Vassil Kanev
We prove the irreducibility of the Hurwitz spaces which parametrize Galois coverings of P^1 whose Galois group is an arbitrary Weyl group and the local monodromies are reflections. This generalizes a classical theorem due to Clebsch and Hurwitz.
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We make some observations concerning the Galois actions on Hurwitz curves and on the closely related but lesser-known Hurwitz origamis.
We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched over the disc. Our results addresses the problem to decide equivalence and stable equivalence algebraically. We recover a homological invariant, which we show to distinguish the equivalence classes of given boundary monodromy and Nielsen type, if the latter is sufficiently large in the appropriate sense.
We give a thorough study of Hurwitz stacks both in Galois and non galois case. The construction is applied to revisit somme classical examples, the stack of stable curves equipped with a level structure, and the stacks of tamely ramified cyclic covers. We exhibit some tautological cohomology classes, and give some universal relations between them. Applications are given to Cornalba-Harris type relations, and to some Hurwitz-Hodge integrals.
Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve
该研究针对代数曲线的伽罗瓦覆盖模问题，利用代数拓扑与复代数几何工具，证明了不同条件下Hurwitz空间的精细/粗模簇性质并构造万有族。
Hurwitz spaces and Inverse Galois Theory
本文综述了Hurwitz空间在反 Galois 理论中的五十年研究，重点介绍近期进展，包括大域、紧化、降理论和模塔等半现代发展，以及通过研究组件环重塑Hurwitz空间算术的最新成果。
In this paper we study the reduction of Galois covers of curves, from characteristic 0 to characteristic p. The starting point is a is a recent result of Raynaud which gives a criterion for good reduction for covers of the projective line branch at 3 points. Under some condition on the Galois group, we extend this criterion to the case of 4 branch points. Moreover, we describe the reduction of the Hurwitz space of such covers and compute the number of covers with good reduction.
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.
Moduli spaces for Galois covers of p-adic Mumford curves by Mumford curves are constructed using Herrlich's Teichmuller spaces, Andre's orbifold fundamental groups, and Kato's graphs of groups encoding ramification data of charts for Mumford orbifolds.
We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric differential equation. This generalizes the result of Deligne- Rapoport on the reduction of the modular curve X(p).
We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with full Galois group and surprisingly little ramification --- for example, the existence of infinitely many such number fields unramified away from {2,3,5}.
This article is a continuation of the article with the same title (see arXiv:1003.2953v1). Let {\rm $\text{HUR}_{d,t}^{G}(\mathbb P^1)$} be the Hurwitz space of degree $d$ coverings of the projective line $\mathbb P^1$ with Galois group $\mathcal S_d$ and having fixed monodromy type $t$ consisting of a collection of local monodromy types (that is, a collection of conjugacy classes of permutations $σ$ of the symmetric group $\mathcal S_d$ acting on the set $I_d=\{1,...,d\}$). We prove that if the type $t$ contains big enough number of local monodromies belonging to the conjugacy class $C$ of an…
We settle the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.
We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of distinct points in P^1. We determine the monodromy of pi_1(P_{2g+6}) on the points of the fiber. This turns out to be the same problem as the action of pi_1(P_{2g+6}) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, gcd(3,N)=1, in place of Z/2. This in turn allows us to answer a question of E…
Teichmueller curves are geodesic discs in Teichmueller space that project to algebraic curves $C$ in the moduli space $M_g$. Some Teichmüller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group $G_\QQ$ acts faithfully on the set of these embedded curves. We also compare the action of $G_\QQ$ on $π_1(C)$ with the one on $π_1(M_g)$ and obtain a relation in the Grothendieck-Teichmueller group, seemingly independent of the known ones.
In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type (p,p,...,p) between formal germs of p-adic curves and which generalises the formula proven by the first author in the case of Galois covers of degree p. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type (p,p,...,p) between p-adic schemes.
Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. Given a $\mathbb{Z}/p$-Galois cover of a formal disc over $R$, one can derive from it a semi-stable model for which the specializations of branch points are distinct and lie in the smooth locus of the special fiber. The description leads to a combinatorial object which resembles a classical Hurwitz tree in mixed characteristic, which we will give the same name. The existence of a Hurwitz tree is necessary for the existence of a $\mathbb{Z}/p$-cover whose branching data fit into that tree. We show that the conditions i…
We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application, we give new conditions on the ramification type that are sufficient for irreducibility of the Hurwitz spaces, suggest some bounds on the number of irreducibility components under certain more general conditions, and show that the number of irreducible components coincides with the number of topological classes of the coverings if the number of brunch points i…
Let $π: Z \ra X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply-connected Lie group $G$. For any dominant weight $λ$ consider the curve $Y = Z/\Stab(λ)$. The Kanev correspondence defines an abelian subvariety $P_λ$ of the Jacobian of $Y$. We compute the type of the polarization of the restriction of the canonical principal polarization of $\Jac(Y)$ to $P_λ$ in some cases. In particular, in the case of the group $E_8$ we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety…
A classical and beautiful story in geometric representation theory is the construction by Springer of an action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. We establish a natural extension of Springer's theory to arbitrary symplectic resolutions of conical symplectic singularities. We analyse features of the action in the case of affine quiver varieties, constructing Weyl group actions on the cohomology of $ADE$ quiver varieties, and also consider "symplectically dual" examples arising from slices in the affine Grassmannian. Along the way…
On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface
本文研究任意域上K3曲面皮卡格外尔群的伽罗瓦不变部分到皮卡等距群的限制映射的像是否为正规子群，通过有理数域上的反例给出否定回答。
Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav of dimension 6 is a Prym-Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge-Hurwitz ei…
For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants $ev:C[X_1,\dots ,X_n]^G\rightarrow C(z)$. The procedure proposed here does the opposite: it uses a theorem of E.~Compoint and computes the operator $L$ from a given evaluation $h$. Moreover it solves a part of the inverse problem of producing $L$ for a given representation of a finite group $G$. Another part considered here, is finding irreducible $G$-invaria…
By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H^1(R,G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-tor…
We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formula to obtain an integral formula for these Kostka numbers. This rewrites the ineq…
Galois Groups of Symmetric Cubic Surfaces
本文提出用认证数值计算计算三次曲面族的伽罗瓦群（即27条直线的单值群），应用于$S_5$不变族发现其远小于Weyl群$W(E_6)$，并计算判别式及推广到四次曲面。
Let Y be a normal projective variety and p a morphism from X to Y, which is a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def(X) and Def(Y) are both smooth, of the same dimension, and p induces a finite branched cover f from Def(X) to Def(Y). We prove that f is Galois. We proceed to calculate the Galois group G, when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Na…
We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for w…
We unify results of Artin, Brieskorn, Slodowy and others by showing that, in all characteristics, the Artin component of the deformation space of a rational surface singularity has a ramified cover where simultaneous resolution exists and the Galois group of this cover is the Weyl group determined by the configuration of $(-2)$-curves in the minimal resolution. This verifies a conjecture made by Burns and Rapoport. We use an integral version of this to show that certain actions of Weyl groups on polynomial rings over the integers give rings of invariants that are also polynomial, and deduce th…
We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E_7 or E_8. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over Q for most of the entries of fiber configurations and Mordell-Weil…
In a recent paper Ben-Zvi and Nadler proved that the induction map from $B$-bundles of degree 0 to semistable $G$-bundles of degree 0 over an elliptic curve is a small map with Galois group isomorphic to the Weyl group of $G$. We generalize their result to all connected components of $\Bun_G$ for an arbitrary reductive group $G$. We prove that for every degree (i.e. topological type) there exists a unique parabolic subgroup such that any semistable $G$\nobreakdash-bundle of this degree has a reduction to it and moreover the induction map is small with Galois group the relative Weyl group of th…
Complex reflection groups as differential Galois groups
该研究提出一种新方案，可构造以任意给定复反射群G为微分伽罗瓦群的线性微分方程可积系统，并针对Shephard-Todd分类中多个低秩不可约复反射群明确构造了对应系统。
Conformal blocks for Galois covers of algebraic curves
该文研究有限群Γ作用下带标记轨道的代数曲线的扭曲共形块空间，证明其满足传播与因子分解定理、相关层局部自由并具平坦射影联络，还证实了Pappas-Rapoport猜想，建立扭曲共形块与广义θ函数的等价性。
The aim of this paper is to generalize the notion of conformal blocks to the situation in which the Lie algebra they are attached to is not defined over a field, but depends on covering data of curves. The result will be a sheaf of conformal blocks on the Hurwitz stack parametrizing Galois coverings of curves. Many features of the classical sheaves of conformal blocks are proved to hold in this more general setting, in particular the fusion rules, the propagation of vacua and the WZW connection.
