On monodromy in families of elliptic curves over $\mathbb C$

TL;DR

This paper investigates the monodromy groups of families of elliptic curves over complex bases, showing they are typically as large as possible, and contrasts this with hyperelliptic curves where monodromy is smaller.

Contribution

It establishes conditions under which the monodromy group of elliptic curve families equals SL(2,Z) and compares this to hyperelliptic curves where monodromy is strictly less.

Findings

Monodromy equals SL(2,Z) for non-isotrivial elliptic families with connected fibers.

Monodromy has index at most 2m in SL(2,Z) when fibers have m components.

Hyperelliptic curves of genus ≥3 have monodromy strictly less than Sp(2g,Z).

Abstract

We show that if we are given a smooth non-isotrivial family of elliptic curves over~$\mathbb C$ with a smooth base~$B$ for which the general fiber of the mapping $J\colon B\to\mathbb A^1$ (assigning $j$-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on $H^1(\cdot,\mathbb Z)$ of the fibers) coincides with $\mathrm{SL}(2,\mathbb Z)$; if the general fiber has $m\ge2$ connected components, then the monodromy group has index at most~$2m$ in $\mathrm{SL}(2,\mathbb Z)$. By contrast, in \emph{any} family of hyperelliptic curves of genus $g\ge3$, the monodromy group is strictly less than $\mathrm{Sp}(2g,\mathbb Z)$. Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.

Equations

$$j(C)=1728\cdot \frac{4p^3}{4p^3+27q^2}.$$

$$\mu :B_3(u,v,w)\to \mathrm{SL}(H^1(X_{u,v,w},\mathbb{Z}))\cong \mathrm{SL}(2,\mathbb{Z}).$$

$${\tilde{\phi }}_{\gamma }^{\ast }:H^1(X_{u,v,w},\mathbb{Z})\to H^1(X_{u,v,w},\mathbb{Z})$$

$$A=\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right),B=\left(\begin{array}{cc}1& 0\\ y& 1\end{array}\right),x,y\in \mathbb{Z}.$$

$$x=2x+x^{2}y,y=2y+x{y}^2.$$

$$\mu (A)=\left(\begin{array}{cc}1& 0\\ -1& 1\end{array}\right),\mu (B)=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right).$$

$$w^2=z^3+Pz+Q,$$

$$\mu :B_3(u,v,w)\to \mathrm{SL}(H^1(X_{b_0}),\mathbb{Z}))=\mathrm{SL}(2,\mathbb{Z})$$

$${\pi }_1(B,b_0)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{X}}_1:y^2$$

$${\mathcal{X}}_2:y^2$$

$${\mathrm{br}}_{{\mathcal{X}}_2}(\gamma )=\tau {\mathrm{br}}_{{\mathcal{X}}_1}(\gamma )\delta {\tau }^{-1},$$

$$t\mapsto (D(\gamma (t))/|D(\gamma (t)|)\cdot \gamma (t)$$

$$\Pi \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$G_2=\rho (\mathrm{Ker}\chi )\cup \chi \rho (\Pi \setminus \mathrm{Ker}\chi )=\rho (\mathrm{Ker}\chi )\cup (-I)(\rho (\Pi \setminus \mathrm{Ker}\chi ))=G_1\cup (-I)G_1,$$

$$V=\{(p,q)\in {\mathbb{A}}^2:4p^3+27q^2\ne 0\}$$

$$V^{\prime }=J_{\mathcal{B}}^{-1}({\mathbb{A}}_{0,1728}^1)=\{(p,q)\in {\mathbb{A}}^2:4p^3+27q^2\ne 0,p\ne 0,q\ne 0\}.$$

$$V^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(\lambda ,j)\mapsto \left(\frac{{\lambda }^2}{\frac{4}{27}\left(\frac{1728}{j}-1\right)},\frac{{\lambda }^3}{\frac{4}{27}\left(\frac{1728}{j}-1\right)}\right).$$

$$W\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(\mathrm{SL}(2,\mathbb{Z}):G)=\frac{(\mathrm{SL}(2,\mathbb{Z}):\Gamma (2))\cdot (\Gamma (2):G\cap \Gamma (2))}{(G:G\cap \Gamma (2))}\le \frac{12d(r-1)}{d},$$

$$(\mathrm{SL}(2,\mathbb{Z}):\mathrm{Mon}(\mathcal{X}))\le 2\cdot \sum _{s\in C}e(s),$$

$$(\mathrm{Sp}(2g,\mathbb{Z}):\mathrm{Mon}(\mathcal{X}))\ge \frac{2^{g^2}(2^{2g}-1)(2^{2(g-1)}-1)\cdot \cdots \cdot (2^2-1)}{(2g+2)!}.$$

$$(\mathrm{Sp}(2g,\mathbb{Z}):\mathrm{Mon}(\mathcal{X},\mathbb{Z}))\ge (\mathrm{Sp}(2g,\mathbb{Z}/2\mathbb{Z}):\mathrm{Mon}(\mathcal{X},\mathbb{Z}/2\mathbb{Z})),$$

$$(\mathrm{Sp}(2g,\mathbb{Z}/2\mathbb{Z}):\mathrm{Mon}(\mathcal{X},\mathbb{Z}/2\mathbb{Z}))\ge \frac{2^{g^2}(2^{2g}-1)(2^{2(g-1)}-1)\cdot \cdots \cdot (2^2-1)}{(2g+2)!}.$$

$$(\mathrm{Sp}(2g,\mathbb{Z}/2\mathbb{Z}):1)=2^{g^2}(2^{2g}-1)(2^{2(g-1)}-1)\cdot \cdots \cdot (2^2-1),$$

$$P(x)=x^{2g+2}+a_{2g}{x}^{2g}+\cdots +a_{1}x+a_0,$$

$$[A_{1},A_{2},A_{3},A_{4}]=\frac{a_{3}-a_{1}}{a_{3}-a_{2}}\bigg{/}\frac{a_{4}-a_{1}}{a_{4}-a_{2}}.$$

$${\mathcal{U}}_X=\{(x,\alpha )\in X\times (({\mathbb{P}}^n{)}^{\ast }\setminus X^{\ast }):x\in H_{\alpha }\},$$

$$V=\{\alpha \in ({\mathbb{P}}^{n-1}{)}^{\ast }:\text{p,hα is transversal to X}\},$$

Full text

Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.

The study has been funded by the Russian Academic Excellence Project ’5-100’.

Introduction

It is believed that if fibers in a family of algebraic varieties “vary enough” then the monodromy group acting on the cohomology of the fiber should be in some sense big. Quite a few results have been obtained in this direction. See for example [4] for families of elliptic curves, [8] for families of hyperelliptic curves, [2] for families of abelian varieties (see also [9] for abelian varieties in the arithmetic situation). In the cited papers cohomology means “étale cohomology with finite coefficients”. In this paper we address the question of “big monodromy” for families of elliptic curves over $\mathbb{C}$ and singular cohomology.

The main result of the paper (Proposition 4.2) asserts that if $\pi\colon\mathcal{X}\to B$ is a smooth non-isotrivial family of elliptic curves over $\mathbb{C}$ and if the general fiber of its “$J$-map” $J_{\mathcal{X}}\colon B\to\mathbb{A}^{1}$ (assigning to each point of the base the $j$-invariant of the fiber) is connected, then the monodromy group of the family $\mathcal{X}$ is the entire group $\operatorname{SL}(2,\mathbb{Z})$, and if the general fiber has $m\geq 2$ connected components, then the monodromy group of the family $\mathcal{X}$ is a subgroup of index at most $2m$ in $\operatorname{SL}(2,\mathbb{Z})$. Here, by monodromy group we mean that acting on $H^{1}(\cdot,\mathbb{Z})$ of the fiber.

An immediate consequence of this proposition is that, in any non-isotrivial family of elliptic curves, the monodromy group has finite index in $\operatorname{SL}(2,\mathbb{Z})$ (Corollary 4.5). This requires some comments.

The above assertion is similar to a well-known result about elliptic curves over number fields, viz. to Serre’s Theorem 3.2 from Chapter IV of [14]. It is possible that one can prove our Corollary 4.5 by imitating, mutatis mutandis, Serre’s proof of this theorem or even derive it from Serre’s theorem or similar arithmetical results. One merit of the approach presented in this paper is that the proofs are very simple and elementary. One should add that the similarity between arithmetic and geometric situations is not absolute. For example, Theorem 5.1 from [8] could suggest that, over $\mathbb{C}$, the monodromy group for some families of hyperelliptic curves of genus $g$ should be the entire $\operatorname{Sp}(2g,\mathbb{Z})$. However, as we show in Proposition 5.2, for any family of hyperelliptic curves of genus $g\geq 3$ over $\mathbb{C}$ the monodromy group acting on $H^{1}(\cdot,\mathbb{Z})$ of the fiber is a proper subgroup of $\operatorname{Sp}(2g,\mathbb{Z})$.

Our main result has three simple consequences, which are presented in Section 4. First, any smooth (i.e., without degenerate fibers) family of elliptic curves over a smooth base with commutative fundamental group, must be isotrivial (Proposition 4.6). Second, for non-isotrivial families we obtain an upper bound on the index of the monodromy group in $\operatorname{SL}(2,\mathbb{Z})$ (Proposition 4.7). Third, in the case of smooth elliptic surfaces we use Miranda’s results from [12] to obtain an upper bound on the index of monodromy group in terms of singular fibers (it turns out that only fibers of the types $\mathrm{I}_{n}$ and $\mathrm{I}^{*}_{N}$ count); see Proposition 4.11.

