Explicit Mumford isomorphism for hyperelliptic curves

通过芒福德表示法研究超椭圆曲线上MDS码和LCD码的显式LCP

Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation

研究超椭圆曲线上代数几何码，通过芒福德表示刻画非特殊除子，构造LCP码并给出MDS准则，在特定条件下得到LCD码，应用于极大超椭圆曲线给出特定参数的MDS LCD码示例并推测其对所有$q \geq 4$存在。

Fast computation of isomorphisms of hyperelliptic curves and explicit Galois descent

We show how to speed up the computation of isomorphisms of hyperelliptic curves by using covariants. We also obtain new theoretical and practical results concerning models of these curves over their field of moduli.

Atiyah-Segal theorem for Deligne-Mumford stacks and applications

We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.

The movable fan of the Horrocks-Mumford quintic

This paper explores the birational geometry of a general Horrocks-Mumford quintic threefold, describing the set of all minimal models up to marked isomorphism, the movable fan (the way in which the nef cones of all these models are arranged to form the movable cone), and the birational automorphism group. There are infinitely many models up to marked isomorphism, falling into just eight (unmarked-) isomorphism classes.

The moduli spaces of hyperelliptic curves and binary forms

There is a canonical isomorphism between the coarse moduli spaces of somooth hyperelliptic curves of genus g and binary forms of degree 2g+2 with nonzero discriminant. In this paper, we study the extension of this isomorphism to the compactification of these moduli spaces. The main result is that the canonical isomorphism above extends to a holomorphic map f_g from the compactication of the moduli space of hyperelliptic curves of genus g to the compactification of the moduli space of binary forms of degree 2g+2. Moreover, we work out how the boundary components are contracted by the map f_g.

Division by 2 on odd degree hyperelliptic curves and their jacobians

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements $\mathfrak{a} \in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. M. Stoll constructed an a…

Prym varieties of étale covers of hyperelliptic curves

It is well known that the Prym variety of an étale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. We compute the degree of this isogeny in the remaining cases and show that only in the case of coverings of degree 4 it is an isomorphism.

Modifications of Hodge bundles and enumerative geometry : the stable hyperelliptic locus

We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated to a family of stable curves, we construct a bundle map ('degree-2 Brill-Noether') from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton-Macpherson residual intersection theory, th…

Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive Limit of Teichmüller Spaces

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain …

Algorithms for Mumford curves

Mumford showed that Schottky subgroups of $PGL(2,K)$ give rise to certain curves, now called Mumford curves, over a non-Archimedean field K. Such curves are foundational to subjects dealing with non-Archimedean varieties, including Berkovich theory and tropical geometry. We develop and implement numerical algorithms for Mumford curves over the field of p-adic numbers. A crucial and difficult step is finding a good set of generators for a Schottky group, a problem solved in this paper. This result allows us to design and implement algorithms for tasks such as: approximating the period matrices …

Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichmüller Space

There exists on each Teichmüller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmüller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q…

Measure-theoretic rigidity for Mumford curves

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only im…

Point-like limit of the hyperelliptic Zhang-Kawazumi invariant

The behavior near the boundary in the Deligne-Mumford compactification of many functions on the moduli space of pointed Riemann surfaces can be conveniently expressed using the notion of "point-like limit" that we adopt from the string theory literature. In this note we study a function on the moduli space of Riemann surfaces that has been introduced by N. Kawazumi and S. Zhang, independently. We show that the point-like limit of the Zhang-Kawazumi invariant in a family of hyperelliptic Riemann surfaces in the direction of any hyperelliptic stable curve exists, and is given by evaluating a com…

Division by $1-ζ$ on superelliptic curves and jacobians

In 2016, Yuri Zarhin gave formulas for "dividing a point on a hyperelliptic curve by 2." Given a point $P$ on a hyperelliptic curve $\mathcal{C}$, Zarhin gives the Mumford's representation of every degree $g$ divisor $D$ such that $2(D - g \infty) \sim P - \infty$. The aim of this paper is to generalize Zarhin's result to the superelliptic situation; instead of dividing by 2, we divide by $1 - ζ$. Even though there is no Mumford's representation for superelliptic curves, we give a formula for functions which cut out $D$.

Geometric aspects on Humbert-Edge's curves of type 5, Kummer surfaces and hyperelliptic curves of genus 2

In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-…

Stratifications of the Singular Fibers of Mumford Systems

An integrable system is a dynamic system characterized by the existence of constants of motion and the existence of algebraic invariants, having an origin in algebraic geometry. In the 1970s, Mumford introduced a new completely integrable system defined on a smooth hyperelliptic curve. In the 2000s, Vanhaecke completed the description of the Munford integrable system by defining a Poisson structure on the phase space of the Mumford system. In this article we will study the singular Mumford system. The starting point is to determine when and why the Mumford system is singular. For this we will …

Division by 2 on hyperelliptic curves and jacobians

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)\in C(K)$ there are $2^{2g}$ points $\frac{1}{2}P \in J(K)$. We describe explicitly the Mumford represesentations of all $\frac{1}{2}P$. The rationality question…

Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered.

