Bounds on the Torsion Subgroup Schemes of Néron–Severi Group Schemes

In this paper, we work over an algebraically closed base field $k$. Although $\operatorname{char}k$ is arbitrary, we are mostly interested in the case where $\operatorname{char}k>0$. The Néron–Severi group $\operatorname{NS}X$ of a smooth projective variety $X$ is the group of divisors modulo algebraic equivalence. Thus, we have an exact sequence

Néron [27, p. 145, Théorème 2] and Severi [31] proved that $\operatorname{NS}X$ is a finitely generated abelian group. Hence, its torsion subgroup $(\operatorname{NS}X)_{\mathrm{tor}}$ is a finite abelian group. Poonen, Testa and van Luijk gave an algorithm for computing $(\operatorname{NS}X)_{\mathrm{tor}}$ [29, Theorem 8.32]. The author gave an explicit upper bound on the order of $(\operatorname{NS}X)_{\mathrm{tor}}$ [21, Theorem 4.12].

As in [32, 7.2], define the Néron–Severi group scheme $\operatorname{\mathbf{NS}}X$ of $X$ by the exact sequence

If $\operatorname{char}k=0$, then $\operatorname{\mathbf{Pic}}^{0}X$ is an abelian variety, so $\operatorname{\mathbf{NS}}X$ is a disjoint union of copies of $\operatorname{Spec}k$. However, if $\operatorname{char}k=p>0$, then $\operatorname{\mathbf{Pic}}^{0}X$ might not be reduced, and Igusa gave the first example [18]. Thus, the Néron–Severi group scheme may have additional infinitesimal $p$-power torsion.

The torsion subgroup scheme $(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$ of $\operatorname{\mathbf{NS}}X$ is a finite commutative group scheme. It is a birational invariant for smooth proper varieties due to [28, p. 92, Proposition 8] and [1, Proposition 3.4]. The first main goal of this paper is to give an explicit upper bound on the order of $(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$. Let $\exp_{a}b\coloneqq a^{b}$. {restatable*}thmNNSbound Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth connected projective variety defined by homogeneous polynomials of degree $\leq d$. Then

One motivation for studying $(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$ is its relationship with fundamental groups. Recall that if $k=\mathbb{C}$, then

However, if $\operatorname{char}k=p>0$, then $\pi^{1}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$ is not determined by $(\operatorname{NS}X)_{\mathrm{tor}}$. Therefore, an upper bound on $\#(\operatorname{NS}X)_{\mathrm{tor}}$ does not give an upper bound on $\#\pi^{1}_{\mathrm{\acute{e}t}}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$. Nevertheless, $\pi^{1}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$ is isomorphic to the group of $k$-points of the Cartier dual of $(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$ [32, Proposition 69]. Hence, we give an upper bound on $\#\pi^{1}_{\mathrm{\acute{e}t}}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$ as a corollary of Section 1. As far as the author knows, this is the first explicit upper bound on $\#\pi^{1}_{\mathrm{\acute{e}t}}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$.

The second main goal of this paper is to give an upper bound on the number of generators of $(\operatorname{NS}X)_{\mathrm{tor}}$. If $\ell\neq\operatorname{char}k$, then $(\operatorname{NS}X)[\ell^{\infty}]$ is generated by at most $(\deg X-1)(\deg X-2)$ elements by [21, Corollary 6.4]. The main tool of this bound is the Lefschetz hyperplane theorem on étale fundamental groups [11, XII. Corollaire 3.5]. Similarly, we prove that the Lefschetz hyperplane theorem on $\operatorname{\mathbf{Pic}}^{\tau}X$ gives an upper bound on the number of generators of $(\operatorname{NS}X)[p^{\infty}]$. However, the author does not know whether this is true in general. Nevertheless, Langer [22, Theorem 11.3] proved that if $X$ has a smooth lifting over $W_{2}(k)$, then the Lefschetz hyperplane theorem on $\operatorname{\mathbf{Pic}}^{\tau}X$ is true. This gives a bound on the number of generators of $(\operatorname{NS}X)_{\mathrm{tor}}$.

thmNSGenLifting If $X\hookrightarrow\mathbb{P}^{r}$ is the reduction of a smooth connected projective scheme $\mathcal{X}\hookrightarrow\mathbb{P}_{W_{2}(k)}^{r}$ over $W_{2}(k)$, then $(\operatorname{NS}X)_{\mathrm{tor}}$ is generated by $(\deg X-1)(\deg X-2)$ elements.

