $X$ is canonical and $X_{0}$ is not smooth but is reduced. Then the following hold.

(1)

We have an $A$ -isomorphism

$X\simeq{\operatorname{Proj}}\,A[x,y,z]/(x^{2}+y^{2}+t^{n+1}z^{2})$

for some $n\geq 0$ .
(2)

$\deg\Delta_{f}=n+1$ .
(3)

If $n=0$ , then $X$ is regular. If $n\geq 1$ , then $X$ has the unique singularity and it is of type $A_{n}$ .

Proof.

Let us show (1). By Corollary 2.13(2), we can write

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(x^{2}+y^{2}+cz^{2})

for some $c\in A=k[[t]]$ . Then we have $c=dt^{m}$ for some $m\in\mathbb{Z}_{\geq 0}$ and $d\in k[[t]]^{\times}$ . Since $X_{0}$ is not smooth, we get $m>0$ . By Hensel’s lemma, we have $\sqrt{d}\in k[[t]]$ . Thus (1) holds.

The assertion (2) follows from Definition 3.1 and Definition 3.4. The assertion (3) holds by either direct computation, or (2) and Theorem 6.3. ∎

Proposition 6.5.

Let $k$ be an algebraically closed field of characteristic $\neq 2$ . Set $A:=k[[t]]$ and let $0$ be the closed point of ${\operatorname{Spec}}\,A$ . Let $f:X\to{\operatorname{Spec}}\,A$ be a conic bundle, where $X$ is canonical and $X_{0}$ is not reduced. Then the following hold.

(1)

We have an $A$ -isomorphism

$X\simeq{\operatorname{Proj}}\,A[x,y,z]/(x^{2}+ty^{2}+t^{n-1}z^{2})$

for some $n\geq 2$ .
(2)

$\deg\Delta_{f}=n$ .
(3)

If $n=2$ , then $X$ has exactly two singular points and both of them are of type $A_{1}$ . If $n\geq 3$ , then $X$ has the unique singular point and it is of type $D_{n}$ .

Proof.

Let us show (1). By Corollary 2.13(1), we can write

X={\operatorname{Proj}}\,A[x,y,z]/(x^{2}+by^{2}+cz^{2}+\alpha yz)

for some $b,c,\alpha\in\mathfrak{m}:=tk[[t]]$ .

We now reduce the problem to the case when $\alpha=0$ . For a suitable $m\in\mathbb{Z}_{>0}$ , we have

by^{2}+cz^{2}+\alpha yz=t^{m}(b^{\prime}y^{2}+c^{\prime}z^{2}+\alpha^{\prime}yz),

where $b^{\prime},c^{\prime},\alpha^{\prime}\in A$ and one of $b^{\prime},c^{\prime},\alpha^{\prime}$ is a unit of $A$ . If $b^{\prime}\in A^{\times}$ , then we may assume that $b^{\prime}=1$ (Hensel’s lemma) and $\alpha^{\prime}=0$ by completing a square: $y^{2}+c^{\prime}z^{2}+\alpha^{\prime}yz=(y+\frac{\alpha^{\prime}}{2}z)^{2}+(c^{\prime}-\frac{\alpha^{\prime 2}}{4})z^{2}$ . If $c^{\prime}\in A^{\times}$ , then the problem is reduced to the case when $b^{\prime}\in A^{\times}$ by switching $y$ and $z$ . If $b^{\prime}\in\mathfrak{m}$ , $c^{\prime}\in\mathfrak{m}$ , and $\alpha^{\prime}\in A^{\times}$ , then we may assume that $b^{\prime}\in A^{\times}$ by applying a linear transform $(x,y,z)\mapsto(x,y,y+z)$ . In any case, the problem is reduced to the case when $\alpha=0$ .

After possibly switching $y$ and $z$ , Hensel’s lemma enables us to write

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(x^{2}+t^{r}(y^{2}+t^{s}z^{2}))

for some $r\geq 0$ and $s\geq 0$ .

We now show that $r=1$ . Suppose $r\geq 2$ . It suffices to conclude that $X$ is not normal. Take the affine open subset defined by $z\neq 0$ :

X^{\prime}:=D_{+}(z)={\operatorname{Spec}}\,k[[t]][x,y]/(x^{2}+t^{r}(y^{2}+t^{s})).

By $r\geq 2$ , the line defined by $x=t=0$ is contained in the singular locus of $X^{\prime}$ . Hence $X^{\prime}$ is not normal. This completes the proof of $r=1$ .

We then obtain

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(x^{2}+ty^{2}+t^{s+1}z^{2})

with $s\geq 0$ . Thus (1) holds.

The assertion (2) follows from Definition 3.1 and Definition 3.4. The assertion (3) holds by either direct computation, or (2) and Theorem 6.3. ∎

6.3.2. Reduced-fibre case with ${\rm char}\,k=2$

Proposition 6.6.

