Hyperelliptically fibred surfaces with nodes

Let $D$ be a compact complex curve of genus $g_{2}\geq 0$ and $f\colon X\to D$ be a proper holomorphic map with $X$ a compact complex surface and such that a general fibre of $f$ is an elliptic curve.

Let $S$ be the set of all $s\in D$ such that $f^{-1}(s)$ is singular (it may even be a multiple fibre). For any smooth genus $1$ curve, let $j(E)\in\mathbb{C}\setminus\{0\}$ be its $j$-invariant. Recall that the $j$-invariant of an elliptic curve

is

and two smooth elliptic curves are isomorphic if and only if they have the same $j$-invariant.

For each $s\in D$ set $E_{s}\mathrel{\mathop{:}}=f^{-1}(s)$ and note that it is connected [6, p. 200–216]. We will analyse two cases separately, depending on whether there are singular fibres or not.

In this section we take a surface $X$ fibred by curves of genus $g_{1}=2$ over a curve of genus $g_{2}$, and use the results of Xiao [40]. Recall that $X$ is projective for free. We set $q(X)\mathrel{\mathop{:}}=h^{1}(\cal{O}_{X})=h^{0}(\Omega_{X}^{1})$, and $p_{g}(X)\mathrel{\mathop{:}}=h^{0}(\omega_{X})=h^{2}(\cal{O}_{X})$. Since $\chi(\cal{O}_{X})\mathrel{\mathop{:}}=1-q(X)+p_{g}(X)$, we have

and equality holds if and only $f$ has no singular fibre and all fibres are isomorphic, see [7] or [40, p. 7].

Recall that $g_{2}\leq q\leq g_{2}+2$ and that $X$ is known if $q=g_{2}+2$. Recall that $E:=f_{\ast}(\omega_{X/D})$ is a rank $2$ vector bundle on $D$ with nonnegative degree and with all rank $1$ quotients of nonnegative degree. Let $E_{1}$ a rank $1$ subsheaf of $E$ with maximal degree. The maximality of the integer $\deg(E_{1})$ means that the coherent sheaf $E/E_{1}$ has no torsion. Since $D$ is a smooth curve and $E/E_{1}$ is a rank $1$ torsion free sheaf, $E/E_{1}$ is a line bundle. Set $\epsilon:=\deg(E_{1})-\deg(E/E_{1})$. The vector bundle $E$ is said to be unstable (resp. properly semistable, resp. stable) if and only if $\epsilon>0$ (resp. $\epsilon=0$ (resp. $\epsilon<0$) (in [21, Ex. V.2.8]) the integer $e$ is our integer $-\epsilon$). Since $\deg(E)=\deg(E_{1})+\deg(E/E_{1})$ and $\epsilon=\deg(E_{1})-\deg(E_{2})$, we have $\epsilon\equiv\deg(E)\pmod{2}$. If $\epsilon>0$, i.e. if $E$ is stable, then the maximal degree rank $1$ subsheaf $E_{1}$ of $E$ is unique, because it corresponds to a unique section of the associated ruled surface over $D$ with negative self-intersection [21, Prop. V.2.21]. The $\mathbb{P}^{1}$-bundle $P$ is associated to a rank $2$ vector bundle $F$ on $D$ which is strongly related to $E$. If $\epsilon>0$ this is the Hirzebruch surface $F_{\epsilon}$ and the uniqueness of $E_{1}$ corresponds to the fact that the ruled surface $F_{\epsilon}\to\mathbb{P}^{1}$, $\epsilon>0$, has a unique section $D_{0}$ with negative self-intersection, i.e. with $D_{0}^{2}=-\epsilon$ [21, Th. V.2.17].

The following result is a summary of [40, p. 16], see eq. (9), Thm. 2.1 and the Rmq. that follows it.

With this notation, $\epsilon>0$ is equivalent to the instability of the vector bundle $F$ and in this case there is a unique section $D_{0}$ of the $\mathbb{P}^{1}$-bundle with negative self-intersection. There is also an effective divisor $R\subset P$ [40, p. 12].

Corollary 2.9 shows that the invariants of $X$ are far from extremal: for surfaces of general type the bound is $K_{X}^{2}\leq 9\chi(\cal{O}_{X})$ [6, Ch. VII, §4].