In Section 3, we prove that $\#(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}\leq\dim_{k}\Gamma(\operatorname{\mathbf{Hilb}}_{Q}X,\mathcal{O}_{\operatorname{\mathbf{Hilb}}_{Q}X})$ for some explicit Hilbert polynomial $Q$. Section 4 bounds $\dim_{k}\Gamma(Y,\mathcal{O}_{Y})$ for an arbitrary projective scheme $Y\hookrightarrow\mathbb{P}^{r}$ defined by homogeneous polynomials of degree at most $d$. Section 5 gives an upper bound on $\#(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$. Section 6 gives an upper bound on $\#\pi_{1}^{\mathrm{\acute{e}t}}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$ and $\#{\pi^{N}_{1}}(X,x_{0})^{\mathrm{ab}}_{\mathrm{tor}}$. Section 7 discusses the Lefschetz-type theorem on $\operatorname{\mathbf{Pic}}^{\tau}X$. Section 8 proves that $(\operatorname{NS}X)_{\mathrm{tor}}$ is generated by less than or equal to $(\deg X-1)(\deg X-2)$ elements if $X$ has a smooth lifting over $W_{2}(k)$.

Given a scheme $X$ over $k$, let $\mathcal{O}_{X}$ be the structure sheaf of $X$. We sometimes denote $\Gamma(X)=\Gamma(X,\mathcal{O}_{X})$. Let $\operatorname{\mathrm{Pic}}X$ be the Picard group of $X$. If $X$ is integral and locally noetherian, then $\operatorname{\mathrm{Cl}}X$ denotes the Weil divisor class group of $X$. If $Y$ is a closed subscheme of $X$, let $\mathscr{I}_{Y\subset X}$ be the sheaf of ideal of $Y$ on $X$. If $X$ is a projective space and there is no confusion, then we may let $\mathscr{I}_{Y}=\mathscr{I}_{Y\subset X}$. Given a $k$-scheme $T$ and a $k$-algebra $A$, let $X(T)=\operatorname{Hom}(T,X)$ and $X(A)=\operatorname{Hom}(\operatorname{Spec}A,X)$.

Suppose that $X$ is a projective scheme in $\mathbb{P}^{r}$. Let $\operatorname{\mathbf{Hilb}}_{Q}X$ be the Hilbert scheme of $X$ corresponding to a Hilbert polynomial $Q$. Let $\operatorname{\mathbf{Hilb}}^{P}X$ be the Hilbert scheme of $X$ parametrizing closed subschemes $Z$ of $X$ such that $\mathscr{I}_{Z}$ has Hilbert polynomial $P$. In particular,

Now, suppose that $X$ is smooth. Let $\operatorname{\mathbf{CDiv}}_{Q}X$ be the subscheme of $\operatorname{\mathbf{Hilb}}_{Q}X$ parametrizing divisors with Hilbert polynomial $Q$. If $D\subset\mathbb{P}^{r}$ is an effective divisor on $X$, let $\operatorname{HP}_{D}$ be the Hilbert polynomial of $D$ as a subscheme.