Let $π: Z \to X$ be Galois cover of smooth projective curves with Galois group $W$ a Weyl group of a simple Lie group $G$. For a dominant weight $λ$, we consider the intermediate curve $Y_λ= Z/\Stab(λ)$. One can realise a Prym variety $P_λ\subset \Jac(Y_λ)$ and we denote $φ_λ$ the restriction of the principal polarisation of $\Jac(Y_λ)$ upon $P_λ$. For two dominant weights $λ$ and $μ$, we construct a correspondence $Δ_{λμ}$ on $Y_λ\times Y_μ$ and calculate the pull-back of $φ_μ$ by $Δ_{λμ}$ in terms of $φ_λ$.
Mirror symmetry for extended affine Weyl groups
本文为扩展仿射Weyl群轨道空间上的Frobenius流形提供了统一的Lie理论镜像对称构造，并以此推导了平坦坐标与预势闭式表达式，计算了Lyashko-Looijenga映射的拓扑度，构造了双Hamiltonian层级。
The purpose of this paper is to prove the upper bound in Malle's conjecture on the distribution of finite extensions of $\mathbb{F}_q(t)$ with specified Galois group. As in previous work of Ellenberg-Venkatesh-Westerland, our result is based upon computations of the homology of braid groups with certain (exponential) coefficients. However, the approach in this paper is new, relying on a connection between the cohomology of Hurwitz spaces and the cohomology of quantum shuffle algebras.
Hodge classes on the moduli space of W(E_6)-covers and the geometry of A_6
本文确定 W(E_6)-覆盖 Hurwitz 空间上对应 Weyl 群25个不可约表示的 Hodge 类，并给出 A_6 经 W(E_6) 型 Prym-Tyurin 簇单值化的初等证明。
Landau-Ginzburg models of generalised Dubrovin-Zhang form and pole collision: Dynkin-type A
本文研究了广义Dubrovin-Zhang形式的Landau-Ginzburg模型，通过极点碰撞框架在不同strata中比较Frobenius结构，并证明了预势能的结构结果，验证了Ma和Zuo关于双扩展仿射Weyl群预势能形式的猜想。
We propose two new approaches to the Tannakian Galois groups of holonomic D-modules on abelian varieties. The first is an interpretation in terms of principal bundles given by the Fourier-Mukai transform, which shows that they are almost connected. The second constructs a microlocalization functor relating characteristic cycles to Weyl group orbits of weights. This explains the ubiquity of minuscule representations, and we illustrate it with a Torelli theorem and with a bound for decompositions of a given subvariety as a sum of subvarieties. The appendix sketches a twistor variant that may be …
On monodromy of monodromy surfaces
该研究将Painlevé方程关联的单值群曲面实现为仿射del Pezzo曲面，明确其单值群为对应仿射外尔群的有限部分，还给出Painlevé方程参数空间的模空间解释。
The central heat trace on large compact classical groups
研究大N极限下紧致经典群热核中心迹的渐近展开，利用最高权与划分对应及拉普拉斯-贝尔特拉米算子的稳定性，并建立随机曲面表示，应用于Casimir谱计数和杨-米尔斯/赫维茨对偶。
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