Let $k$ be an algebraically closed field of characteristic two. Set $A:=k[[t]]$ and let $0$ be the closed point of ${\operatorname{Spec}}\,A$ . Let $f:X\to{\operatorname{Spec}}\,A$ be a conic bundle, where $X$ is canonical and $X_{0}$ is not smooth but is reduced. Then the following hold.

(1)

We have an $A$ -isomorphism

$X\simeq{\operatorname{Proj}}\,A[x,y,z]/(xy+t^{n+1}z^{2})$

for some $n\geq 0$ .
(2)

$\deg\Delta_{f}=n+1$ .
(3)

If $n=0$ , then $X$ is regular. If $n\geq 1$ , then $X$ has the unique singularity and it is of type $A_{n}$ .

Proof.

Let us show (1). By Proposition 2.14(2), we can write

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(xy+cz^{2})

for some $c\in\mathfrak{m}:=tk[[t]]$ . Then we can write $c=c^{\prime}t^{m}$ for some $c^{\prime}\in A^{\times}$ and $m\in\mathbb{Z}_{\geq 0}$ . By the following equality of ideals of $A[x,y,z]$

(xy+cz^{2})=(xy+c^{\prime}t^{m}z^{2})=((c^{\prime-1}x)y+t^{m}z^{2}),

we may assume that $c=t^{m}$ . Since $X_{0}$ is not smooth, we get $m>0$ . Thus (1) holds.

The assertion (2) follows from Definition 3.1 and Definition 3.4. The assertion (3) holds by either direct computation, or (2) and Theorem 6.3. ∎

6.3.3. Non-reduced-fibre case with ${\rm char}\,k=2$

Lemma 6.7.

Let $k$ be an algebraically closed field of characteristic two. Set $A:=k[[t]]$ and let $0$ be the closed point of ${\operatorname{Spec}}\,A$ . Let $f:X\to{\operatorname{Spec}}\,A$ be a conic bundle, where $X$ is canonical and $X_{0}$ is not reduced. Then an $A$ -isomorphism

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(ax^{2}+by^{2}+cz^{2}+t^{n}xy)

holds for some $a,b,c\in A$ and $n\geq 1$ .

Proof.

We have

X={\operatorname{Proj}}\,A[x,y,z]/(ax^{2}+by^{2}+cz^{2}+\alpha yz+\beta zx+\gamma xy)

for some $a,b,c\in A=k[[t]]$ and $\alpha,\beta,\gamma\in\mathfrak{m}=tk[[t]]$ . We may assume that $v_{t}(\alpha)\geq v_{t}(\beta)\geq v_{t}(\gamma)=:n$ , where $v_{t}:k[[t]]\to\mathbb{Z}\cup\{\infty\}$ denotes the discrete valuation with $v_{t}(t)=1$ . Then we can write

\alpha=t^{n}\alpha^{\prime},\quad\beta=t^{n}\beta^{\prime},\quad\gamma=t^{n}\gamma^{\prime}

for some $\alpha^{\prime},\beta^{\prime}\in A$ and $\gamma^{\prime}\in A^{\times}$ . Taking the multiple with $\gamma^{\prime-1}$ , we get

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(a^{\prime}x^{2}+b^{\prime}y^{2}+c^{\prime}z^{2}+t^{n}(\alpha^{\prime}yz+\beta^{\prime}zx+xy))

for some $a^{\prime},b^{\prime},c^{\prime}\in A=k[[t]]$ . It holds that

\displaystyle\alpha^{\prime}yz+\beta^{\prime}zx+xy=(x+\alpha^{\prime}z)(y+\beta^{\prime}z)-\alpha^{\prime}\beta^{\prime}z^{2}.

Applying the coordinate change $(x+\alpha^{\prime}z,y+\beta^{\prime}z,z)\mapsto(x,y,z)$ , we obtain

X\simeq{\operatorname{Proj}}\,A[x,y,z]/(a^{\prime\prime}x^{2}+b^{\prime\prime}y^{2}+c^{\prime\prime}z^{2}+t^{n}xy)

for some $a^{\prime\prime},b^{\prime\prime},c^{\prime\prime}\in A$ and $n\in\mathbb{Z}_{\geq 0}$ . Since $X_{0}$ is not reduced, we obtain $n>0$ . ∎

Notation 6.8.

