The fundamental group of a Galois cover of CP^1 X T

A cusp singularity with no Galois cover by a complete intersection

With an explicit example, we confirm a conjecture by Neumann and Wahl that there exist cusps with no Galois cover by a complete intersection. Some computational techniques are reviewed, and a method for deciding whether a given cusp has a complete intersection Galois cover is developed.

Hurwitz spaces of Galois coverings of P^1 with Galois groups Weyl groups

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.

On metabelian galois coverings of complex algebraic varieties

Let $f:X\to Y$ be a finite ramified Galois covering of algebraic varieties defined over the complex numbers. In this paper, we prove some structure theorems for such coverings in the case that the non-abelian Galois group of the cover is metabelian (in particular metacyclic). Our results extend the previous ones, obtained by several authors, in the case of abelian and dihedral Galois coverings. In analogy with the abelian (and generalizing the dihedral) case, we find "building data" of metabelian covers from which the cover can be reconstructed. In addition to analyzing several examples, we co…

Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors

We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack $X$ over a field $k$. This is done by describing the Nori fundamental gerbe of an essentially finite cover of $X$. A similar result is also obtained for the $S$-fundamental gerbe.

Remarks on Galois rational covers

In this note we improve the theorem on Galois rational covers $X\dashrightarrow V$ for primitive Fano varieties $V$, recently proven by the author, in the two directions: we extend to the maximum the class of Galois groups $G$, for which the proof works, and relax the conditions that must be satisfied by the variety $V$ -- the divisorial canonicity alone is sufficient.

关于$R_k$型Zappatic曲面并集的伽罗瓦覆盖

On Galois covers of a union of Zappatic surfaces of type $R_k$

本文研究两个$R_k$型Zappatic曲面并集的伽罗瓦覆盖，证明其是一般型单连通曲面，并计算了陈数与拓扑指标。

Ramified Galois covers via monoidal functors

We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and étale group schemes and we prove that, if $G$ is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of $G$-covers is reducible.

Purely Inseparable Galois theory I: The Fundamental Theorem

We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).

五重覆盖的伽罗瓦闭包及其雅可比簇的分解

Galois closure of a 5-fold covering and decomposition of its Jacobian

本文针对紧黎曼曲面间的五重分歧覆盖，利用分歧数据确定所有伽罗瓦闭包，并通过中间覆盖的雅可比簇与 Prym 簇给出其群代数分解，同时计算各阿贝尔簇的维数和诱导极化。

关于极小次数曲面退化的 Galois 覆盖

On the Galois covers of degenerations of surfaces of minimal degree

本文研究复射影空间中极小次数曲面的 Galois 覆盖拓扑，并证明 n≥5 时这些覆盖均为单连通的一般类型曲面。

Finite groups acting on coherent sheaves and Galois covers

Let G be a finite group and ρ: G--> End(E) be a group representation of G on a coherent sheaf over an integral scheme. The purpose of this paper shall give a decomposition theorem of such representations in non-splitting components and apply this results to the studie of Galois covers

Non-constant curves of genus 2 with infinite pro-Galois covers

For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.

The fundamental group of Galois cover of the surface ${\mathbb T} \times {\mathbb T}$

This is the final paper in a series of four, concerning the surface $T \times T$ embedded in $\mathbb{CP}^8$, where $T$ is a the one dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto $\mathbb{CP}^2$, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

Calabi-Yau流形的Galois覆盖

Galois covers of Calabi-Yau manifolds

本文研究了Calabi-Yau流形之间的Galois有理映射，探讨了其birational几何和Galois闭包，并应用于典型例子。

On the existence of a torsor structure for Galois covers over a complete discrete valuation ring

In this note we investigate the problem of existence of a torsor structure for Galois covers of (formal) schemes over a complete discrete valuation ring of residue characteristic $p>0$ in the case of abelian Galois groups of type (p,p,...,p).

Automorphisms of Galois Coverings of Generic $m$-Canonical Projections

The automorphism group of the Galois covering induced by a pluri-canonical generic covering of a projective space is investigated. It is shown that by means of such coverings one obtains, in dimensions one and two, serieses of specific actions of the symmetric groups $S_d$ on curves and surfaces not deformable to an action of $S_d$ which is not the full automorphism group. As an application, new DIF $\ne$ DEF examples for $G$-varieties in complex and real geometry are given.

