The integral monodromy of hyperelliptic and trielliptic curves

Big symplectic or orthogonal monodromy modulo l

Let k be a field not of characteristic two and L be a set of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M_ell is a subgroup of a corresponding isometry group G_ell over F_ell, and we say it has big monodromy if it contains the derived subgroup DG_ell=[G_ell,G_ell]. We prove a theorem which gives sufficient conditions for M_ell to have big monodromy. We apply the theorem to explicit …

On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$ does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is natura…

Dihedral Monodromy and Xiao Fibrations

We construct three new families of fibrations $π: S \to B$ where $S$ is an algebraic complex surface and $B$ a curve that violate Xiao's conjecture relating the relative irregularity and the genus of the general fiber. The fibers of $π$ are certain étale cyclic covers of hyperelliptic curves that give coverings of $P^1$ with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill-Noether range.

Tautological classes on the moduli space of hyperelliptic curves with rational tails

We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…

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

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 mon…

单值群与异常霍奇类，II：Sato-Tate群

Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

本文针对超椭圆曲线 $y^2=x^m+1$ 的雅可比簇 $J_m$ 计算其 Sato-Tate 群，推广了阿贝尔簇 Sato-Tate 群的描述，并证明了 Gross-Koblitz 公式的一个推广。

Disproof of modularity of moduli space of CY 3-folds of double covers of P3 ramified along eight planes in general positions

We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole $Sp(20,R)$. We construct an interesting submoduli, which we call \emph{hyperelliptic locus}, over which the weight 3 $Q$-Hodge structure is the third wedge product of the weight 1 $Q$-Hodge structure on the corresponding hyperelliptic curve. The non-extendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved.

Massey Products and Fujita decompositions on fibrations of curves

Let $f:S\to B$ be a fibration of curves and let $f_\astω_{S/B}=\cU\oplus \cA$ be the second Fujita decomposition of $f.$ In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $\cU$. The main result states that their vanishing on a general fibre of $f$ implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we p…

Mean field equations, hyperelliptic curves and modular forms: II

A pre-modular form $Z_n(σ; τ)$ of weight $\tfrac{1}{2} n(n + 1)$ is introduced for each $n \in \Bbb N$, where $(σ, τ) \in \Bbb C \times \Bbb H$, such that for $E_τ= \Bbb C/(\Bbb Z + \Bbb Z τ)$, every non-trivial zero of $Z_n(σ; τ)$, namely $σ\not\in E_τ[2]$, corresponds to a (scaling family of) solution to the mean field equation \begin{equation} \tag{MFE} \triangle u + e^u = ρ\, δ_0 \end{equation} on the flat torus $E_τ$ with singular strength $ρ= 8πn$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(τ) \subset {\rm Sym}^n E_τ$, the Lamé curve, associated to the MFE wa…

通过分支覆盖的扰动分裂代数奇异纤维化

Splitting Algebraic Singular Fibrations via Perturbation of Branch Covers

本文针对亏格二纤维化中另外两类奇异纤维，改用非超椭圆多项式族重建曲线，通过分支覆盖扰动将其形变为 Lefschetz 纤维化，并恢复中心纤维紧致化行为及相应单值分解。

The kernel of the monodromy of the universal family of degree $d$ smooth plane curves

We consider the parameter space $\mathcal U_d$ of smooth plane curves of degree $d$. The universal smooth plane curve of degree $d$ is a fiber bundle $\mathcal E_d\to\mathcal U_d$ with fiber diffeomorphic to a surface $Σ_g$. This bundle gives rise to a monodromy homomorphism $ρ_d:π_1(\mathcal U_d)\to\mathrm{Mod}(Σ_g)$, where $\mathrm{Mod}(Σ_g):=π_0(\mathrm{Diff}^+(Σ_g))$ is the mapping class group of $Σ_g$. The main result of this paper is that the kernel of $ρ_4:π_1(\mathcal U_4)\to\mathrm{Mod}(Σ_3)$ is isomorphic to $F_\infty\times\mathbb{Z}/3\mathbb{Z}$, where $F_\infty$ is a free group of …

Mean field equations, hyperelliptic curves and modular forms: I

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = ρ\, δ_0$, $ρ\in \mathbb R_{>0}$ on flat tori $E_τ= \mathbb C/(\mathbb Z + \mathbb Zτ)$, (b) the classical Lamé equations and (c) modular forms. A major theme in part I is a classification of developing maps $f$ attached to solutions $u$ of the mean field equation according to the type of transformation laws (or monodromy) with respect to $Λ$ satisfied by $f$. We are especially interested in the case when the parameter $ρ$ in the mean field equation is an integer multiple of $4π$. In…

可容许覆盖上的Hodge积分与Prym类的tautological投影

Hodge integrals on admissible covers and tautological projections of Prym classes

本文提出一种算法，利用可容许覆盖模空间上三个Hodge丛的比较公式，计算Prym轨迹类的tautological投影，并应用于计算Prym映射次数及证明低亏格下雅可比类的不可整除性。

Twelve points on the projective line, branched covers, and rational elliptic fibrations

The following divisors in the space Sym^{12} P^1 of twelve points on P^1 are actually the same: (A) the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to P^1; (E) the branch locus of a degree 3 map from a genus 4 curve to P^1 induced by a theta-characteristic; and several more. The corr…