Let $k$ be an algebraically closed field of characteristic two. Set $A:=k[[t]]$ , $\mathfrak{m}:=tk[[t]]$ , and $\kappa:=A/\mathfrak{m}(\simeq k)$ . Let $0$ be the closed point of ${\operatorname{Spec}}\,A$ . Let

f:X={\operatorname{Proj}}\,A[x,y,z]/(Q)\to{\operatorname{Spec}}\,A

be a conic bundle, where $X$ is canonical and $X_{0}$ is not reduced. Assume that

Q=ax^{2}+by^{2}+cz^{2}+t^{n}xy

for some $a,b,c\in A$ and $n\geq 1$ . We often write $a(t):=a,b(t):=b,c(t):=c$ . For $a(t)=a_{0}+a_{1}t+a_{2}t^{2}+\cdots$ , it holds that $a(0)=a_{0},a^{\prime}(t)=a_{1}+2a_{2}t+3a_{3}t^{2}+\cdots$ , and $a^{\prime}(0)=a_{1}$ .

Lemma 6.9.

We use Notation 6.8. Assume $n\geq 2$ . Then the following hold.

(1)

The singular locus of $X$ is given by the intersection of the following two lines on $\mathbb{P}^{2}_{\kappa}$ :

$\sqrt{a(0)}x+\sqrt{b(0)}y+\sqrt{c(0)}z=\sqrt{a^{\prime}(0)}x+\sqrt{b^{\prime}(0)}y+\sqrt{c^{\prime}(0)}z=0.$

In particular, these lines are distinct and $X$ has the unique singular point.
(2)

We may assume that the singular point is either $[0:1:0]\in\mathbb{P}^{2}_{\kappa}$ or $[0:0:1]\in\mathbb{P}^{2}_{\kappa}$ . More precisely, there exist $\widetilde{a},\widetilde{b},\widetilde{c}\in A$ such that an $A$ -isomorphism

$X\simeq{\operatorname{Proj}}\,A[x,y,z]/(\widetilde{a}x^{2}+\widetilde{b}y^{2}+\widetilde{c}z^{2}+t^{n}xy)$

holds and the singular point of ${\operatorname{Proj}}\,A[x,y,z]/(\widetilde{a}x^{2}+\widetilde{b}y^{2}+\widetilde{c}z^{2}+t^{n}xy)$ , which is unique by (1), is either $[0:1:0]\in\mathbb{P}^{2}_{\kappa}$ or $[0:0:1]\in\mathbb{P}^{2}_{\kappa}$ .

Proof.

Let us show (1). Let $[x_{0}:y_{0}:z_{0}]\in\mathbb{P}^{2}_{\kappa}$ be a singular point of $X$ . By Proposition 4.6, this singular point $(t,[x:y:z])=(0,[x_{0}:y_{0}:z_{0}])$ is a solution of

Q=\partial_{t}Q=\partial_{x}Q=\partial_{y}Q=\partial_{z}Q=0.

By $t=0$ , the following equalities automatically hold:

\partial_{x}Q=\partial_{y}Q=\partial_{z}Q=0.

By $n\geq 2$ , the remaining conditions $Q=\partial_{t}Q=0$ can be written as

a(0)x^{2}+b(0)y^{2}+c(0)z^{2}=a^{\prime}(0)x^{2}+b^{\prime}(0)y^{2}+c^{\prime}(0)z^{2}=0.

Since both are double lines, the singular locus is either a line or a point. If the singular locus is a line, then $X$ would not be normal. Thus (1) holds.

Let us show (2). Note that the unique singular point can be written as $[\alpha:\beta:\gamma]\in\mathbb{P}^{2}_{\kappa}$ . If $[\alpha:\beta:\gamma]=[0:0:1]$ , then there is nothing to show. By symmetry, we may assume that $\beta\neq 0$ , and hence $\beta=1$ . Applying $[x:y:z]\mapsto[x+\alpha y:y:z+\gamma y]$ , we may assume that $\alpha=\gamma=0$ . Thus (2) holds. ∎

Proposition 6.10.

We use Notation 6.8. Assume that $n\geq 2$ and $[0:1:0]$ is the unique singularity of $X$ . Then one and only one of (I) and (II) holds.

(I)
1. (1)
  
  We have an $A$ -isomorphism
  
  $X\simeq{\operatorname{Proj}}\,A[x,y,z]/(tx^{2}+z^{2}+t^{n}xy)$
  
  for some $n>0$ .
2. (2)
  
  $\deg\Delta_{f}=2n$ .
3. (3)
  - •
    
    If $n=1$ , then $X$ has exactly two singular points and both of them are of type $A_{1}$ .
  - •
    
    If $n\geq 2$ , then $X$ has the unique singular point and it is of type $D^{0}_{2n}$ in the sense of [Art77, Page 16].
(II)
1. (1)
  
  We have an $A$ -isomorphism
  
  $X\simeq{\operatorname{Proj}}\,A[x,y,z]/(x^{2}+tz^{2}+t^{n}xy)$
  
  for some $n>0$ .
2. (2)
  
  $\deg\Delta_{f}=2n+1$ .
3. (3)
  
  $X$ has the unique singular point and it is of type $D^{0}_{2n+1}$ in the sense of [Art77, Page 16].