直线的伽罗瓦覆盖族的一种拓扑构造

A topological construction of families of Galois covers of the line

本研究提出直线伽罗瓦覆盖族的新拓扑构造，以配置空间性质、覆盖理论和Grauert-Remmert延拓定理为基础，替代了既有构造并修正了非阿贝尔伽罗瓦群情形下的不准确之处。

On the arithmetic Galois covers of higher relative dimensions

In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.

Fano-Mori 纤维空间的有理连通 Galois 覆盖

Rationally connected Galois covers of Fano-Mori fibre spaces

本文证明了在特定稳定性与扭转条件下，Fano-Mori 纤维空间上的有理连通 Galois 循环有理覆盖必从底拉回，并据此给出 $V$ 不含此类覆盖的判据。

Moduli of Galois p-covers in mixed characteristics

We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to \cX$.

Foliations and webs inducing Galois coverings

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a com…

Zappatic稳定曲面的形变及其Galois覆盖

Deformations of Zappatic stable surfaces and their Galois covers

本文研究可形变为具有唯一En型奇点的平面Zappatic稳定曲面的代数曲面，证明其Galois覆盖在n≥4时为一般型单连通曲面并给出Chern数公式，进而证明n>30时此类曲面不存在。

Galois covers of type (p,...,p), vanishing cycles formula, and the existence of torsor structures

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.

具有八面体包络的曲面的伽罗瓦覆盖的基本群

The fundamental group of Galois covers of surfaces with octahedral envelope

本文计算了具有4次奇点、退化包络同构于八面体的8次曲面的伽罗瓦覆盖的基本群，确定其为2³³阶亚交换群，通过融合局部群的迭代方法完成计算，并通过三个简化曲面说明非平凡循环对计算的影响。

On the existence of a torsor structure for Galois covers

Let $R$ be a complete discrete valuation ring with residue characteristic $p>0$. In this note we give an example of a Galois cover $f:Y\to X$ between flat and normal formal $R$-schemes of finite type which is étale above the generic fibre of $X$ such that the special fibre of $Y$ is reduced and such that $f$ doesn't have the structure of a torsor under a finite and flat $R$ group scheme. In the example $R$ has equal characteristic $p$ and the Galois group of the cover is cyclic of order $p^2$. In the example the formal scheme $X$ is affine and can be choosen to be even smooth.

On the topology of the complements of reducible plane curves via Galois covers

Let ${\mathcal {B}}$ be a reducible reduced plane curve. We introduce a new point of view to study the topology of $(\PP^2, {\mathcal {B}})$ via Galois covers and Alexander polynomials. We show its effectiveness through examples of Zariski $N$-plets for conic and conic-quartic configurations.

On some Galois covers of the Suzuki and Ree curves

We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. We also show that $\tilde{\mathcal{S}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^4}$, and $\tilde{\mathcal{R}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^6}$. Finally, we compute the genera of many Galois subcovers of $\tilde{\mathcal{…

Mackey-functor structure on the Brauer groups of a finite Galois covering of schemes

Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite étale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object "Brauer-Mackey functor". We realize this as a {\it cohomological Mackey functor} on the Galois category of finite étale coverings. For any finite étale covering of schemes, we can associate two homomorphisms for Brauer groups, namely the pull-back and the norm map. These homomorphisms make Brauer groups into a bivariant functor ($=$ Mackey functor) on the Galois…

On Spaces Associated with Invariant Divisors on Galois Covers of Riemann Surfaces and Their Applications

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.

Fake Liftings of Galois Covers between Smooth Curves

In this paper we investigate the problem of lifting of Galois covers between algebraic curves from characteristic p>0 to characteristic 0. We prove a refined version of the main result of Garuti concerning this problem in [Ga]. We formulate a refined version of the Oort conjecture on liftings of cyclic Galois covers between curves. We introduce the notion of fake liftings of cyclic Galois covers between curves, their existence would contradict the Oort conjecture, and we study the geometry of their semi-stable models. Finally, we introduce and investigate on some examples the smoothening proce…