In Section 5 we prove the above mentioned result about families of hyperelliptic curves of genus $3$ or higher.

In Section 6, we derive from our main result that the hyperplane monodromy group of a smooth Del Pezzo surface (or, for Del Pezzos of degree $2$, the monodromy group acting on $H^{1}(\cdot,\mathbb{Z})$ of smooth elements of the anticanonical linear system) is the entire $\operatorname{SL}(2,\mathbb{Z})$ (Proposition 6.1). I realize that it is not the only way to obtain this result. Observe that, in view of Proposition 5.2, Proposition 6.1 cannot be extended to surfaces with hyperelliptic hyperplane sections.

Sections 1 through 3 are devoted to auxiliary material (Proposition 3.3 may be of some independent interest).

Acknowledgements

I am grateful to Yu. Burman, Andrey Levin, Sergey Rybakov, Ossip Schwarzman, and Yuri Zarhin for useful discussions.

Notation and conventions

All our algebraic varieties are defined over $\mathbb{C}$ and reduced, so they are essentially identified with their sets of closed points; the only exception is the discussion of the notion of quadratic twist in Section 3. If $X$ is an algebraic variety, then $X_{\mathrm{sm}}$ is its smooth locus and $X_{\mathrm{sing}}$ is its singular locus.

When we say “a general $X$ has property $Y$”, this always means “property $Y$ holds for a Zariski open and dense set of $X$’s”. The word “generic” is used in the scheme-theoretic sense.

If $B$ is an algebraic variety and $\pi\colon\mathcal{X}\to B$ is a proper and flat morphism such that a general fiber of $\pi$ is, say, a smooth curve of genus $1$, we will say that $\pi$ is a family of curves of genus $1$. If, in addition, the morphism $f$ is smooth, we will say that $\pi$ (or just $\mathcal{X}$ if there is no danger of confusion) is a smooth family, or a family of smooth varieties. If $\pi\colon\mathcal{X}\to B$ is a family over $B$ and $f\colon B^{\prime}\to B$ is a morphism, then by $p^{\prime}\colon f^{*}\mathcal{X}\to B^{\prime}$ we mean the pullback of $\mathcal{X}$ along $f$.

By $\pi_{1}$ of an algebraic variety over $\mathbb{C}$ we always mean fundamental group in the classical (complex) topology.

As usual, we put $\Gamma(2)=\left\{A\in\operatorname{SL}(2,\mathbb{Z})\colon A\equiv I\pmod{2}\right\}$, where $I$ is the identity matrix.

Finally, we fix some terminology and notation concerning elliptic curves.

Following Miranda [12], we distinguish between curves of genus $1$ and elliptic curves: by elliptic curve over a field $K$ we mean a smooth projective curve over $K$ of genus $1$ with a distinguished $K$-rational point.

Similarly, by a smooth family of curves of genus $1$ we will mean a smooth family $\pi\colon\mathcal{X}\to B$ such that its fibers are curves of genus $1$, and by a smooth family of elliptic curves we mean a pair $(\mathcal{X},s)$, where $\mathcal{X}\to B$ is a smooth family of curves of genus $1$ and $s\colon B\to\mathcal{X}$ is a section.

To each curve $C$ of genus $1$ over a filed $K$ one can assign its $j$-invariant $j(C)\in K$; recall that if $C$ is (the smooth projective model of) the curve defined by the Weierstrass equation $y^{2}=x^{3}+px+q$, then

[TABLE]

Two curves of genus $1$ over $\mathbb{C}$ are isomorphic if and only if their $j$-invariants are equal.

We say that a family over $B$ is isotrivial if it becomes trivial after a pullback along a generically finite morphism $B_{1}\to B$. For families of curves of genus $1$ this is equivalent to the condition that $j$-invariants of all fibers are the same.

1. Generalities on monodromy groups

Suppose that $B$ is an irreducible variety and $\pi\colon\mathcal{X}\to B$ is a family of smooth varieties.

If $b\in B_{\mathrm{sm}}$, $k\in\mathbb{N}$, and $G$ is an abelian group, then the fundamental group $\pi_{1}(B_{\mathrm{sm}},b)$ acts on $H^{k}(p^{-1}(b),G)$.

Definition 1.1**.**

The image (corresponding to this action) of $\pi_{1}(B_{\mathrm{sm}},b)$ in $\operatorname{Aut}(H^{k}(p^{-1}(b),G))$ will be called monodromy group of the family $\mathcal{X}$ at $b$ and denoted $\operatorname{Mon}(\mathcal{X},b)$ (we suppress the mention of $k$ and $G$; there will be no danger of confusion).

Since $B$ is irreducible, $B_{\mathrm{sm}}$ is path connected. Hence, if we fix once and for all the group $A=\operatorname{Aut}(H^{k}(p^{-1}(b_{0}),G))$ for some $b_{0}\in B_{\mathrm{sm}}$, then all the groups $\operatorname{Mon}(\mathcal{X},b)$ define the same conjugacy class of subgroups of $A$; this class (or, abusing the language, any subgroup belonging to this class) will be denoted by $\operatorname{Mon}(\mathcal{X})$.

In the sequel we will be working with families of smooth curves of genus $g$ (in most cases $g$ will be equal to $1$) as fibers and monodromy action on $H^{1}$ of the fiber. Since monodromy preserves the intersection form, the subgroups $\operatorname{Mon}(\mathcal{X})$, where $\mathcal{X}$ is such a family, will be defined up to an inner automorphism of the group $\operatorname{Sp}(2g,\mathbb{Z})$ ($\operatorname{SL}(2,\mathbb{Z})$ if $g=1$).

Convention 1.2**.**

If $\pi\colon\mathcal{X}\to B$ is a non-smooth family, then by $\operatorname{Mon}(\mathcal{X})$ we mean $\operatorname{Mon}(\mathcal{X}|_{U})$, where $U\subset B$ is the Zariski open subset over which $\pi$ is smooth.

Below we list some simple properties of monodromy groups.

Proposition 1.3**.**

Suppose that $B$ is an irreducible variety, $U\subset B$ is a non-empty Zariski open subset, and $\mathcal{X}$ is a smooth family over $B$. Then $\operatorname{Mon}(\mathcal{X}|_{U})=\operatorname{Mon}(\mathcal{X})$.

Proof.

The result follows from the fact that, for any $b\in U\cap B_{\mathrm{sm}}$, the natural homomorphism $\pi_{1}(U\cap B_{\mathrm{sm}},b)\to\pi_{1}(B_{\mathrm{sm}},b)$ is epimorphic (see for example [7, 0.7(B) ff.]). ∎

Proposition 1.4**.**

Suppose that $B^{\prime}$ and $B$ are smooth irreducible varieties and $\mathcal{X}$ is a smooth family over $B$. If $f\colon B^{\prime}\to B$ is a dominant morphism such that a general fiber of $f$ has $m$ connected components, then $\operatorname{Mon}(f^{*}\mathcal{X})$ is conjugate to a subgroup of $\operatorname{Mon}(\mathcal{X})$, of index at most $m$.

Corollary 1.5**.**

Suppose that $B^{\prime}$ and $B$ are smooth irreducible varieties and $\mathcal{X}$ is a smooth family over $B$. If $f\colon B^{\prime}\to B$ is a dominant morphism such that a general fiber of $f$ is connected, then $\operatorname{Mon}(f^{*}\mathcal{X})$ is conjugate to $\operatorname{Mon}(\mathcal{X})$.

Proof of Proposition 1.4.

It follows from [16, Corollary 5.1] and the algebraic version of Sard’s theorem that there exists a Zariski open non-empty $U\subset B$ such that all the fibers of $f$ over points of $U$ are smooth and the induced mapping $f^{\prime}\colon f^{-1}U\to U$ is a locally trivial bundle in the complex topology. Proposition 1.3 implies that $\operatorname{Mon}(\mathcal{X}|_{U})=\operatorname{Mon}(\mathcal{X})$. Since $f^{-1}(U)$ is (path) connected, each fiber of this bundle has $m$ connected components, and the base is locally path connected, $f^{\prime}_{*}(\pi_{1}(f^{-1}U,x))$ is a subgroup of index at most $m$ in $\pi_{1}(U,f(x))$ for any $x\in f^{-1}(U)$. This implies the proposition. ∎

2. Some remarks on 3-braids

In this section, all topological terms will refer to the classical (complex) topology.

We begin with some remarks on 3-braids (not claiming to novelty).

Let $\mathbb{C}^{(3)}$ stand for the configuration space of unordered triples of distinct points in the complex plane. It is well known that $\pi_{1}(\mathbb{C}^{(3)})\cong B_{3}$, where $B_{3}$ is braid group with $3$ strands. If $(u,v,w)\in\mathbb{C}^{(3)}$ is an unordered triple, we will write $B_{3}(u,v,w)$ instead of $\pi_{1}(\mathbb{C}^{(3)},(u,v,w))$.