Proof.

Let us show (1). For $\gamma:=t^{n}$ , we have

X={\operatorname{Proj}}\,A[x,y,z]/(ax^{2}+by^{2}+cz^{2}+\gamma xy).

Note that the condition $\gamma=t^{n}$ is not stable under the following argument. Since $[0:1:0]$ is a singular point of $X$ , we obtain $b(0)=b^{\prime}(0)=0$ (Lemma 6.9(1)). Recall that the singular point $[0:1:0]$ coincides with the solution of the following equation (Lemma 6.9):

(6.10.1)

\sqrt{a(0)}x+\sqrt{c(0)}z=\sqrt{a^{\prime}(0)}x+\sqrt{c^{\prime}(0)}z=0.

In particular,

{\rm(I)}\,\,c(0)\neq 0\qquad\qquad\text{or}\qquad\qquad{\rm(II)}\,\,a(0)\neq 0.

In what follows, we only treat (I), since the proofs are very similar.

Assume (I), i.e., $c(0)\neq 0$ . We get $c\in A^{\times}$ . Taking the multiplication with $c^{-1}$ , the defining equation of $X$ becomes

ax^{2}+by^{2}+z^{2}+\gamma xy=0.

By the fact that $[0:1:0]$ is the unique solution of (6.10.1), we obtain $a^{\prime}(0)\neq 0$ . Applying $z\mapsto z+\sqrt{a(0)}x$ , we may assume that $a(0)=0$ . Hence we can write

a=a(t)=a_{1}t+a_{2}t^{2}+\cdots\in k[[t]],

where $a_{1},a_{2},...\in k$ and $a_{1}\neq 0$ .

We now erase the term $by^{2}$ . Let $b_{m}t^{m}$ be the leading term of $b$ , i.e., $b=b_{m}t^{m}+b_{m+1}t^{m+1}+\cdots\in k[[t]]$ with $b_{m},b_{m+1},...\in k$ and $b_{m}\neq 0$ . If $m=2\ell$ for some $\ell\in\mathbb{Z}_{\geq 0}$ (i.e., $m$ is even), then we can erase $b_{2\ell}t^{2\ell}$ by applying $z\mapsto z+\lambda y$ for suitable $\lambda\in k$ . Hence we may assume that $m=2\ell+1$ for some $\ell\in\mathbb{Z}_{\geq 0}$ (i.e., $m$ is odd). We can write

\gamma=\gamma_{s}t^{s}+\gamma_{s+1}t^{s+1}+\cdots

for some $\gamma_{s},\gamma_{s+1},...\in k$ with $\gamma_{s}\neq 0$ . Since $X_{0}$ is non-reduced, we have $s\geq 1$ . We treat the following two cases separately.

(i)

$2\ell+1\leq s+\ell$ .
(ii)

$2\ell+1>s+\ell$ .

Assume (i). In this case, we apply the $A$ -linear transform $(x,y,z)\mapsto(x+\mu t^{\ell}y,y,z)$ for some $\mu\in k$ , so that the defining polynomial of $X$ becomes

a(x+\mu t^{\ell}y)^{2}+by^{2}+z^{2}+\gamma(x+\mu t^{\ell}y)y=ax^{2}+(a\mu^{2}t^{2\ell}+b+\gamma\mu t^{\ell})y^{2}+z^{2}+\gamma xy.

If $\mu\neq 0$ , then the leading term of $\gamma\mu t^{\ell}$ is equal to $\gamma_{s}\mu t^{\ell+s}$ , which is of degree $\geq 2\ell+1$ by the assumption $2\ell+1\leq s+\ell$ . Therefore, the leading term of $a\mu^{2}t^{2\ell}+b+\gamma\mu t^{\ell}$ is of degree $\geq 2\ell+1$ . It is enough to find $\mu\in k$ that makes this inequality strict. The coefficient of $t^{2\ell+1}$ in $a\mu^{2}t^{2\ell}+b+\gamma\mu t^{\ell}\in k[[t]]$ is equal to

a_{1}\mu^{2}+b_{2\ell+1}+\gamma_{\ell+1}\mu,

where we set $\gamma_{\ell+1}:=0$ when $\ell+1<s$ . By $a_{1}\neq 0$ , there is a solution $\mu\in k$ of the equation $a_{1}\mu^{2}+b_{2\ell+1}+\gamma_{\ell+1}\mu=0$ . This completes the case when (i). Hence we may assume (ii). In this case, we apply the $A$ -linear transform $(x,y,z)\mapsto(x+\nu t^{2\ell+1-s}y,y,z)$ for some $\nu\in k$ , so that the defining polynomial of $X$ becomes

a(x+\nu t^{2\ell+1-s}y)^{2}+by^{2}+z^{2}+\gamma(x+\nu t^{2\ell+1-s}y)y

=ax^{2}+(a\nu^{2}t^{4\ell+2-2s}+b+\gamma\nu t^{2\ell+1-s})y^{2}+z^{2}+\gamma xy.