Non hyperelliptic curves of genus three over finite fields of characteristic two

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non hyperelliptic locus M_3^{nh} of the moduli space M_3 of curves of genus 3. By adding to these computations the knowed results on the hyperelliptic locus we obtain a complete picture of …

具有自同构群 $Q_8$ 的亏格 4 超特殊超椭圆曲线的枚举算法

Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$

本文提出枚举自同构群为四元数群 $Q_8$ 的亏格 4 超特殊超椭圆曲线的算法，并通过 Magma 实现得到特征 $p<10000$ 下的同构类数量，进而提出关于同构类个数与 $p$ 模 8 关系的猜想。

德利尼-里曼-罗奇同构

The Deligne-Riemann-Roch isomorphism

本文通过发展新的陈类范畴和相对交理论，确立了德利尼计划中关于精细格罗滕迪克-里曼-罗奇公式的几何部分，并展示了其在Knudsen-Mumford展开式和格里菲斯丛中的应用。

Computational Aspects of Hyperelliptic Curves

We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the reduced automorphism group of $\p$ has at least two involutions, there exists a representative $C$ of the isomorphism class $\p$ which is defined over $L$.

Cyclic Automorphisms groups of genus 10 non-hyperelliptic curves

In this paper, we list all cyclic automorphisms subgroups $H$ for which there exists a smooth projective non-hyperelliptic sextic curve $C$ with $H\preceq Aut(C)$. Furthermore, we attach to each group a defining equation of a plane sextic curve having exactly this group as cyclic automorphisms subgroup. These equations cover up to isomorphism all plane non-singular sextic curves having some non-trivial automorphism.

导出的德林费尔德-穆尔德堆栈的霍奇-康斯坦-罗斯伯格同构

Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks

本文研究了导出德林费尔德-穆尔德堆栈的霍奇-康斯坦-罗斯伯格同构，建立了其与微分形式上同调之间的自然同构，并扩展了相关同调理论的结果。

The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $Ω\mathcal{M}_{g}$

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of th…

The moduli space of real vector bundles of rank two over a real hyperelliptic curve

The Desale-Ramanan Theorem is an isomorphism between the moduli space of rank two vector bundles over complex hyperelliptic curve and the variety of linear subspaces in an intersection of two quadrics. We prove a real version of this theorem for the moduli space of real vector bundles over a real hyperelliptic curve. We then apply this result to study the topology of the moduli space, proving that it is relatively spin and identifying the diffeomorphism type for genus two curves. Our results lay the groundwork for future study of the quantum homology of these moduli spaces.

Ekedahl-Oort strata of hyperelliptic curves in characteristic 2

Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the $2$-torsion group scheme $J_X[2]$ of the Jacobian of $X$ in terms of the Ekedahl-Oort type. The interesting feature is that $J_X[2]$ depends only on some discrete invariants of $X$, namely, on the ramification invariants associated with the branch points. We give a complete classification of the gr…

The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are t…

热方程与通过自守形式在p-adic Mumford曲线上听出亏格

Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms

本文在p-adic Mumford曲线的K-有理点上构造自伴算子，生成Feller半群，通过热方程谱的极限点对应微分形式零点，从而从谱恢复亏格，超椭圆情形可显式提取。

射影 Deligne–Mumford 栈上半稳定对的模空间

Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

本文把半稳定对模空间构造推广至射影 Deligne–Mumford 栈，研究其形变与障碍理论，并在三维光滑情形定义 PT 不变量。

An effective proof of the hyperelliptic Shafarevich conjecture

Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.

Finiteness of hyperelliptic and superelliptic curves with CM Jacobians

In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of fixed genus $g\geq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.

超椭圆曲线、扫描映射与二次L函数族的矩

Hyperelliptic curves, the scanning map, and moments of families of quadratic L-functions

计算辫子群在应用于积分约化Burau表示的任何Schur函子中的稳定同调，这是带扭曲系数的Mumford猜想的超椭圆类似物，还将结果与二次L函数族矩的猜想联系起来，证实了相关预测。

Fiber products of hyperelliptic curves

Let $k$ be a number field, and let $S$ be a finite set of maximal ideals of the ring of integers of $k$. In his 1962 ICM address, Shafarevich asked if there are only finitely many $k$-isomorphism classes of algebraic curves of a fixed genus $g\ge 1$ with good reduction outside $S$. He verified this for $g=1$ by reducing the problem to Siegel's theorem. Parshin extended this argument to all hyperelliptic curves (cf. also the work of Oort). The general case was settled by Faltings' celebrated work. In this note we give a short proof of Shafarevich's conjecture for hyperelliptic curves, by reduci…