For any triple $(u,v,w)\in\mathbb{C}^{(3)}$, we denote by $X_{u,v,w}$ the elliptic curve which is the smooth projective model of the curve with equation $y^{2}=(x-u)(x-v)(x-w)$. We are going to define a homomorphism

[TABLE]

To wit, it is well known that any braid $\gamma\in B_{3}(u,v,w)$ can be represented by a homeomorphism $\varphi_{\gamma}\colon\mathbb{C}\to\mathbb{C}$ such that $\varphi_{\gamma}(\{u,v,w\})=\{u,v,w\}$ and $\varphi_{\gamma}$ is identity outside a bounded set. Putting $\mathbb{P}^{1}=\mathbb{C}\cup\{\infty\}\supset\mathbb{C}$, we extend $\varphi_{\gamma}$ to a homeomorphism from $\mathbb{P}^{1}$ to itself by putting $\varphi_{\gamma}(\infty)=\infty$. If $\pi\colon X_{u,v,w}\to\mathbb{P}^{1}\supset\mathbb{C}$ is the morphism induced by the projection $(x,y)\mapsto x$, then there exists a unique homeomorphism $\tilde{\varphi}_{\gamma}\colon X_{u,v,w}\to X_{u,v,w}$ such that $\pi\circ\tilde{\varphi}_{\gamma}=\varphi_{\gamma}\circ\pi$ and $\tilde{\varphi}_{\gamma}=\mathrm{id}$ on $\pi^{-1}(\mathbb{P}^{1}\setminus K)$, where $K\subset\mathbb{C}$ is the compact set outside of which $\pi_{\gamma}=\mathrm{id}$. The automorphism

[TABLE]

does not depend on the choice of the $\phi_{\gamma}$ representing $\gamma$, and we put $\mu(\gamma)=\tilde{\varphi}_{\gamma}^{*}$.

Proposition 2.1**.**

If $\Gamma\in B_{3}(u,v,w)$ is the braid represented by the loop in $\mathbb{C}^{(3)}$ defined by the formula $t\mapsto(ue^{2\pi it},ve^{2\pi it},ve^{2\pi it})$, $t\in[0;1]$, then $\mu(\Gamma)=\left(\begin{smallmatrix}-1&0\\ 0&-1\end{smallmatrix}\right)$.

Proof.

To prove the proposition, we choose generators of $B_{3}(u,v,w)$ and a basis in $H_{1}(X_{u,v,w},\mathbb{Z})$.

To fix generators of the braid group, we choose the points $u,v,w\in\mathbb{C}$ so that they are collinear and $v$ lies between $u$ and $w$. Now let $A$ and $B$ be the braids corresponding to the following closed paths in $\mathbb{C}^{(3)}$: in the path defining $A$, the point $w$ stays where it is while $u$ and $v$ are swapped, $u$ and $v$ moving along small arcs close to the segment $[p,q]$ so that the composition of paths traveled by $u$ and $v$ defines a positively oriented simple closed curve. The braid $B$ is defined similarly, with the point $u$ staying put and the points $v$ and $w$ being exchanged; see Figure 1.

The group $B_{3}(u,v,w)$ is generated by $A$ and $B$, and these braids satisfy the relation $ABA=BAB$.

For a basis in $H_{1}(X_{u,v,w},\mathbb{Z})$ we choose the $1$-cycles $\alpha$ and $\beta$ that are obtained by lifting the closed paths $\alpha$ and $\beta$ on Fig. 2 from $\mathbb{C}$ to $X_{u,v,w}$.

Abusing the language, we will denote the action on $H_{1}(X_{u,v,w},\mathbb{Z})$ of a homeomorphism representing the braid $A$ by the same letter $A$, and similarly for $B$. Since the homeomorphisms representing $A$ and $B$ can be chosen to be identity outside the corresponding dashed ovals on Fig. 1, it is clear that $A(\alpha)=\alpha$ and $B(\beta)=\beta$. Taking into account that $A$ and $B$ preserve the intersection pairing on $H_{1}(X_{u,v,w},\mathbb{Z})$, one concludes that, in the basis $(\alpha,\beta)$, the action of $A$ and $B$ on $H_{1}$ is given by matrices of the form

[TABLE]

The relation $ABA=BAB$ implies that

[TABLE]

Equations (2) imply that either $x=y=0$ or $xy=-1$. The first case is impossible: if both $A$ and $B$ act as identity, then the entire braid group acts identically, which is absurd since its action is non-trivial (see for example [1]). So, one of the integers $x$ and $y$ is equal to $1$ and the other is equal to $-1$. Dualizing, we see that either $\mu(A)$ acts on $H^{1}(X_{u,v,w},\mathbb{Z})$ as $\left(\begin{smallmatrix}1&0\\ -1&1\end{smallmatrix}\right)$ (in the basis dual to $(\alpha,\beta)$) and $\mu(B)$ acts as $\left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)$, or vice versa.

Now it is well known that $\Gamma=(AB)^{3}$. Plugging the possible values of $A$ and $B$, one obtains the result. ∎

Remark 2.2*.*

One can show that, with the choice of signs as on figures 1 and 2, one has

[TABLE]

We do not need to be that precise.

3. Quadratic twists and monodromy

In this and the following section we will be studying monodromy groups acting on $H^{1}(\cdot,\mathbb{Z})$ of fibers in families of smooth curves of genus $1$. In such families, the monodromy group acting on $H^{1}$ of the fiber is contained in $\operatorname{SL}(2,\mathbb{Z})$.

If $\mathcal{X}\to B$ is a smooth family of elliptic curves, then the morphism $B\to\mathbb{A}^{1}$ assigning the $j$-invariant $j(p^{-1}(b))$ to a point $b\in B$, will be denoted by $J_{\mathcal{X}}$. Following Miranda [12, Lecture V], we will say that $J_{\mathcal{X}}$ is the $J$-map of the family $\mathcal{X}$ (in Kodaira’s paper [11], the morphism $J_{\mathcal{X}}$ is called analytic invariant of the family $\mathcal{X}$).

Notation 3.1**.**

If $\mathcal{X}$ is a smooth family of elliptic curves over $B$ and if $b_{0}\in B$, then the monodromy representation $\pi_{1}(B,b_{0})\to\operatorname{SL}(H^{1}(\mathcal{X}_{b_{0}},\mathbb{Z}))$ will be denoted by $\rho_{\chi}$.

Suppose now that $p\colon X\to B$ is a family of elliptic curves over a smooth and connected base. Since its fiber over the generic point of $B$ is an elliptic curve over the field of rational functions $\mathbb{C}(B)$, and since this elliptic curve can be reduced to the Weierstrass normal from, there exists a Zariski open subset $U\subset B$ such that the restriction $\mathcal{X}|_{U}$ is isomorphic to the family

[TABLE]

where $P$ and $Q$ are regular functions on $U$, the fiber over $b\in B$ being the smooth projective model of the curve defined by the equation $w^{2}=z^{3}+P(b)z+Q(b)$, and discriminant of the right-hand side of (3) does not vanish on $U$. Proposition 1.3 shows that $\operatorname{Mon}(\mathcal{X})=\operatorname{Mon}(\mathcal{X}|_{U})$, so, as far as monodromy groups are concerned, we may and will assume that $U=B$ and that the family is defined by (3) with non-vanishing discriminant.

Any such family of the form (3) defines a morphism $\operatorname{Br}_{\mathcal{X}}\colon B\to\mathbb{C}^{(3)}$ assigning to each point $b\in B$ the collection of roots of $z^{3}+P(b)z+Q(b)$. If $b_{0}\in B$ and if $\{u,v,w\}$ is the set of roots of the polynomial $z^{3}+P(b)z+Q(b)$, then the morphism $\operatorname{Br}_{\mathcal{X}}$ induces a homomorphism $\operatorname{br}_{\mathcal{X}}\colon\pi_{1}(B,b_{0})\to B_{3}(u,v,w)$. If $\mathcal{X}_{b_{0}}$ is the fiber of $\mathcal{X}$ over $b_{0}$, and if

[TABLE]

is the homomorphism defined in Section 2, then the diagram

[TABLE]

is commutative.

Suppose that $p_{1}\colon\mathcal{X}_{1}\to B$ and $p_{2}\colon\mathcal{X}_{2}\to B$ are two families of elliptic curves over a base $B$. One says that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ differ by a quadratic twist if their scheme-theoretic generic fibers (which are elliptic curves over the field of rational functions $K=\mathbb{C}(B)$) are isomorphic over a quadratic extension of $K$. It it clear that this is the case if and only if there exists a morphism of degree $2$ (not necessarily finite or étale) $f\colon U\to B$ such that $f^{*}\mathcal{X}_{1}$ and $f^{*}\mathcal{X}_{2}$ are isomorphic smooth families. If the families $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ differ by a quadratic twist, then they can be represented by Weierstrass equations

[TABLE]

where $D$ is a rational function on $B$ (see [15, Chapter X, Proposition 5.4]).

Being interested only in the monodromy groups $\operatorname{Mon}(\mathcal{X}_{1})$ and $\operatorname{Mon}(\mathcal{X}_{2})$, we can, replacing $B$ by a Zariski open subset if necessary, assume that the families $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are smooth; in particular, this implies that $D$ is a regular function on $B$ without zeroes.

Suppose that $B$ is a smooth algebraic variety, $D$ is a regular function on $B$ without zeroes, and $b_{0}\in B$ is a point. In the definition that follows we regard $B$ as a complex manifold and $D$ as a holomorphic function on $B$.