The leading term of $a\nu^{2}t^{4\ell+2-2s}$ is of degree $1+4\ell+2-2s$ , which is larger than $2\ell+1$ by the assumption $2\ell+1>s+\ell$ . Hence the coefficient of $t^{2\ell+1}$ in $a\nu^{2}t^{4\ell+2-2s}+b+\gamma\nu t^{2\ell+1-s}\in k[[t]]$ is equal to $b_{2\ell+1}+\gamma_{s}\nu$ . Hence it is enough to set $\nu:=\gamma_{s}^{-1}b_{2\ell+1}$ . Therefore, we may assume that $b=0$ .

Then the defining equation of $X$ becomes

ax^{2}+z^{2}+\gamma xy=0.

By using $z^{2}$ , we may assume that $a$ only has odd terms, i.e., $a=a_{1}t+a_{3}t^{3}+a_{5}t^{5}+\cdots\in k[[t]]$ . We can write

ax^{2}=(a_{1}t+a_{3}t^{3}+a_{5}t^{5}+\cdots)x^{2}=t((\sqrt{a_{1}}+\sqrt{a_{3}}t+\sqrt{a_{5}}t^{2}+\cdots)x)^{2},

and hence we may assume that $a=t$ . Finally, we can assume that $\gamma=t^{n}$ after replacining $y$ with $uy$ for some unit $u\in A^{\times}$ . Since $X_{0}$ is not reduced, we get $n>0$ . Thus (1) holds.

The assertion (2) follows from Definition 3.1 and Definition 3.4. The assertion (3) holds by Theorem 6.3 and [Art77, page 16]. ∎

Example 6.11.

Let $k$ be an algebraically closed field of characteristic two. Set $A:=k[[t]]$ and let $0\in{\operatorname{Spec}}\,A$ be the closed point. Fix $r\in\mathbb{Z}_{>0}$ and $s\in\mathbb{Z}_{>0}$ .

(I)
Consider a conic bundle:

$f:X:={\operatorname{Proj}}\,A[x,y,z]/(x^{2}+ty^{2}+t^{2r}z^{2}+t^{s}xy)\to{\operatorname{Spec}}\,A.$

Then the following hold.
1. (1)
  
  $\deg\Delta_{f}=2r+2s$ .
2. (2)
  
  $[0:0:1]\in X_{0}$ is the unique singularity of $X$ and it is of type $D_{2r+2s}^{r}$ in the sense of [Art77, Page 16].
(II)
Consider a conic bundle:

$f:X:={\operatorname{Proj}}\,A[x,y,z]/(x^{2}+ty^{2}+t^{2r+1}z^{2}+t^{s}xy)\to{\operatorname{Spec}}\,A.$

Then the following hold.
1. (1)
  
  $\deg\Delta_{f}=2r+2s+1$ .
2. (2)
  
  $[0:0:1]\in X_{0}$ is the unique singularity of $X$ and it is of type $D_{2r+2s+1}^{r}$ in the sense of [Art77, Page 16].

Proof.

We only prove (I). The assertion (1) follows from Definition 3.1 and Definition 3.4. Let us show (2). We now prove that $[0:0:1]\in X_{0}$ is the unique singularity of $X$ . Take a singular point $[x_{0}:y_{0}:z_{0}]\times t_{0}\in\mathbb{P}^{2}_{x,y,z}\times\mathbb{A}^{1}_{t}$ of $\widetilde{X}:=\{x^{2}+ty^{2}+t^{2r}z^{2}+t^{s}xy=0\}\subset\mathbb{P}^{2}_{x,y,z}\times\mathbb{A}^{1}_{t}$ . By the Jacobian criterion (Proposition 4.6), we obtain the following equation:

x_{0}^{2}+t_{0}y_{0}^{2}+t_{0}^{2r}z_{0}^{2}+t_{0}^{s}x_{0}y_{0}=0,

t^{s}_{0}y_{0}=0,\qquad t_{0}^{s}x_{0}=0,\qquad y_{0}^{2}+st_{0}^{s-1}x_{0}y_{0}=0.

If $t_{0}\neq 0$ , then we would get $x_{0}=y_{0}=0$ , which leads to a contradiction $z_{0}=0$ . Hence $t_{0}=0$ . Then we get $x_{0}=y_{0}=0$ . Thus $[0:0:1]\times\{0\}\in\mathbb{P}^{2}_{x,y,z}\times\mathbb{A}^{1}_{t}$ is the unique singularity of $\widetilde{X}:=\{x^{2}+ty^{2}+t^{2r}z^{2}+t^{s}xy=0\}$ . Thus $[0:0:1]\in X_{0}$ is the unique singularity of $X$ .