Let $\operatorname{\mathbf{Pic}}X$ be the Picard scheme of $X$. Let $\operatorname{\mathbf{Pic}}^{0}X$ be the identity component of $\operatorname{\mathbf{Pic}}^{0}X$. Let $\operatorname{\mathbf{Pic}}^{\tau}X$ be the disjoint union of the connected components of $\operatorname{\mathbf{Pic}}X$ corresponding to $(\operatorname{NS}X)_{\mathrm{tor}}$. Let $\operatorname{\mathrm{Pic}}^{0}X=(\operatorname{\mathbf{Pic}}^{0}X)(k)$ and $\operatorname{\mathrm{Pic}}^{\tau}X=(\operatorname{\mathbf{Pic}}^{\tau}X)(k)$. Let $\operatorname{\mathbf{NS}}X=\operatorname{\mathbf{Pic}}X/(\operatorname{\mathbf{Pic}}^{0}X)_{\mathrm{red}}$, and let $\operatorname{NS}X=\operatorname{\mathrm{Pic}}X/\operatorname{\mathrm{Pic}}^{0}X$. Given $x_{0}\in X(k)$, let $\pi^{\mathrm{\acute{e}t}}_{1}(X,x_{0})$ and ${\pi^{N}_{1}}(X,x_{0})$ be the étale fundamental group and Nori’s fundamental group scheme [28] of $(X,x_{0})$, respectively. Given a vector space $V$, let $\operatorname{\mathbf{Gr}}(n,V)$ be the Grassmannian parametrizing $n$-dimensional linear subspaces of $V$.

Let $S$ be a separated $k$-scheme of finite type, and let $X$ be a $S$-scheme, with the structure map $f\colon X\rightarrow S$. Then given a point $t\in S$, let $X_{t}$ be the fiber over $t$. Let $\mathscr{F}$ be a quasi-coherent sheaf on $X$, then let $\mathscr{F}_{t}$ be the sheaf on $X_{t}$ which is the fiber of $\mathscr{F}$ over $t$. Let $g\colon T\rightarrow S$ be a morphism. Let $p_{1}\colon X\times_{S}T\rightarrow X$ and $p_{2}\colon X\times_{S}T\rightarrow T$ be the projections. Then as in as in [19, Definition 9.3.12], for an invertible sheaf $\mathscr{L}$ on $X$, let $\operatorname{\mathbf{LinSys}}_{\mathscr{L}/X/S}$ be the scheme given by

Given a set $S$ and $T$, let $S\amalg T$ be the disjoint union of $S$ and $T$. Let $\#S$ be the cardinality of $S$. Let $\mathbb{N}$ be the set of nonnegative integers, and $\mathbb{N}_{<n}$ be the set of nonnegative integers strictly less than $n$. We regard an $n$-tuple $\mathbf{a}$ on $S$ as a function $\mathbf{a}\colon\mathbb{N}_{<n}\rightarrow S$. Thus, the $i$th component of $\mathbf{a}$ is denoted by $\mathbf{a}(i)$.

Given a group $G$, let $G^{\mathrm{ab}}$ be the abelianization of $G$. Suppose that $G$ is abelian. Then let $G_{\mathrm{tor}}$, $G[n]$ and $G[p^{\infty}]$ be the torsion subgroup, $n$-torsion subgroup and $p$-power torsion subgroup of $G$, respectively. For a finite abelian group $G$, let $G^{\vee}$ equals $\operatorname{Hom}(G,\mathbb{Q}/\mathbb{Z})$.

Given a commutative group scheme $G$, let $G_{\mathrm{tor}}$, $G[n]$ and $G[p^{\infty}]$ be the torsion subgroup scheme, $n$-torsion subgroup scheme, and $p$-power torsion subgroup scheme of $G$, respectively. Suppose that $G$ is finite. Then let $\operatorname{{\#}}G$ be the order of $G$, let $G^{\mathrm{ab}}$ be the abelianization of $G$, and let $G^{\vee}$ be the Cartier dual of $G$.

Given a finite module $M$, let $\operatorname{Fitt}_{i}(M)$ be the $i$th fitting ideal of $M$ [7, Proposition 20.4]. If $M$ is a graded module, its degree $t$ part is denoted as $M_{t}$. If $M$ is a finite $k[x_{0},x_{1},\dots,x_{r}]$-module, then let $\widetilde{M}$ be the coherent sheaf on $\mathbb{P}^{r}$ associated to $M$. Given a monomial order $\preceq$ and an ideal $I\subset k[x_{0},x_{1},\dots,x_{r}]$, let $\operatorname{in}_{\preceq}(I)$ be the initial ideal of $I$ with respect to $\preceq$. We write an $m\times n$ matrix as

Let $A$ and $B$ be are commutative rings, and $f\colon A\rightarrow B$ be a homomorphism. Let $M=(a_{i,j})_{i<m,j<n}\in A^{m\times n}$. Let $\operatorname{im}M$ be the $A$-module generated by the columns of $M$, and let

Given a linear map $\mu\colon A^{m}\rightarrow A^{l}$, let $\mu^{\oplus n}(M)$ be the $l\times n$ matrix obtained by applying $\mu$ to every column vector of $M$.