Definition 3.2**.**

In the above setting, by $\chi_{D}\colon\pi_{1}(B,b)\to\{\pm 1\}$ we denote the homomorphism defined as follows. If $B_{0}\in B$, $\gamma\in\pi_{1}(B,b_{0})$, we put $\chi_{D}(\gamma)=-1$ if the function $\sqrt{D}$ changes after the analytic continuation along a loop representing $\gamma$, and we put $\chi_{D}(\gamma)=1$ otherwise. In other words, if a loop representing $\gamma$ is of the form $t\mapsto\varphi(t)$, $t\in[0;1]$, then $\chi_{D}(\gamma)=(-1)^{k}$, where $k$ is the number of times the loop $D\circ\varphi$ winds around the origin.

We will say that $\chi_{D}$ is the quadratic character associated to $D$.

Proposition 3.3**.**

In the above setting, suppose that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are smooth families of elliptic curves that differ by a quadratic twist as in (4). Then the monodromy homomorphism $\rho_{\chi_{2}}\colon\pi_{1}(B)\to\operatorname{SL}(2,\mathbb{Z})$ differs from $\chi_{D}\rho_{\chi_{1}}$ by an inner automorphism of $\operatorname{SL}(2,\mathbb{Z})$.

Proof.

Suppose that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are defined by the equations (4), where $D$ has no zeroes or poles on $B$ and discriminants of the left-hand sides of never vanish. If $u(b)$, $v(b)$, and $w(b)$ are the roots of the polynomial $x^{3}+P(b)x+Q(b)$, where $b\in B$, then roots of the polynomial $x^{3}+D(b)^{2}P(b)x+D(b)^{3}Q(b)$ are $D(b)u(b)$, $D(b)v(b)$, and $D(b)w(b)$.

In the argument that follows we will not distinguish between path and loops in $\mathbb{C}^{(3)}$ and their homotopy classes; this will not lead to a confusion. That said, choose a base point $b_{0}\in B$ and fix a path $\tau$ in $\mathbb{C}^{(3)}$ joining the points (unordered triples) $(D(b_{0})u(b_{0}),D(b_{0})v(b_{0}),D(b_{0})w(b_{0}))$ and $(u(b_{0}),v(b_{0}),w(b_{0}))$. If $\gamma\in\pi_{1}(B,b_{0})$, then

[TABLE]

where $\delta\in B_{3}(u(b_{0}),v(b_{0}),w(b_{0}))$ is the loop defined by the formula

[TABLE]

in which we use the following notation: if $\lambda\in\mathbb{C}^{*}$ and $\gamma=(u,v,w)\in\mathbb{C}^{(3)}$, then $\lambda\cdot\gamma$ is the unordered triple $(\lambda u,\lambda v,\lambda w)$.

If the loop $\delta$ winds $k$ times around the origin, then Proposition 2.1 implies that $\mu(\delta)=(-1)^{k}I=\chi_{D}(\gamma)I$, where $I$ is the identity matrix, whence the result. ∎

Lemma 3.4**.**

Suppose that $\Pi$ is a group and that $\rho\colon\Pi\to\operatorname{SL}(2,\mathbb{Z})$ and $\chi\colon\Pi\to\{\pm 1\}$ are homomorphisms. Put $G_{1}=\operatorname{Im}\rho$, $G_{2}=\operatorname{Im}(\chi\rho)$. Then one of the following cases holds:

(i)* $G_{1}=G_{2}$;*

(ii)* there exists a subgroup $H\subset G_{1}$, $(G_{1}:H)=2$, such that $G_{2}=H\cup(-I)(G_{1}\setminus H)$;*

(iii)* $G_{2}$ is the subgroup of $\operatorname{SL}(2,\mathbb{Z})$ generated by $G_{1}$ and $-I$.*

Proof.

If the quadratic character $\chi$ is trivial, then $G_{2}=G_{1}$ and case (i) holds. Suppose now that $\chi$ is non-trivial; then $\operatorname{Ker}\chi$ is a subgroup of index $2$ in $\Pi$.

If $\operatorname{Ker}\rho\subset\operatorname{Ker}\chi$, then the character $\chi$ factors through the group $G_{1}$:

[TABLE]

now if $H=\operatorname{Ker}\bar{\chi}$, then $G_{2}=H\cup(-I)(G_{1}\setminus H)$.

Suppose finally that $\operatorname{Ker}\rho\not\subset\operatorname{Ker}\chi$. Then $G_{1}=\rho(\operatorname{Ker}\chi)=\rho(\Pi\setminus\operatorname{Ker}\chi)$ and

[TABLE]

so $G_{2}$ is the subgroup generated by $G_{1}$ and $-I$ and case (iii) holds. ∎

Suppose now that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are smooth families of elliptic curves over the same base $B$. If we fix a base point $b\in B$, we can identify (not canonically) first integer cohomology groups of the fibers $(\mathcal{X}_{1})_{b}$ and $(\mathcal{X}_{2})_{b}$ and identify them both with $\mathbb{Z}^{2}$.

Proposition 3.5**.**

Suppose that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are smooth families of elliptic curves over the same base $B$ and that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ differ by a quadratic twist with a non-vanishing regular function $D$ as in (4). Put $G_{i}=\operatorname{Mon}(\mathcal{X}_{i},b)\subset\operatorname{SL}(2,\mathbb{Z})$ (these subgroups are only defined up to a conjugation). Then:

(i)* either $G_{1}$ and $G_{2}$ are conjugate, or one of these groups contains a subgroup of index $2$ that is conjugate to the other subgroup, or each $G_{i}$ contains a subgroup $H_{i}$ of index $2$ and the subgroups $H_{1}$ and $H_{2}$ are conjugate.*

(ii)* if $G_{1}=\operatorname{SL}(2,\mathbb{Z})$, then $G_{2}=\operatorname{SL}(2,\mathbb{Z})$.*

Proof.

Put $\Pi=\pi_{1}(B)$. Proposition 3.3 implies that, conjugating the subgroups if necessary, one may assume that there exist homomorphisms $\rho\colon\Pi\to\operatorname{SL}(2,\mathbb{Z})$ and $\chi\colon\Pi\to\{\pm 1\}$ such that $G_{1}=\operatorname{Im}(\rho)$, $G_{2}=\operatorname{Im}(\chi\rho)$.

We prove part (i) first; to that end, we invoke Lemma 3.4. If case (i) or case (iii) of this lemma holds, we are done. In case (ii) there exists a subgroup $H\subset G_{1}$, $(G_{1}:H)=2$, such that $G_{2}=H\cup(-I)(G_{1}\setminus H)$.

If $-I\in H$, this implies that $G_{2}=G_{1}$; if $-I\in G_{1}\setminus H$, this implies that $G_{2}=H$, so $G_{2}$ is a subgroup of index $2$ in $G_{1}$; in the remaining case $-I\notin G_{1}$, one has $G_{2}\neq G_{1}$, $G_{2}\supset H$ and $(G_{2}:H)=(G_{1}:H)=2$. Thus, part (i) is proved.

To prove part (ii), suppose that $G_{1}=\operatorname{SL}(2,\mathbb{Z})$. If case (i) or case (iii) of Lemma 3.4 holds, it is clear that $G_{2}=\operatorname{SL}(2,\mathbb{Z})$. If case (ii), observe that $\operatorname{SL}(2,\mathbb{Z})$ contains a unique subgroup $H$ of index $2$: this follows from the fact that the abelianization of $\operatorname{SL}(2,\mathbb{Z})$ is $\mathbb{Z}/12\mathbb{Z}$. Expressing the corresponding epimorphism $\varphi\colon\operatorname{SL}(2,\mathbb{Z})\to\mathbb{Z}/12\mathbb{Z}$ in terms of its action on the generators $\left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0&-1\\ 1&1\end{smallmatrix}\right)$, one sees that $\varphi(-I)\in 6\mathbb{Z}/12\mathbb{Z}$, whence $-I\in H$, whence $G_{2}=G_{1}=\operatorname{SL}(2,\mathbb{Z})$. ∎

Corollary 3.6** (from the proof).**

Suppose that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are families of elliptic curves over the same smooth base that differ by a quadratic twist. Then

(i)* if the monodromy group $\operatorname{Mon}(\mathcal{X}_{1})$ has finite index in $\operatorname{SL}(2,\mathbb{Z})$, then either the indices $(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}_{1}))$ and $(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}_{2}))$ are equal or one of them is twice greater than the other;*

(ii)* images of $\operatorname{Mon}(\mathcal{X}_{1})$ and $\operatorname{Mon}(\mathcal{X}_{2})$ in $\operatorname{PSL}(2,\mathbb{Z})$ are conjugate.∎*

Proposition 3.7**.**

Suppose that $\mathcal{X}$ and $\mathcal{Y}$ are smooth families of elliptic curves over the same base $B$ and that their $J$-maps $J_{\mathcal{X}},J_{\mathcal{Y}}\colon B\to\mathbb{A}^{1}$ are equal and non-constant. Then

(i)* either $(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}))=(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{Y}))$ or one of these indices is twice greater than the other;*

(ii)* images of $\operatorname{Mon}(\mathcal{X})$ and $\operatorname{Mon}(\mathcal{Y})$ in $\operatorname{PSL}(2,\mathbb{Z})$ are conjugate;*

(iii)* if $\operatorname{Mon}(\mathcal{X})=\operatorname{SL}(2,\mathbb{Z})$ then $\operatorname{Mon}(\mathcal{Y})=\operatorname{SL}(2,\mathbb{Z})$.*

Notation 3.8**.**

In the next section we will see that if the $J$-map is not constant then the monodromy group has finite index in $\operatorname{SL}(2,\mathbb{Z})$. In the statement of this proposition we allow indices of subgroups to be infinite and assume that $2\cdot\infty=\infty$.