The singularity is the origin of

x^{2}+ty^{2}+t^{2r}+t^{s}xy\in k[[t]][x,y].

Applying the $k[[t]]$ -linear transform $(x,y)\mapsto(x+t^{r},y)$ , the equation becomes

x^{2}+ty^{2}+t^{s}xy+t^{r+s}y\in k[[t]][x,y].

This is of type $D_{2r+2s}^{r}$ in the sense of [Art77, Page 16]. Thus (2) holds. ∎

7. Special phenomena in characteristic two

Although $\Delta_{f}$ can be non-reduced under our definition (e.g., Theorem 6.3), $\Delta_{f}$ was classically defined as its reduced structure $(\Delta_{f})_{{\operatorname{red}}}$ [Bea77], [MM83]. In this section, we shall observe that both $\Delta_{f}$ and $(\Delta_{f})_{{\operatorname{red}}}$ can behave worse in characteristic two than characteristic $\neq 2$ (Subsection 7.2). We start by establishing some results which hold in characteristic $\neq 2$ (Subsection 7.1).

7.1. Results in characteristic $\neq 2$

Proposition 7.1.

Let $f:X\to T$ be a conic bundle of noetherian regular $\mathbb{Z}[\frac{1}{2}]$ -schemes. Assume that $\dim T\leq 1$ . Then $\Delta_{f}$ is regular and any fibre of $f$ is geometrically reduced.

Proof.

If $\dim T=0$ , then there is nothing to show. In what follows, we assume that $\dim T=1$ and $T$ is an integral scheme. Note that $f$ is generically smooth (Lemma 2.17). By Proposition 4.2, it is enough to show that any fibre of $f$ is geometrically reduced. Suppose that there exists a point $0\in T$ such that $X_{0}$ is not geometrically reduced. Let us derive a contradiction. Taking the strict henselisation of $\mathcal{O}_{T,0}$ , we may assume that $T={\operatorname{Spec}}\,A$ , where $(A,\mathfrak{m},\kappa)$ is a strictly henselian local ring with $\dim A=1$ . In particular, $A$ is a discrete valuation ring. Let $t$ be a uniformiser, i.e., $\mathfrak{m}=tA$ . By Corollary 2.13, we can write

X={\operatorname{Proj}}\,A[x,y,z]/(Q),\qquad Q=x^{2}+by^{2}+cz^{2}+\alpha yz,

for some $b,c,\alpha\in\mathfrak{m}$ . Applying a suitable linear transform, we may assume that $b\neq 0$ (cf. the proof of Proposition 6.5(1)). We can write $b=t^{\ell}\widetilde{b}$ for some $\ell\in\mathbb{Z}_{>0}$ and $\widetilde{b}\in A^{\times}$ . By Hensel’s lemma, we may assume $\widetilde{b}=1$ :

Q=x^{2}+t^{\ell}y^{2}+cz^{2}+\alpha yz.

Since the open subset $D_{+}(y)=\{x^{2}+t^{\ell}+cz^{2}+\alpha z=0\}$ of $X$ is regular, we obtain $\ell=1$ . We have $\alpha=t\widetilde{\alpha}$ for some $\widetilde{\alpha}\in A$ . By completing a square:

t^{\ell}y^{2}+\alpha yz=ty^{2}+t\widetilde{\alpha}yz=t\left(y+\frac{\widetilde{\alpha}}{2}z\right)^{2}-\frac{t\widetilde{\alpha}^{2}}{4}z^{2},

we may assume that $\alpha=0$ , i.e.,

Q=x^{2}+ty^{2}+cz^{2}.

By the same argument as above, we may assume that $c=t$ :

Q=x^{2}+ty^{2}+tz^{2}.

Then $X$ is not regular at the closed point $[0:1:\sqrt{-1}]\in\mathbb{P}^{2}_{\kappa}$ over $0\in T$ , which is absurd. ∎

Proposition 7.2.

Let $f:X\to T$ be a conic bundle of noetherian regular $\mathbb{Z}[\frac{1}{2}]$ -schemes. Then the following hold.

(1)

$\Delta_{f}$ is reduced.
(2)

If $T$ is a smooth surface over an algebraically closed field $k$ of characteristic $\neq 2$ (in particular, $X$ is a smooth threefold over $k$ ), then $\Delta_{f}$ is normal crossing.

Proof.