There is a very long description of surfaces with $q=g_{2}+1$ in [40, §3] and almost all genus $2$ fibrations have $q=g_{2}$. For such fibrations the upper bounds in Thm. 2.8 give

Moreover, [40, p. 19] gives a picture showing some forbidden parts on the plane $(K_{X}^{2},\chi)$. In a small range of integers $g_{2}$, $\chi:=\chi(\cal{O}_{X})$ and $\epsilon>0$ all numerical invariant are the numerical invariants of some genus $2$ fibration.

By definition, a smooth curve $C$ of genus $g\geq 2$ is hyperelliptic if there is a degree $2$ morphism $u:C\to\mathbb{P}^{1}$. Equivalently, by the universal property of the projective line, $C$ is hyperelliptic if and only if there exist a degree $2$ line bundle $L$ on $C$ and a $2$-dimensional linear space $V\subseteq H^{0}(L)$ with no base points (hence inducing the map $u$).

Around 1817, N. Abels showed that such genus $g$ curves may be understood as follows. Fix a degree $2g+2$ polynomial $f(x)\in\mathbb{C}[x]$ without multiple roots. Consider the affine curve

Its unique smooth projective completion is a hyperelliptic curve of genus $g$, the morphism $u$ being induced by the projection $(x,y)\to x$. If we view $\mathbb{C}^{2}$ as an affine chart of the plane $\mathbb{P}^{2}$, then the corresponding closure of the affine curve is singular.

A natural ambient for hyperelliptic curves is the weighted projective plane $\mathbb{P}(1,1,g+1)$. We take coordinates $x_{0},x_{1},y$ and call $h(x_{0},x_{1})$ the homogeneous degree $g+1$ polynomial such that $h(1,x_{1})=f(x)$. Giving weight $g+1$ to the variable $y$, the equation $y^{2}=h(x_{0},x_{1})$ defines our curve in $\mathbb{P}(1,1,g+1)$. Furthermore, fixing any integer $t$ such that $1\leq t\leq g$, we may take as $f(x)$ a degree $2g+2$ polynomial with $2g+2-2t$ simple roots and $t$ roots of multiplicity $2$. Then the curve $y^{2}=h(x_{0},x_{1})$ is an irreducible and nodal curve of arithmetic genus $g$ with exactly $t$ nodes.

Now we explain two other constructions containing all smooth genus $g$ hyperelliptic curves within an ambient surface (here a smooth surface) and, containing in the same linear system (so the same arithmetic genus $g$) irreducible curves with exactly $t$ nodes for all $t=1,\dots,g$.

First construction: Set $F_{0}:=\mathbb{P}^{1}\times\mathbb{P}^{1}$. We have $\mathrm{Pic}(S)\cong\mathbb{Z}^{2}$ and we may take as a generators of $\mathrm{Pic}(F_{0})$ the isomorphism classes $\cal{O}_{F_{0}}(1,0)$ and $\cal{O}_{F_{0}}(0,1)$ of the fibres of the $2$ projections $F_{0}\to\mathbb{P}^{1}$. For all $(a,b)\in\mathbb{N}^{2}$ we have $h^{0}(\cal{O}_{F_{0}}(a,b))=(a+1)(b+1)$.

Now assume $a>0$ and $b>0$. In this case $\cal{O}_{F_{0}}(a,b)$ is very ample. Hence, a general element $|\cal{O}_{F_{0}}(a,b)|$ is smooth and connected. We have $h^{0}(\cal{O}_{C})=1$ for any $C\in|\cal{O}_{F_{0}}(a,b)|$, even for the ones with multiple components.

Since $\omega_{F_{0}}\cong\cal{O}_{F_{0}}(-2,-2)$, $\omega_{C}\cong\cal{O}_{C}(a-2,b-2)$ for any $C\in|\cal{O}_{F_{0}}(a,b)|$. Thus, all $C\in|\cal{O}_{F_{0}}(a,b)|$ have arithmetic genus $1+ab-a-b$.

(a) The parameter space: Fix an integer $t$ such that $0\leq t\leq 1+ab-a-b$. Let $V(a,b,t)$ denote the set of all irreducible and nodal $C\in|\cal{O}_{F_{0}}(a,b)|$ with exactly $t$ nodes. We have

and $V(a,b,t)$ is irreducible with $\dim V(a,b,t)=\dim|\cal{O}_{F_{0}}(a,b)|-t=ab+a+b-t$, see [37, 39]. Take $a=2$ and $b=g+1$. Each $C\in|\cal{O}_{F_{0}}(2,g+1)|$ has arithmetic genus $g$. Each smooth $C$ is hyperelliptic (use either of the $2$ projections $F_{0}\to\mathbb{P}^{1}$).