Proof.

In view of Propositions 3.6 and 3.5 it suffices to show that $\mathcal{X}$ and $\mathcal{Y}$ differ by a quadratic twist. To that end put $K=\mathbb{C}(B)$ (the field of rational functions). The (scheme-theoretic) generic fibers of the families $\mathcal{X}$ and $\mathcal{Y}$ over $\operatorname{Spec}K$ are elliptic curves over $K$. They have the same $j$-invariant $J_{\mathcal{X}}=J_{\mathcal{Y}}\in\mathbb{C}(B)$, and this $j$-invariant is not equal to [math] or $1728$ since $J_{\mathcal{X}}=J_{\mathcal{Y}}$ is not constant. Hence, these elliptic curves differ by a quadratic twist by virtue of Proposition 5.4 from [15, Chapter X], and so are the corresponding families. ∎

4. Main result and applications

We begin with a folklore result for which I do not know an adequate reference.

Proposition 4.1**.**

Suppose that $\pi\colon\mathcal{X}\to B$ is a smooth family of curves of genus $1$, where $B$ is a variety (i.e., a reduced scheme of finite type over $\mathbb{C}$). Then the mapping from $B$ to $\mathbb{C}$ that assigns $j$-invariant $j(f^{-1})(b)$ to a point $b\in B$, is induced by a morphism from $B$ to $\mathbb{A}^{1}$.

Proof.

If $\mathcal{X}^{\prime}=\operatorname{Pic}^{0}(\mathcal{X}/B)$ (relative Picard variety, see [10, Section 5]), then the family $p^{\prime}\colon\mathcal{X}^{\prime}\to\mathbb{C}$ has a section (to wit, the zero section) and induces the same mapping from $B$ to $\mathbb{C}$ since $\operatorname{Pic}^{0}(C)\cong C$ if $C$ is a smooth curve of genus $1$ over $\mathbb{C}$. Thus, without loss of generality one may assume that the family in question has a section; in this case see [6, § 5]. ∎

Proposition 4.2**.**

Suppose that $\pi\colon\mathcal{X}\to B$ is a smooth family of curves of genus $1$ over a smooth and connected base $B$ (the ground field is $\mathbb{C}$); let $J_{\mathcal{X}}\colon B\to\mathbb{A}^{1}$ be the $J$-map, attaching to any point $a\in B$ the $j$-invariant of the fiber of $\mathcal{X}$ over $a$.

(i)* If the morphism $J_{\mathcal{X}}$ is not constant and its general fiber is connected, then $\operatorname{Mon}(X)=\operatorname{SL}(2,\mathbb{Z})$.*

(ii)* If the morphism $J_{\mathcal{X}}$ is not constant and its general fiber has $m\geq$ connected components, then $\operatorname{Mon}(\mathcal{X})$ is a subgroup of index at most $2m$ in $\operatorname{SL}(2,\mathbb{Z})$ and the image of $\operatorname{Mon}(\mathcal{X})$ in $\operatorname{PSL}(2,\mathbb{Z})$ is a subgroup of index at most $m$ in $\operatorname{PSL}(2,\mathbb{Z})$.*

We begin with a lemma.

Lemma 4.3**.**

Suppose that $\pi\colon\mathcal{X}\to B$ is a smooth family of curves of genus $1$. Then there exists a smooth family of elliptic curves $p^{\prime}\colon\mathcal{X}^{\prime}\to B$ such that the $J$-maps $J_{\mathcal{X}},J_{\mathcal{X}^{\prime}}\colon B\to\mathbb{A}^{1}$ are the same and $\operatorname{Mon}(\mathcal{X}^{\prime})\subset\operatorname{SL}(2,\mathbb{Z})$ is conjugate to $\tau(\operatorname{Mon}(\mathcal{X}))$, where $\tau\colon\operatorname{SL}(2,\mathbb{Z})\to\operatorname{SL}(2,\mathbb{Z})$ is the automorphism defined by the formula $M\mapsto(M^{t})^{-1}$.

Proof.

Put $\mathcal{X}^{\prime}=\operatorname{Pic}^{0}(\mathcal{X}/B)$. As we have seen in the proof of Proposition 4.1, $J_{\mathcal{X}}=J_{\mathcal{X}^{\prime}}$ and the family $p^{\prime}\colon\mathcal{X}^{\prime}\to B$ has a section. Finally, $R^{1}p^{\prime}_{*}\mathbb{Z}\cong\operatorname{\underline{Hom}}(R^{1}p_{*}\mathbb{Z},\mathbb{Z})$, where by $\mathbb{Z}$ we mean the constant sheaf with the stalk $\mathbb{Z}$ (see for example [13, § 9]), and this implies the assertion about monodromy. ∎

Proof of Proposition 4.2.

Lemma 4.3 implies that we may assume that the family $\pi\colon\mathcal{X}\to B$ in question has a section. Assuming that, put

[TABLE]

and consider the smooth family of elliptic curves $\mathcal{B}\to V$ in which the fiber over $(p,q)$ is the smooth projective model $C_{p,q}$ of the curve with equation $y^{2}=x^{3}+px+q$ and the section assigns to $(p,q)$ the “point at infinity” of this model. It is well known (see for example [1, Corollary to Theorem 1]) that $\operatorname{Mon}(\mathcal{B})=\operatorname{SL}(2,\mathbb{Z})$.

Now put $\mathbb{A}^{1}_{0}=\mathbb{A}^{1}\setminus\{0\}$, $\mathbb{A}^{1}_{0,1728}=\mathbb{A}^{1}\setminus\{0,1728\}$ and

[TABLE]

Let $\mathcal{B}^{\prime}$ be the restriction of the family $\mathcal{B}$ to $V^{\prime}$; put $B^{\prime}=J_{\mathcal{X}}^{-1}(\mathbb{A}^{1}_{0,1728})$, and let $\mathcal{X}^{\prime}$ be the restriction of $\mathcal{X}$ to $B^{\prime}$. Proposition 1.3 implies that $\operatorname{Mon}(\mathcal{X}^{\prime})=\operatorname{Mon}(\mathcal{X})$ and $\operatorname{Mon}(\mathcal{B}^{\prime})=\operatorname{Mon}(\mathcal{B})=\operatorname{SL}(2,\mathbb{Z})$.

Observe that there exists an isomorphism $g\colon V\to\mathbb{A}^{1}_{0}\times\mathbb{A}^{1}_{0,1728}$ such that the diagram

[TABLE]

is commutative. Indeed, one can define $g$ by the formula $(p,q)\mapsto(q/p,j(C_{p,q}))$, and the inverse morphism will be

[TABLE]

Hence, in the fibered product

[TABLE]

(we mean fibered product in the category of reduced algebraic variates, so $W$ is the scheme theoretic fibered product modulo nilpotents) the variety $W$ is isomorphic to $(\mathbb{A}^{1}_{0})\times B^{\prime}$ (in particular, $W$ is smooth and irreducible) and fibers of $f$ are isomorphic to fibers of $J_{\mathcal{X}_{0}}$. Thus, the hypothesis implies that a general fiber of the morphism $J_{\mathcal{X}_{0}}$ has $m$ connected components. On the other hand, any fiber of the morphism $u$ is irreducible since it is isomorphic to $\mathbb{A}^{1}_{0}$. Now Proposition 1.4 and Corollary 1.5 imply that for the pullback families $f^{*}\mathcal{B}^{\prime}$ and $u^{*}\mathcal{X}^{\prime}$ on $W$, the group $\operatorname{Mon}(f^{*}\mathcal{B}^{\prime})$ is a subgroup of index at most $m$ in $\operatorname{Mon}(\mathcal{B})=\operatorname{SL}(2,\mathbb{Z})$ and $\operatorname{Mon}(u^{*}\mathcal{X}_{0})=\operatorname{Mon}(\mathcal{X}_{0})$ (as usual, this equation holds up to a conjugation).

Since $J_{f^{*}\mathcal{B}^{\prime}}=J_{\mathcal{B}^{\prime}}\circ f=J_{\mathcal{X}^{\prime}}\circ u=J_{u^{*}\mathcal{X}^{\prime}}$, Proposition 3.7 implies the result. ∎

Remark 4.4*.*

I do not know whether the bound in this proposition can be improved to $(\operatorname{PSL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}))\leq m$ for $m>1$.

Proposition 4.2 implies the following fact.

Corollary 4.5**.**

If $\mathcal{X}\to B$ is a non-isotrivial smooth family of curves of genus $1$, then its monodromy group is a subgroup of finite index in $\operatorname{SL}(2,\mathbb{Z})$.

Here is the first application of what we proved.

Proposition 4.6**.**

If $B$ is a smooth algebraic variety with abelian fundamental group, then any smooth family of curves of genus $1$ over $X$ must be isotrivial.

Proof.

Suppose that $\pi\colon\mathcal{X}\to B$ is a smooth family of curves of genus $1$, where $B$ is smooth and irreducible and $\pi_{1}(B)=G$ is abelian.