The assertion (1) holds by applying Proposition 7.1 after taking the base change ${\operatorname{Spec}}\,\mathcal{O}_{T,\xi}\to T$ for a generic point $\xi$ of $\Delta_{f}$ . The assertion (2) holds by (1) and the proof of [Bea77, Ch. I, Proposition 1.2(iii)]. ∎

7.2. Examples in characteristic two

The following example shows that singularities of $\Delta_{f}$ and $(\Delta_{f})_{{\operatorname{red}}}$ can be arbitrarily bad even if $X$ is a smooth threefold and $T$ is a smooth surface (cf. Proposition 7.2(2)).

Example 7.3.

We work over an algebraically closed field $k$ of characteristic two. Set

T:=\mathbb{A}^{2}_{u,v}={\operatorname{Spec}}\,k[u,v]\qquad\text{and}\qquad\mathbb{P}^{2}_{x,y,z}:={\rm Proj}\,k[x,y,z].

Fix $\varphi(u,v)\in k[u,v]\setminus\{0\}$ . Set

X:=\{x^{2}+uy^{2}+vz^{2}+\varphi(u,v)^{2}yz\}\subset\mathbb{A}^{2}_{u,v}\times\mathbb{P}^{2}_{x,y,z}=T\times\mathbb{P}^{2}_{x,y,z}.

Then the induced morphism $f:X\to T$ is a conic bundle. The discriminant scheme $\Delta_{f}\subset T$ of $f$ is given as follows (Definition 3.1 and Definition 3.4):

\Delta_{f}=\{\varphi(u,v)^{4}=0\}\subset T=\mathbb{A}^{2}_{u,v}.

In particular, the conic bundle $f:X\to T$ is generically smooth by $\varphi(u,v)\neq 0$ .

Let us show, by using a Jacobian criterion (Proposition 4.6), that $X$ is smooth over $k$ . Suppose that a closed point $(u_{0},v_{0})\times[x_{0}:y_{0}:z_{0}]\in\mathbb{A}^{2}_{u,v}\times\mathbb{P}^{2}_{x,y,z}$ is a solution of

Q=\partial_{u}Q=\partial_{v}Q=\partial_{x}Q=\partial_{y}Q=\partial_{z}Q=0,

\text{where}\qquad Q:=x^{2}+uy^{2}+vz^{2}+\varphi(u,v)^{2}yz.

By $\partial_{u}Q=0$ and $\partial_{v}Q=0$ , we have $y_{0}=z_{0}=0$ . Then $[x_{0}:y_{0}:z_{0}]=[1:0:0]$ . However, this is not a solution of $Q=0$ , which is absurd. Hence $X$ is smooth. This conic bundle $f:X\to T$ satisfies the following properties.

(1)

For any point $t\in\Delta_{f}$ , its fibre $X_{t}$ is not geometrically reduced.
(2)
The discriminant divisor $\Delta_{f}\subset T$ is set-theoretically given by $\{\varphi(u,v)=0\}$ In particular, the following phenomena happen in characteristic two, each of which does not occur in any other characteristic.
- •
  
  Assume that $\varphi(u,v)=u$ . In this case, $(\Delta_{f})_{{\operatorname{red}}}$ is a smooth curve, and $X_{t}$ is non-reduced for any $t\in(\Delta_{f})_{{\operatorname{red}}}$ (cf. Theorem 4.4, Proposition 7.2).
- •
  
  Assume that $\varphi(u,v)=uv(u+v)$ . In this case, $(\Delta_{f})_{{\operatorname{red}}}$ is not normal crossing (cf. Proposition 7.2).

As $\varphi(u,v)$ is chosen to be an arbitrary nonzero element of $k[u,v]$ , the singularities of $\Delta_{f}$ and $(\Delta_{f})_{{\operatorname{red}}}$ can be arbitrarily bad.

Example 7.4.

We use the same notation as in Example 7.3. Set $\varphi(u,v):=u$ . We have

X=\{x^{2}+uy^{2}+vz^{2}+u^{2}yz\}\subset T\times\mathbb{P}^{2}_{x,y,z}.

Take the localisation $T^{\prime}$ of $T={\operatorname{Spec}}\,k[u,v]$ at the generic point of $\{u=0\}$ , i.e.,

T^{\prime}:={\operatorname{Spec}}\,k[u,v]_{(u)}={\operatorname{Spec}}\,k(v)[u]_{(u)}.

Then $k(v)[u]_{(u)}$ is a discrete valuation ring. Take the base change $X^{\prime}:=X\times_{T}T^{\prime}$ and let $f^{\prime}:X^{\prime}\to T^{\prime}$ be the induced conic bundle. Then the following hold.

(1)

$X^{\prime}$ and $T^{\prime}$ are regular.
(2)

$\dim T^{\prime}=1$ .
(3)

$f^{\prime}:X^{\prime}\to T^{\prime}$ is generically smooth.
(4)

$\Delta_{f^{\prime}}={\operatorname{Spec}}\,(k(v)[u]_{(u)}/(u^{4}))$ is not reduced.