Let $D$ be a smooth hyperelliptic curve of genus $g$. Fix a general $L\in\mathrm{Pic}^{g+1}(D)$. Since $g\geq 2$ and $L$ is general, $h^{0}(L)=2$ and $L$ is base point free. Thus $|L|$ induces a degree $g+1$ morphism $v:D\to\mathbb{P}^{1}$. Let $u:D\to\mathbb{P}^{1}$ be the degree $2$ map given by the hyperellipticity of $D$ and $R\in\mathrm{Pic}^{2}(D)$ the associated line bundle. Let $w=(u,v)\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ be the morphism induced by $u$ and $v$. Since $L$ is not a multiple of $R$ and $4\deg(u)=2$, $w$ is birational onto its image. The curve $\mathrm{Im}(w)\in|\cal{O}_{F_{0}}(2,g+1)|$ has arithmetic genus $g$ and the smooth genus $D$ curve as its normalization. Thus $\mathrm{Im}(w)\cong D$ and $w(D)\in|\cal{O}_{F_{0}}(2,g+1)|$.

In general, we may take any Hirzebruch surface $F_{e}$, $e\geq 0$ [21, Ch. V,§2], and as in step (b) below we may use explicit equations for the linear systems in $F_{e}$, many of which contain hyperelliptic curves. For any $0\leq t\leq p_{a}$, where $p_{a}$ is the arithmetic genus of any element of $|\cal{O}_{F_{e}}(ah+bf)|$, the existence, irreducibility, and the dimension of the family all irreducible nodal curves in a prescribed linear system $|\cal{O}_{F_{e}}(ah+bf)|$ with exactly $t$ nodes are given in [37, 39].

(b) Hyperelliptic families with irreducible fibres having 1 or 2 nodes: Fix an integer $g\geq 2$. Let $\cal{H}(g)$ be the stack of all smooth hyperelliptic curves of genus $g$. We recall that any smooth genus $2$ curve is hyperelliptic. For any $X\in\cal{H}(g)$ call $h_{X}\colon X\to\mathbb{P}^{1}$ the degree 2 morphism, usually denoted $g^{1}_{2}$, corresponding (by definition) to the hyperelliptic curve $X$.

We recall that such degree 2 morphism is unique for each hyperelliptic curve of genus $g>1$ and for $g=2$ it is the canonical map. Denote by $R$ the only spanned degree $2$ line bundle on $X$, so that $h_{X}$ is the morphism associated to the complete linear system $|R|$. Since $\deg(L)=2g+2\geq 2g+1$, the line bundle $L:=R^{\otimes(g+1)}$ is very ample and non-special. Thus $h^{0}(L)=\deg(L)+1-g$ and $|L|$ induces an embedding $f\colon X\to\mathbb{P}^{g+1}$. Fix $D_{1},D_{2},D_{3}\in|R|$ such that $D_{i}\neq D_{j}$ for all $i\neq j$. Since $f$ is an embedding and $\deg(D_{i})=2$, $f(D_{i})$ spans a line $\langle D_{i}\rangle$. The line bundle $R^{\otimes(g-1)}$ induces a degree $2$ map with as image $\mathbb{P}^{1}$ (case $g=2$) or a rational normal curve of $\mathbb{P}^{g-1}$ (case $g\geq 2$). We have $h^{0}(R^{\otimes g-2})=g-1$. We see that the lines $\langle D_{i}\rangle$ and $\langle D_{j}\rangle$ span a plane and hence they meet. The $3$ lines $\langle D_{1}\rangle\cup\langle D_{2}\rangle\cup\langle D_{3}\rangle$ span a $\mathbb{P}^{3}$, because $h^{0}(R^{\otimes g-2})=g-1=h^{0}(R^{\otimes(g+1)})-4$. We obtain that $f(X)$ is contained in a cone $T$ with vertex $o\notin f(X)$ and as a base a rational normal curve of $\mathbb{P}^{g+1}$, see [12]. Let $u\colon Y\to T$ be the minimal resolution of $T$. The surface $Y$ is isomorphic to the Hirzebruch surface $F_{g+1}$ and $h:=u^{-1}(o)\cong\mathbb{P}^{1}$. We have $\mathrm{Pic}(Y)\cong\mathbb{Z}^{2}$ with $h$ and a fibre $f$ of its ruling (i.e. the strict transform of a line of $T$ passing through $o$) as a basis over $\mathbb{Z}$, with intersection numbers