We are to show that the $J$-map $J_{\mathcal{X}}\colon B\to\mathbb{A}^{1}$ is constant. If this is not the case, then Corollary 4.5 asserts that the monodromy group $\operatorname{Mon}(\mathcal{X})$ of the family $\mathcal{X}$ has finite index in $\operatorname{SL}(2,\mathbb{Z})$. Since $\Gamma(2)$ has finite index in $\operatorname{SL}(2,\mathbb{Z})$, one has $(\Gamma(2):\Gamma(2)\cap\operatorname{Mon}(\mathcal{X}))<\infty$. If $G$ is the image of $\Gamma(2)\cap\operatorname{Mon}(\mathcal{X})$ in $\Gamma(2)/\{\pm I\}$, then $G$ is an abelian subgroup of finite index in $\Gamma(2)/\{\pm I\}$. The latter group is isomorphic to the free group $F_{2}$ with two generators, and Schreier’s theorem on subgroups of free groups implies that $F_{2}$ contains no abelian subgroup of finite index. We arrived at the desired contradiction. ∎

For the case of non-commutative $\pi_{1}$ of the base, one can obtain an upper bound on the index of monodromy groups in non-isotrivial families.

Proposition 4.7**.**

Suppose that $\mathcal{X}\to B$ is a smooth non-isotrivial family of curves of genus $1$ over a smooth base $B$ and that $\pi_{1}(B)$ can be generated by $r\geq 2$ elements. Then $(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}))\leq 12(r-1)$.

Corollary 4.8**.**

Suppose that $\mathcal{X}\to C$ is a non-isotrivial family of elliptic curves over a smooth curve of genus $g$, with $s$ degenerate fibers. Then $(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}))\leq 12(2g+s-1)$.

Proposition 4.7 is a consequence of the following elementary lemma.

Lemma 4.9**.**

Suppose that $G\subset\operatorname{SL}(2,\mathbb{Z})$ is a subgroup of finite index and that $G$ can be generated by $r$ elements. Then $(\operatorname{SL}(2,\mathbb{Z}):G)\leq 12(r-1)$.

Proof of the lemma.

Throughout the proof, free group with $m$ generators will be denoted by $F_{m}$.

Since $G$ can be generated by $r$ elements, there exists an epimorphism $\pi\colon F_{r}\twoheadrightarrow G$. Putting $H=p^{-1}(G\cap\Gamma(2))$, one obtains the following commutative diagram of embeddings and surjections:

[TABLE]

If $d\leq 6$ is the order of $\operatorname{Im}(j\circ i)$, then $(G:G\cap\Gamma(2))=(F_{r}:H)=d$, so by Schreier’s theorem $H\cong F_{d(r-1)+1}$. Since the morphism $p^{\prime}$ is surjective, the group $G\cap\Gamma(2)$ can be generated by $d(r-1)+1$ elements. Put $F=\Gamma(2)/\{\pm I\}$, and let $\pi\colon\Gamma(2)\to F$ be the natural projection. The subgroup $\pi(G\cap\Gamma(2))\subset F$ can be also generated by $d(r-1)+1$ elements; since $F\cong F_{2}$, Schreier’s theorem implies that $\pi(G\cap\Gamma(2))\cong F_{m}$, where $m\leq d(r-1)+1$ (indeed, the free group $F_{n}$ cannot be generated by $k<n$ elements). Applying Schreier’s for the third time, we obtain that $(F:\pi(H\cap\Gamma(2)))=m-1\leq d(r-1)$, whence $(\Gamma(2):G\cap\Gamma(2))\leq 2(m-1)\leq 2d(r-1)$. It follows from the right-hand square of the diagram (5) that

[TABLE]

whence the result. ∎

Proof of Proposition 4.7.

Put $\operatorname{Mon}(\mathcal{X})=G\subset\operatorname{SL}(2,\mathbb{Z})$. Since $\pi_{1}(B)$ can be generated by $r$ elements, the same is true for $G$; now Corollary 4.5 implies that $(\operatorname{SL}(2,\mathbb{Z}):G)<+\infty$, and Lemma 4.9 applies. ∎

Using Proposition 4.2 one can obtain other lower bounds for monodromy groups. Observe first that the named proposition immediately implies the following corollary, in the statement of which we use Convention 1.2.

Corollary 4.10**.**

If $\pi\colon\mathcal{X}\to C$ is a family of elliptic curves over a smooth projective curve $C$ and if $J_{\mathcal{X}}\colon C\dasharrow\mathbb{A}^{1}$ is its $J$-map, and if $J_{\mathcal{X}}$ is not constant, then $(\operatorname{SL}(2,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}))\leq 2\deg J_{\mathcal{X}}$.

If $\mathcal{X}$ is smooth and $\pi$ has a section, one can be more specific.

Proposition 4.11**.**

Suppose that $\pi\colon\mathcal{X}\to C$ is a minimal smooth elliptic surface with section (it means that $\mathcal{X}$ is a smooth projective surface, $C$ is a smooth projective curve, the general fiber of $\pi$ is a smooth curve of genus $1$, no fiber of $\pi$ contains a rational $(-1)$-curve, and $p$ has a section) and that $J_{\mathcal{X}}$ is not constant.

Then

[TABLE]

where $e(s)=n$ if the fiber over $s$ is a cycle of $n$ smooth rational curves or the nodal rational curve if $n=1$ (type $\mathrm{I}_{n}$ in Kodaira’s classification [11, 12]), $e(s)=n$ if the fiber over $s$ consists of $n+5$ smooth rational curves with intersection graph isomorphic to the extended Dynkin graph $\tilde{D}_{n+4}$, $n\geq 1$ (type $\mathrm{I}^{*}_{n}$ in Kodaira’s classification), and $e(s)=0$ otherwise.

Proof.

In view of Corollary 4.10 the index in the left-hand side of (6) is less or equal to $2\deg J_{\mathcal{X}}$, and $\deg J_{\mathcal{X}}$ equals $\sum_{s\in C}e(s)$ by virtue of Corollary IV.4.2 from [12]. ∎

Similarly, one can express $\deg J_{\mathcal{X}}$ (and obtain a lower bound for $\operatorname{Mon}(\mathcal{X})$) using the information about the points where $j$-invariant of the fiber (smooth or not) equals [math] or $1728$, see for example [12, Lemma IV.4.5, Table IV.3.1] and Table. In the notation if [12], $j$-invariant is 1728 times less than that defined by (1); of course, this does not affect multiplicities of poles.

5. A remark on families of hyperelliptic curves

Proposition 5.1**.**

If $\pi\colon\mathcal{X}\to B$ is a smooth family of hyperelliptic curves of genus $g>2$, then

[TABLE]

Corollary 5.2**.**

If $\pi\colon\mathcal{X}\to B$ is a smooth family of hyperelliptic curves of genus $g>2$, then $\operatorname{Mon}(\mathcal{X})$ is a proper subgroup of $\operatorname{Sp}(2g,\mathbb{Z})$.

Proof of Proposition 5.1.

In this proof, $\operatorname{Mon}(\mathcal{X},\mathbb{Z})$ will denote the monodromy group acting on the integer $H^{1}$ of a fiber of $\mathcal{X}$, and $\operatorname{Mon}(\mathcal{X},\mathbb{Z}/2\mathbb{Z})$ will stand for the monodromy group acting on cohomology with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

Since the reduction modulo $2$ mapping $\operatorname{Sp}(2g,\mathbb{Z})\to\operatorname{Sp}(2g,\mathbb{Z}/2\mathbb{Z})$ is surjective, one has

[TABLE]

so it suffices to show that

[TABLE]

To that end, let $X$ be a hyperelliptic curve of genus $g\geq 2$ that is a fiber of $\mathcal{X}$; denote its Weierstrass points by $P_{1},\ldots,P_{2g+2}$. It is well known (see for example [5, Lemma 2.1]) that the $2$-torsion subgroup $(\operatorname{Pic}(X))_{2}\subset\operatorname{Pic}(X)$ is generated by classes of divisors $P_{i}-P_{j}$. Since $\operatorname{Pic}(X)_{2}\cong H^{1}(X,\mathbb{Z}/2\mathbb{Z})$, the action of $\pi_{1}(B_{\mathrm{sm}})$ on $H^{1}(X,\mathbb{Z}/2\mathbb{Z})$ is completely determined by the permutations of the Weierstrass points $P_{1},\ldots,P_{2g+2}$ it induces. Thus, order of $\operatorname{Mon}(\mathcal{X},\mathbb{Z}/2\mathbb{Z})$ is at most $(2g+2)!$. Since

[TABLE]

the proposition follows. ∎

Remark 5.3*.*

The bound in Proposition 5.1 is sharp, which follows from A’Campo’s paper [1]. To wit, for any $g\geq 2$ let us regard $\mathbb{A}^{2g+1}$ as the space of polynomials

[TABLE]

and denote the space of polynomials $P\in\mathbb{A}^{2g+1}$ with a multiple root by $\Sigma\subset\mathbb{A}^{2g+1}$. If $\mathcal{X}$ is a family over $\mathbb{A}^{2g+1}\setminus\Sigma$ in which the fiber over $P$ is the smooth projective model of the curve with equation $y^{2}=P(x)$ (which is hyperelliptic of genus $g$), then part of the corollary on page 319 of [1] can be restated to the effect that index $(\operatorname{Sp}(2g,\mathbb{Z}):\operatorname{Mon}(\mathcal{X}))$ is equal to the right-hand side of (7). Actually, not only the order of $\operatorname{Mon}(\mathcal{X})$ is known: a description of this group can be found in the appendix to [3], which (the appendix) is devoted to the exposition of results of A.Varchenko.

6. Appendix: an application to Del Pezzo surfaces

Robinson said, ‘It was only to be expected.’

–Muriel Spark, Robinson

In this section, by way of an application of Proposition 4.2, we prove the following fact.