Torsion points of small order on hyperelliptic curves

Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian variety over $K$. Let us consider the canonical embedding of $C$ into $J$ that sends $O$ to the zero of the group law on $J$. This embedding allows us to identify $C(K)$ with a certain subset of the commutative group $J(K)$. A special case of the famous theorem of Raynaud (Manin--Mumford conjecture) asserts that the set of torsion points in $C(K)$ is finite. It i…

亏格4曲线的不变量

Invariants of genus 4 curves

本文利用由transvectants定义的不变量，对特征0代数闭域上位于P³秩3或秩4二次曲面上的非超椭圆亏格4曲线进行显式同构分类，分别给出60个和65个可高效计算的不变量。

Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\m…

双曲曲线的周期-指数问题

Period -Index problem for hyperelliptic curves

本文研究双曲曲线的周期-指数问题，证明了2- torsion元素中指数与幂次相等，并通过同构建立与二次曲面 pencils 的关系，推导出弱Hasse原理。

Local invariants attached to Weierstrass points

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities. We prove an exact formula relating the two invariants with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the sem…

离散周期Toda流的算术方面

Arithmetic aspects of discrete periodic Toda flows

本文通过Mumford的代数描述，提出离散周期Toda流的新线性化方法，并揭示其与周期盒球流的p-进描述及数论的联系。

Counting hyperelliptic curves that admit a Koblitz model

Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})…

Combinatorial Classes, Hyperelliptic Loci, and Hodge Integrals

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstraß points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli space of curves $\mathcal{M}_{g,1}$ and the combinatorial moduli space $\mathcal{M}^{comb}_{g,1}$, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles $W_a\subset\mathcal{M}^{comb}_{g,1}$ whose…

Rosenhain形式的超特殊超椭圆曲线

On superspecial hyperelliptic curves of Rosenhain forms

本文研究特征p≥3的代数闭域上超特殊超椭圆曲线的Rosenhain形式，证明其参数满足平方条件，并提出一种枚举小特征下此类曲线的新算法，成功计算了低亏格曲线的同构类数量。

The moduli space of stable supercurves and its canonical line bundle

We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.

带有Ramond奇点的超黎曼曲面模空间上的测度

A measure on the moduli space of super Riemann surfaces with Ramond punctures

本文利用超Mumford同构与超周期映射，在带Ramond奇点的超黎曼曲面模空间上构造了一个测度。

非奇异性消解与绝对远阿贝尔猜想

Resolution of non-singularities and the absolute anabelian conjecture

本文针对p进局部域上双曲曲线的绝对远阿贝尔猜想，将S. Mochizuki针对准Belyi型曲线的证明结果推广到包含双曲Mumford曲线在内的更广泛曲线族。

关于特征为2的有限域上超椭圆曲线的Weil多项式

On Weil Polynomials of Hyperelliptic Curves over Finite Fields of Characteristic 2

本文通过模2剩余类约束给出特征2有限域上超椭圆雅可比矩阵存在性的新阻碍条件，证明Weil系数奇偶性渐近等分布，并据此排除大量3维和4维同源类，同时提出运行时间显著改进的任意亏格超椭圆曲线枚举算法。

Computing binary curves of genus five

Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in $\mathbb{P}^4$. We present and explain algorithms we used to determine, up to isomorphism over $\mathbb{F}_2$, all genus 5 curves defined over $\mathbb{F}_2$, and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over $\mathbb{F}_2$ of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over $\mathbb{F}_2$ weighted by the size of their $…

Counting isomorphism classes of superspecial curves

A superspecial curve is a (non-singular) curve over a field of positive characteristic whose Jacobian variety is isomorphic to a product of supersingular elliptic curves over the algebraic closure. It is known that for given genus and characteristic, there exist only finitely many superspecial curves, up to isomorphism over an algebraically closed field. In this article, we give a brief survey on results of counting isomorphism classes of superspecial curves. In particular, this article summarizes some recent results in the case of genera four and five, obtained by the author and S.\ Harashita…

Complete Intersections of Two Quadrics and Galois Cohomology

For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.

具有阶为6的循环自同构群的亏格3非超椭圆曲线的$a$-数

The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6

本文研究了具有阶为6循环自同构群的亏格3非超椭圆曲线，给出了其同构的充分必要条件，并利用高斯超几何级数描述Hasse-Witt矩阵，确定了可能的$a$-数及达到最大$a$-数的同构类数量。

Periods of generalized Tate curves

A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing moduli and deformation parameters. By considering each generalized Tate curve as a family of degenerating Riemann surfaces, we give explicit formulas of the period isomorphism between its de Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. A remarkable fact is that similar formulas hold also for families of Mumfor…

Stable log surfaces, admissible covers, and canonical curves of genus 4

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a degeneration of a curve of class $(3,3)$. We prove that the compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of $\math…

Bi-algebraicity in the rank one Riemann--Hilbert correspondence via o-minimality

For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham moduli space' of rank $n$ vector bundles with flat connection on $X$. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are `bi-algebraic' for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We ad…