Let $\imath\colon X\hookrightarrow\mathbb{P}^{r}$ be a closed embedding of a smooth connected projective variety $X$. Let $H\subset X$ be a hyperplane section of $X$. The goal of this section is to describe an integer $m$ such that

For a positive integer $m$, consider the diagram

in which the middle morphism is yet to be defined. The natural embedding $\operatorname{\mathbf{CDiv}}_{\operatorname{HP}_{mH}}X\hookrightarrow\operatorname{\mathbf{Hilb}}_{\operatorname{HP}_{mH}}X$ is open and closed [20, Theorem 1.13]. The natural quotient $\operatorname{\mathbf{Pic}}^{\tau}X\twoheadrightarrow(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$ is faithfully flat [5, VI_A, Théorème 3.2]. Suppose that there is a faithfully flat morphism $\operatorname{\mathbf{CDiv}}_{\operatorname{HP}_{mH}}X\rightarrow\operatorname{\mathbf{Pic}}^{\tau}X$ for some large $m$. Then $\Gamma((\operatorname{\mathbf{NS}}X)_{\mathrm{tor}})\hookrightarrow\Gamma(\operatorname{\mathbf{CDiv}}_{\operatorname{HP}_{mH}})$ is injective by [10, Corollaire 2.2.8], and $\Gamma(\operatorname{\mathbf{Hilb}}_{\operatorname{HP}_{mH}})\twoheadrightarrow\Gamma(\operatorname{\mathbf{CDiv}}_{\operatorname{HP}_{mH}})$ is surjective, so we get (1). Such $m$ will be obtained as a byproduct of the construction of Picard schemes. The theorem and the proof below are a modification of the proof of [19, Theorem 9.4.8].

We now need to find an integer $m$ satisfying (2). Recall the definition of Castelnuovo–Mumford regularity.

Mumford [26, p. 99] showed that if $\mathscr{F}$ is $m$-regular, then it is also $t$-regular for every $t\geq m$.

Given a homogeneous ideal $I\subset k[x_{0},\dots,x_{r}]$, Hoa gave an explicit finite upper bound on $\varphi(\operatorname{HP}_{I})$ [16, Theorem 6.4(i)]. We will describe $m$ in terms of Gotzmann numbers of Hilbert polynomials.

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

Therefore, if we bound $\dim_{k}\Gamma(Y,\mathcal{O}_{Y})$ for a general projective scheme $Y$, and explicitly describe Hilbert schemes in projective spaces, then we can bound $\operatorname{{\#}}(\operatorname{\mathbf{NS}}X)_{\mathrm{tor}}$.

Let $Y\hookrightarrow\mathbb{P}^{r}$ be a projective scheme defined by a homogeneous ideal $I$ generated by homogeneous polynomials of degree at most $d$. The aim of the section is to give an upper bound on $\dim_{k}\Gamma(Y,\mathcal{O}_{Y})$ in terms of $r$ and $d$. First, we reduce to the case where $I$ is a monomial ideal.

Hence, we will focus on the case where $I$ is a monomial ideal. Let $m$ be a monomial and $I$ be a monomial ideal of $k[x_{0},\dots,x_{r}]$. Then by abusing notation, we denote

Suppose that $I$ is generated by monomials of degree at most $d$. We want to show that $\dim_{k}\Gamma(Y,\mathcal{O}_{Y})\leq d^{r}$. By induction on $\mu(I)$, we will show a slightly stronger proposition: for every monomial $m\in k[x_{0},\dots,x_{r}]$ whose exponents are at most $d$, we have

Since $\Gamma(Y,\mathcal{O}_{Y})=\Gamma(\mathbb{P}^{r},\widetilde{(1)/I})$ and $\#M(m,I)\leq d^{r+1}$, it follows that $\dim_{k}\Gamma(Y,\mathcal{O}_{Y})\leq d^{r}$.