Proposition 6.1**.**

If $X\subset\mathbb{P}^{n}$ is a Del Pezzo surface embedded by (a subsystem of) the anticanonical linear system $|-K_{X}|$, then the monodromy group acting on $H^{1}(\cdot,\mathbb{Z})$ of its smooth hyperplane sections is the entire $\operatorname{SL}(2,\mathbb{Z})$.

First recall some notation and definitions.

If $X$ is an algebraic variety and $\mathcal{R}$ is a coherent sheaf of reduced $\mathcal{O}_{X}$-algebras, we denote its relative spectrum (which is a scheme over $X$) by $\operatorname{\mathbf{Spec}}\mathcal{R}$ (under our assumptions $\operatorname{\mathbf{Spec}}\mathcal{R}$ is an algebraic variety and the canonical morphism $\operatorname{\mathbf{Spec}}\mathcal{R}\to X$ is finite).

If $p\in\mathbb{P}^{n}$ is a point and $L\subset\mathbb{P}^{n}$ is a linear subspace, then $\overline{p,L}$ denotes the linear span of $\{p\}\cup L$.

If $A_{1},\ldots,A_{4}$ are points on the affine line with coordinates $a_{1},\ldots,a_{4}$, then by their cross-ratio we mean

[TABLE]

If $X\subset\mathbb{P}^{n}$ is a smooth projective variety and $X^{*}\subset(\mathbb{P}^{n})^{*}$ is its projective dual, one can define the “universal smooth hyperplane section of $X$”, that is, the family

[TABLE]

where $H_{\alpha}\subset\mathbb{P}^{n}$ is the hyperplane corresponding to the point $\alpha\in(\mathbb{P}^{n})^{*}$. The morphism $\pi\colon(x,\alpha)\mapsto\alpha$ makes $\mathcal{X}$ a smooth family of $n$-dimensional projective varieties over $(\mathbb{P}^{n})^{*}\setminus X^{*}$; for any natural $d$, this family induces a monodromy action of $\pi_{1}((\mathbb{P}^{n})^{*}\setminus X^{*})$ on $H^{d}(Y,\mathbb{Z})$, where $Y$ is a smooth hyperplane section of $X$.

In the above setting, the image of $\pi_{1}((\mathbb{P}^{n})^{*}\setminus X^{*})$ in the group $\operatorname{Aut}(H^{n}(Y,\mathbb{Z}))$ will be called hyperplane monodromy group of $X$.

Lemma 6.2**.**

Suppose that $X\subset\mathbb{P}^{n}$ is a smooth projective variety and that $p\in\mathbb{P}^{n}\setminus X$ is a point such that the projection with center $p$ induces an isomorphism $\pi_{p}\colon X\to X^{\prime}\subset\mathbb{P}^{N-1}$. If $H\ni p$ is a hyperplane that is transversal to $X$, then, after identifying $Y=X\cap H$ with $Y^{\prime}=\pi_{p}(Y)=X^{\prime}\cap\pi_{p}(H)$, the hyperplane monodromy groups acting on $H^{n}(Y,\mathbb{Z})$ and $H^{n}(Y^{\prime},\mathbb{Z})$, are the same.

The proof that is sketched below was suggested to me by Jason Starr.

Sketch of proof.

Denote by $H_{p}\subset(\mathbb{P}^{n})^{*}$ the hyperplane corresponding to the point $p\in\mathbb{P}^{n}$. It is clear that $H_{p}$ is naturally isomorphic to $(\mathbb{P}^{N-1})^{*}$ and that $(X^{\prime})^{*}=X^{*}\cap H_{p}$. Moreover, the hyperplane $H_{p}$ is transversal to $X^{*}$ at any smooth point of $X^{*}$ (indeed, if $H_{p}$ is tangent to $X^{*}$ at a smooth point, then $p\in(X^{*})^{*}=X$, which contradicts the hypothesis).

To prove the lemma it suffices to show that $\pi_{1}(H_{p}\setminus(X^{\prime})^{*})$ surjects onto $\pi_{1}((\mathbb{P}^{n})^{*}\setminus X^{*})$. To that end observe that there exists a line $\ell\subset H_{p}$ that is transversal to the smooth part of $X^{*}\cap H_{p}=(X^{\prime})^{*}$ (in particular, $\ell$ does not pass through singular points of $X^{*}\cap H_{p}$). It follows from the transversality of $H_{p}$ to the smooth part of $X^{*}$ that $\ell$ is transversal to the smooth part of $X^{*}$, too. Thus, $\pi_{1}(\ell\setminus X^{*})$ surjects both onto $\pi_{1}(H_{p}\setminus(X^{\prime})^{*})$ and onto $\pi_{1}((\mathbb{P}^{n})^{*}\setminus X^{*})$, whence the desired surjectivity. ∎

Lemma 6.2 implies that when studying hyperplane monodromy groups one may always assume that the variety in question is embedded by a complete linear system. Recall that if a Del Pezzo surface $X\subset\mathbb{P}^{n}$ is embedded by the complete linear system $|-K_{X}|$ then $\deg X=n\leq 9$; besides, if $n>3$, $p\in X$ is a general point, and $\bar{X}$ is the blow-up of $X$ at $p$, then the projection $\pi_{p}\colon X\dasharrow\mathbb{P}^{n-1}$ induces an isomorphism $\bar{\pi}_{p}\colon\bar{X}\to X^{\prime}=\overline{\pi_{p}(X)}\subset\mathbb{P}^{n-1}$ and $X^{\prime}\subset\mathbb{P}^{n-1}$ is a Del Pezzo surface embedded by $|-K_{X^{\prime}}|$.

Lemma 6.3**.**

In the above setting, suppose that the hyperplane monodromy group of $X^{\prime}$ is the entire $\operatorname{SL}(2,\mathbb{Z})$. Then the hyperplane monodromy group of $X$ is the entire $\operatorname{SL}(2,\mathbb{Z})$ as well.

Proof.

Informally the proof may be summed up in one phrase: if variation of hyperplanes passing through $p$ and transversal to $X$ is enough to obtain the entire group $\operatorname{SL}(2,\mathbb{Z})$, then a fortiori this is the case for all hyperplanes transversal to $X$. A formal argument follows.

Assume that the $\mathbb{P}^{n-1}$ into which the surface $X$ is projected is a hyperplane in $\mathbb{P}^{n}$, $\mathbb{P}^{n-1}\not\ni x$. If a hyperplane $H\subset\mathbb{P}^{n}$ contains the point $p\in X$, then $H=\overline{p,h}$, where $h=H\cap\mathbb{P}^{n-1}$ is a hyperplane in $\mathbb{P}^{n-1}$. If $H$ is transversal to $X$, then $\pi_{p}$ induces an isomorphism between the curve $H\cap X$, which is a smooth hyperplane section of $X$, and the curve $h\cap X^{\prime}$, which is a smooth hyperplane section of $X^{\prime}$. Put

[TABLE]

where $h_{\alpha}\subset\mathbb{P}^{n-1}$ is the hyperplane corresponding to the point $\alpha\in(\mathbb{P}^{n-1})^{*}$, and set

[TABLE]

In the diagram

[TABLE]

where $\mathcal{U}_{X^{\prime}}$ and $\mathcal{U}_{X}$ are universal smooth hyperplane sections of $X^{\prime}$ and $X$, $j$ is an open embedding, and $r$ maps a hyperplane $h_{\alpha}\subset\mathbb{P}^{n-1}$ to the hyperplane $\overline{p,h_{\alpha}}$ in $\mathbb{P}^{n}$, both squares are Cartesian. Pick a point $\alpha\in V$; the hyperplane $\overline{p,h_{\alpha}}\subset\mathbb{P}^{n}$ is $H_{r(\alpha)}$, where $r(\alpha)\in(\mathbb{P}^{n})^{*}$. If $Y^{\prime}={q^{\prime}}^{-1}(\alpha)$ and $Y=X\cap H_{r(\alpha)}=q^{-1}(r(\alpha))$, then in the commutative diagram

[TABLE]

the mapping $w$ is an epimorphism since $V$ is Zariski open in $(\mathbb{P}^{n-1})^{*}\setminus(X^{\prime})^{*}$, whence $\operatorname{Im}u\subset\operatorname{Im}v$. This proves the lemma. ∎

Projecting Del Pezzo surfaces in $\mathbb{P}^{n}$, $n>3$, consecutively from general points on them, one arrives at a cubic in $\mathbb{P}^{3}$; Lemma 6.3 implies that it suffices to prove Proposition 6.1 for this surface.

The next lemma reduces the problem to the case of “Del Pezzo surfaces of degree $2$”.

Suppose that $X\subset\mathbb{P}^{3}$ is a smooth cubic and $p\in X$ is a general point. Let $\bar{X}$ be the blow-up of $X$ at $p$. Then the projection $\pi_{p}\colon\mathbb{P}^{3}\dasharrow\mathbb{P}^{2}$ induces a finite morphism $\bar{\pi}_{p}\colon\bar{X}\to\mathbb{P}^{2}$ of degree $2$; the branch locus of this morphism is a smooth curve $C\subset\mathbb{P}^{2}$ of degree $4$. For $\alpha\in(\mathbb{P}^{2})^{*}$, denote the corresponding line by $\ell_{\alpha}\subset\mathbb{P}^{2}$. If $\ell_{\alpha}$ is transversal to $C$ (i.e., $\alpha\notin C^{*}$), then $\bar{\pi}_{p}^{-1}(\ell_{\alpha})$ is smooth, irreducible, and isomorphic to $X\cap\overline{p,\ell_{\alpha}}$.