In particular, Proposition 7.1 does not hold in characteristic two even if we impose the generically smooth assumption.

Example 7.5 (Fano threefolds with non-reduced discriminants).

We work over an algebraically closed field $k$ of characteristic two. Set

X:=\{sx^{2}+ty^{2}+uz^{2}+syz\}\subset\mathbb{P}^{2}_{x,y,z}\times\mathbb{P}^{2}_{s,t,u}.

and let $f:X\to T:=\mathbb{P}^{2}_{s,t,u}$ be the induced morphism, which is a conic bundle. By Proposition 4.6, we can check that $X$ is smooth over $k$ . It follows from $X\in|\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(2,1)|$ that $X$ is a smooth Fano threefold. We have

\Delta_{f}={\operatorname{Proj}}\,k[s,t,u]/(s^{3}),

which is non-reduced. Furthermore, $(\Delta_{f})_{{\operatorname{red}}}$ is a smooth rational curve.

Remark 7.6 (Failure of Bertini).

We use the same notation as in Example 7.5. For any smooth curve $T^{\prime}$ on $T=\mathbb{P}^{2}_{s,t,u}$ , its inverse image $X^{\prime}:=f^{-1}(T^{\prime})$ is not smooth. Indeed, if $X^{\prime}$ is smooth, then the resulting conic bundle $f^{\prime}:X^{\prime}\to T^{\prime}$ is a generically smooth conic bundle from a smooth surface $X^{\prime}$ to a smooth curve $T^{\prime}$ , which implies that $\Delta_{f^{\prime}}$ is reduced (Proposition 4.2, Lemma 5.14). However, this is absurd, because $\Delta_{f}$ is non-reduced and hence so is $\Delta_{f^{\prime}}$ (Remark 3.5).

Note that the original proof of the Mori-Mukai formula (Theorem 1.1) in characteristic zero depends on the Bertini theorem for base point free divisors [MM83, Proposition 6.2]. This example shows that the same argument does not work in characteristic two.

Remark 7.7 (Wild conic bundles).

We work over an algebraically closed field $k$ of characteristic two. If $f:X\to T$ is a wild conic bundle, then it is hard from its definition (Definition 5.9) to know what is $\Delta_{f}^{{\operatorname{bdl}}}$ . When $f:X\to T$ is a wild conic bundle from a smooth projective threefold $X$ to a smooth projective surface $T$ , the following holds:

\Delta^{{\operatorname{bdl}}}_{f}\overset{{\rm(i)}}{\equiv}-f_{*}(K_{X/T}^{2})\overset{{\rm(ii)}}{\equiv}-K_{T},

where (i) follows from Theorem 5.15 and (ii) holds by [MS03, Corollary 4].

Discriminant divisors for conic bundles

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1 (Remark 5.11, Theorem 5.15).

1.1. Overview of proofs and contents

Remark 1.2.

2. Preliminaries

2.1. Notation

2.1.1. Singularities of minimal model program

2.1.2. Linear algebra over local rings

Lemma 2.1.

Proof.

2.2. Definitions of conic bundles

2.2.1. Conic bundles

Definition 2.2.

Definition 2.3.

Remark 2.4.

Lemma 2.5.

Proof.

Remark 2.6.

Proposition 2.7 (cf. [Bea77]*Proposition 1.2).

Proof.

2.2.2. Conics over rings

Definition 2.8.

Proposition 2.9.

Proof.

2.3. Local description of conic bundles

Lemma 2.10.

Proof.

2.3.1. Local description in characteristic ≠2\neq 2

Proposition 2.11.

Proof.

Proposition 2.12.

Proof.

Corollary 2.13.

Proof.

2.3.2. Local description in characteristic 22

Proposition 2.14.

Proof.

Proposition 2.15.

Proof.

2.4. Generic smoothness

Definition 2.16.

Lemma 2.17.

Proof.

Example 2.18.

3. Discriminant divisors

3.1. Definition of discriminant divisors

Definition 3.1 (cf.[ABBB21]*Section 2.b).

Remark 3.2.

Proposition 3.3.

Proof.

Definition 3.4.

Remark 3.5.

3.2. Locus of non-reduced fibres

Definition 3.6.

Remark 3.7.

Lemma 3.8.

Proof.

Step 1.

Proof of Step 1.

Step 2.

Proof of Step 2.

Step 3.

Proof of Step 3.

Proposition 3.9.

Proof.

Definition 3.10.

Remark 3.11.

3.3. Singular fibres

Proposition 3.12.

Proof.

Theorem 3.13.

Proof.

Proposition 3.14.

Proof.

3.4. Case of characteristic two

3.15.

2.3.1. Local description in characteristic $\neq 2$

2.3.2. Local description in characteristic $2$