For any curve $D\subset F$ (even reducible) which is not a member of $|cf|$ for any $c>0$, there are integers $a>0$ and $b\geq a(g+1)$ such that $D\in|\cal{O}_{Y}(ah+bf)|$. The linear system $|\cal{O}_{Y}(ah+bf)|$ contains a curve $D$ such that $D\cap h=\emptyset$, i.e. such that $o\notin u(D)$ if and only if $b=a(g+1)$.

The case $a=2$ and $b=2g+2$ is the linear system corresponding to our genus $g$ hyperelliptic curves. We take an arbitrary integer $a>0$ and give an explicit parameter space for the part of the linear system $|\cal{O}_{Y}(ah+a(g+1)f)|$ not intersecting $h$.

Fix variables $x_{0},x_{1},y$ and give weight $1$ to the variables $x_{0}$ and $x_{1}$ and weight $g+1$ to the variable $y$. Let $V(g+1,a)$ denote the set of all $f\in\mathbb{C}[x_{0},x_{1},y]$ which are weighted homogeneous with total degree $a(g+1)$. The zero-locus of any $v\in V(a+1,g)$ is an element $D\in|\cal{O}_{Y}(ah+(g+1)f|$ such that $D\cap h=\emptyset$. Those belonging to the linear system $|\cal{O}_{Y}(ah+a(g+1)f)|$ such that $D\cap h\neq\emptyset$ have $h$ as a component, because the intersection number of $h+(g+1)f$ and $h$ is $0$.

Take again $a=2$. In this case we get a linear system whose smooth elements are smooth genus $g$ hyperelliptic curves. Their nodal degenerations may contain $t$ nodes, for any $0\leq t\leq g$ by [37, 39] and also the reducible nodal curves of the form $D_{1}\cup D_{2}$ with $D_{1}\cong D_{2}\cong\mathbb{P}^{1}$ and $\#(D_{1}\cap D_{2})=g+1$. It is sufficient to take as $D_{1}$ and $D_{2}$ two general elements of $|h+(g+1)f|$.

(c) Hyperelliptic families with reducible nodal fibres: For an integer $g\geq 2$, let $\overline{\cal{M}}_{g}$ denote the coarse moduli space of stable genus $g$ curves. In the boundary $\overline{\cal{M}}_{g}\setminus\cal{M}_{g}$ we get all nodal reducible curves $C\cup E$ with $C$ a smooth curve of genus $g$, $E$ an elliptic curve and $C\cap E$ a unique point. If $g=2$ we get a nodal union of $2$ elliptic curves which occurs as a limit of a family of smooth genus $2$ curves. Now assume $g\geq 3$ and assume that $C$ is hyperelliptic. Both the theory of generalised coverings [20] and that of limit linear series [14] give that $C\cup E$ is the flat limit of a family of hyperelliptic curves. See also [19, Ch. 6 C]. In $\overline{\cal{M}}_{g}$ there exists a point representing the nodal reducible curve $D_{1}\cup D_{2}$ described at the end of part (b).

We use another construction to build more examples of hyperelliptic fibrations with nodes.

For completeness we present some further details of the toric case. However, we observe that our constructions provide genus 2 unions of elliptic curves inside toric varieties only in the case just described above, which is a very particular construction on a blowing up of $\mathbb{P}^{2}$.

Fix integers $g\geq 2$ and $0\leq c\leq g$ and a general $S\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ with $\#S=c$. Now we consider the zero-dimensional scheme $Z$ defined by

Elements of $|\cal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(2,g+1)|$ containing $Z$ are exactly the elements of $|\cal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(2,g+1)|$ singular at all points of $S$. We have $h^{0}(\cal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(2,g+1))=3(g+2)$. It is well-known that $h^{0}(\cal{I}_{Z}(2,g+1))=3(g+2)-3c$ ([27]), i.e.

and that a general $D\in|\cal{I}_{Z}(2,c)|$ is an integral curve of arithmetic genus $g$, geometric genus $g-c$ and exactly $c$ ordinary nodes as singularities. If $S$ is defined over $\mathbb{R}$ we may even find $D$ defined over $\mathbb{R}$.