The proof of the following lemma is similar to that of Lemma 6.3.

Lemma 6.4**.**

Put

[TABLE]

and denote the morphism $(\alpha,x)\mapsto\alpha$ by $q\colon\mathcal{X}\to(\mathbb{P}^{2})^{*}\setminus C^{*}$. If $\operatorname{Mon}(\mathcal{X},\mathbb{Z})=\operatorname{SL}(2,\mathbb{Z})$, then the hyperplane monodromy group of $X$ is also equal to $\operatorname{SL}(2,\mathbb{Z})$.

Our next lemma is valid over algebraically closed fields of arbitrary characteristic.

Lemma 6.5**.**

Suppose that $W$ is a smooth irreducible variety of dimension $n$, $L$ is a smooth irreducible curve (we do not assume that $W$ or $L$ is projective), and $\varphi\colon W\to L$ is a proper and surjective morphism with $(n-1)$-dimensional fibers. Put $Z=\operatorname{\mathbf{Spec}}\varphi_{*}\mathcal{O}_{W}$ and let $v\colon Z\to L$ be the natural morphism.

If there exists a point $p\in L$ such that $\varphi^{-1}(p)$ is irreducible and the morphism $\varphi$ has maximal rank at a general point of $\varphi^{-1}(p)$, then the natural morphism $v\colon Z\to L$ is an isomorphism.

Proof.

It is clear that $Z$ is an irreducible and reduced curve. Since $\varphi$ is proper and $\varphi^{-1}(p)$ is connected, the stalk $(\varphi_{*}\mathcal{O}_{W})_{p}$ is a local ring, so $v^{-1}(p)$ consists of one point; denote this point by $z$. I claim that that $z$ is a smooth point of $Z$ and the morphism $v$ is unramified at $z$. Indeed, let $\tau\in\mathcal{O}_{L,p}$ be a generator of the maximal ideal. Its image $v^{*}\tau\in\mathcal{O}_{Z,z}$ can be represented by a regular function $f\in\mathcal{O}_{W}(\varphi^{-1}(U))$, where $U\subset L$ is a Zariski neighborhood of $p$. Since the morphism $\varphi$ has maximal rank at a general point of $\varphi^{-1}(p)$, the function $v^{*}\tau$ vanishes on the irreducible divisor $\varphi^{-1}(p)$ with multiplicity $1$. Since regular functions on $\varphi^{-1}(U)$ must be constant on the fibers of the proper morphism $\varphi$, any element of the maximal ideal of the local ring $\mathcal{O}_{Z,z}$ is representable by a regular function $g\in\mathcal{O}_{W}(\varphi^{-1}(V))$, where $V$ is a Zariski neighborhood of $p$, such that the zero locus of $g$ in $\varphi^{-1}(V)$ coincides with $u^{-1}(z)$. Hence, $v^{*}\tau$ generates the maximal ideal of $\mathcal{O}_{Z,z}$, which proves our claim.

Since $v^{-1}(p)=\{z\}$, $Z$ is smooth at $z$, and $v$ is unramified at $z$, we conclude that the finite morphism $v$ has degree $1$. Since $L$ is smooth, Zariski main theorem implies that $v$ is an isomorphism. ∎

Proposition 6.6**.**

Suppose that $\pi\colon X\to\mathbb{P}^{2}$ is a finite morphism of degree $2$ branched over a smooth quartic $C\subset\mathbb{P}^{2}$, where $X$ is smooth. If $J\colon(\mathbb{P}^{2})^{*}\setminus C^{*}\to\mathbb{A}^{1}$ is the morphism $\alpha\mapsto j(\pi^{-1}(\ell_{\alpha}))$, where $\ell_{\alpha}$ is the line in $\mathbb{P}^{2}$ corresponding to $\alpha\in(\mathbb{P}^{2})^{*}$, then a general fiber of $J$ is irreducible.

Proof.

Let us show that the morphism $J$ extends to a morphism

[TABLE]

Indeed, if $\ell\subset\mathbb{P}^{2}$ is a line and $\ell\cap C=\{P_{1},P_{2},P_{3},P_{4}\}$, then the curve $\pi^{-1}(\ell)$ is a curve of genus $1$ and

[TABLE]

where $\lambda$ is the cross-ratio $[P_{1},P_{2},P_{3},P_{4}]$, in no matter what order (see for example [15, Chapter III, Proposition 1.7b]). If $\alpha$ is a smooth point of $C^{*}\subset(\mathbb{P}^{2})^{*}$, then the line $\ell_{\alpha}$ is tangent to $C$ at exactly one point that is not an inflection point. Thus, as the line $\ell$ tends to $\ell_{\alpha}$, exactly two intersection points from $\ell\cap C$ merge, so the cross-ratio of these four points tends to [math] (or $1$, or $\infty$, depending on the ordering), and formula (11) shows that $j(\pi^{-1}(\ell))$ tends to $\infty$. This proves the existence of the desired extension.

Our argument shows that $J_{1}^{-1}(\infty)=C^{*}\setminus(C^{*})_{\mathrm{sing}}$; if we regard $J_{1}$ as a rational mapping from $(\mathbb{P}^{2})^{*}$ to $\mathbb{P}^{1}$ and if

[TABLE]

is a minimal resolution of indeterminacy for $J_{1}$, then $J_{2}^{-1}(\infty)$ equals the strict transform of $C^{*}$ with respect to $\sigma$.

Now I claim that, at a general point of $J_{2}^{-1}(\infty)$, derivative of $J_{2}$ has rank $1$. It suffices to prove this assertion for $J_{1}$ and a general smooth point of $C^{*}$. To that end it suffices to construct an analytic mapping $\gamma\colon D\to(\mathbb{P}^{2})^{*}$, where $D$ is a disk in the complex plane with center at [math], such that $\gamma(D\setminus\{0\})\subset(\mathbb{P}^{2})^{*}\setminus C^{*}$, $\gamma(0)$ is a smooth point of $C^{*}$, and $|j(\pi^{-1}(\ell_{\gamma(t)}))|\sim\mathrm{const}/|t|$.

Suppose that a point $c\in C$ is not an inflection point nor a tangency point of a bitangent; if $\ell_{\alpha}\subset\mathbb{P}^{2}$ is the tangent line to $C$ at $c$, then $\alpha$ s a smooth point of $C^{*}$. Now choose affine $(x,y)$-coordinates in $\mathbb{P}^{2}$ so that $c=(0,0)$, the tangent $\ell_{\alpha}$ has equation $y=0$, and $\ell_{\alpha}\cap C=\{c,(C,0),(D,0)\}$, where $C,D\neq 0$ (so the remaining two points of $\ell_{\alpha}\cap C$ are in the finite part of $\mathbb{P}^{2}$ with respect to the chosen coordinate system). If $\ell_{\gamma(t)}$ is the line with affine equation $y=t$, then, for all small enough $t$, one has $\ell_{\gamma(t)}\cap C=\{A(t),B(t),C(t),D(t)\}$, where the $x$-coordinates of $A(t)$ and $B(t)$ are $\sqrt{t}+o(\sqrt{|t|})$ (for both values of $\sqrt{t}$), while the $x$ coordinates of $C(t)$ and $D(t)$ tend to finite and non-zero numbers $C$ and $D$. Hence,

[TABLE]

formula (11) implies that $|j(\pi^{-1}(\ell_{t}))|\sim\mathrm{const}/|t|$, as desired.

Let

[TABLE]

be the Stein factorization in which $W$ is a blow-up of $(\mathbb{P}^{2})^{*}$ (see (12)), $Z=\operatorname{\mathbf{Spec}}(J_{2})_{*}\mathcal{O}_{W}$, and $v$ is a finite morphism. Applying Lemma 6.5 with $L=\mathbb{P}^{1}$, $\varphi=J_{2}$, and $p=\infty$, we conclude that $v$ is an isomorphism. Thus, fibers of $J_{2}$ coincide with fibers of $u$; since the latter are connected, fibers of $J_{2}$ are connected as well. Bertini theorem implies that a general fiber of $J_{2}$ is smooth; since it is connected, it must be irreducible. This implies that a general fiber of $J$ is irreducible. ∎

Proof of Proposition 6.1.

In view of Proposition 6.2 and Lemmas 6.3 and 6.4, it suffices to prove that $\operatorname{Mon}(\mathcal{X})=\operatorname{SL}(2,\mathbb{Z})$, where $\mathcal{X}$ is the family defined by (10).

Applying Proposition 6.6 to the surface $\bar{X}$ (blow-up of a cubic at a general point $p$) and the mapping $\bar{\pi}_{p}\colon\bar{X}\to\mathbb{P}^{2}$ (induced by the projection with center $p$), we see that the family $\mathcal{X}$ defined by formula (10) satisfies the hypothesis of Proposition 4.2(i), whence $\operatorname{Mon}(\mathcal{X})=\operatorname{SL}(2,\mathbb{Z})$. ∎

Remark 6.7*.*

Our argument shows as well that if $X$ is a Del Pezzo surface of degree $2$, then the monodromy group acting on $H^{1}(\cdot,\mathbb{Z})$ of non-singular elements of the anticanonical linear system $|-K_{X}|$, is $\operatorname{SL}(2,\mathbb{Z})$. I do not know the answer for Del Pezzo surfaces of degree $1$.