Dihedral Galois covers of algebraic varieties and the simple cases

In this article we investigate the algebra and geometry of dihedral covers of smooth algebraic varieties. To this aim we first describe the Weil divisors and the Picard group of divisorial sheaves on normal double covers. Then we provide a structure theorem for dihedral covers, that is, given a smooth variety Y, we describe the algebraic "building data" on Y which are equivalent to the existence of such covers X --> Y. We introduce then two special very explicit classes of dihedral covers: the simple and the almost simple dihedral covers, and we determine their basic invariants. For the simple…

射影空间的伽罗瓦自覆盖与本质维数

Galois self-covers of projective spaces and essential dimensions

分类射影空间的伽罗瓦自覆盖，并证明每个非平凡自覆盖的本质维数达到最大值n。

Classification of quadruple Galois canonical covers II

In this article we classify quadruple Galois canonical covers $ϕ$ of singular surfaces of minimal degree. This complements the work done in math.AG/0302045, so the main output of both papers is the complete classification of quadruple Galois canonical covers of surfaces of minimal degree, both singular and smooth. Our results show that the covers $X$ studied in this article are all regular surfaces and form a bounded family in terms of geometric genus $p_g$. In fact, the geometric genus of $X$ is bounded by 4. Together with the results of Horikawa and Konno for double and triple covers, a stri…

Inverse Galois problem for ordinary curves

We consider the inverse Galois problem over function fields of positive characteristic p, for example, the inverse Galois problem over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, which are ordinary in the sense that their Jacobian has maximal p-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers.

A note on splitting numbers for Galois covers and $π_1$-equivalent Zariski $k$-plets

In this paper, we introduce \textit{splitting numbers} of subvarieties in a smooth variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. By splitting numbers, we give a necessary and sufficient condition for two plane curves of type $(b,m)$ to be topologically equivalent as pairs of the complex projective plane and plane curves, where a plane curve of type $(b,m)$ is an arrangement of two smooth plane curves of degree $3$ and $b$ defined by I.~Shimada. Consequently, we prove that there are $π_1$-equivalent Zariski $k$-plets for any $k\geq2…

Galois branched covers with fixed ramification locus

We examine conditions under which there exists a non-constant family of Galois branched covers of curves over an algebraically closed field $k$ of fixed degree and fixed ramification locus, under a notion of equivalence derived from considering linear series on a fixed smooth proper source curve $X$. We show such a family exists precisely when the following conditions are satisfied: $\operatorname{char}(k)=p>0$, $X$ is isomorphic to $\mathbb{P}^1_k$, there is a unique ramification point, and the Galois group is $(\mathbb{Z}/p\mathbb{Z})^m$ for some integer $m>0$.

Some remarks on cyclic Galois coverings of the projective line over finite fields

We study cyclic finite Galois extensions of the rational function field of the projective line P^{1}(F_q) over a finite field F_q with q elements defined by considering quotient curves by finite subgroups of the projective linear group PGL(2,q), and we enumerate them expressing the count in terms of Stirling numbers.

A note on embeddings of $S_4$ and $A_5$ into the Cremona group and versal Galois coverings

In this paper we introduce two versal $S_4$ coverings, and show that the two are birationaly distinct, i.e. there exists no $S_4$ equivariant birational map between them. We also introduce two versal $A_5$ coverings and show that the two are birationaly distinct also. As a corollary we obtain that there are at least three non-conjugate embbedings of $S_4$ (resp. $A_5$) into $Cr_2(\mathbb{C})$, the Cremona group.

关于曲线及其族的 Galois 覆盖

On Galois Coverings of curves and their Families

本文利用结构态射对层前推关联的局部系统描述代数曲线及其族的Galois覆盖，通过Galois群作用将层分解为不可约局部系统以刻画群在同调群上的作用，并详细研究了$¶^1$的二面体覆盖。

Branch Stabilisation for the Components of Hurwitz Moduli Spaces of Galois Covers

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.

The fields of definition of branched Galois covers of the projective line

In this paper I explore the structure of the fields of definition of Galois branched covers of the projective line over \bar Q. The first main result states that every mere cover model has a unique minimal field of definition where its automorphisms are defined, and goes on to describe special properties of this field. One corollary of this result is that for every G-Galois branched cover there is a field of definition which is Galois over its field of moduli, with Galois group a subgroup of Aut(G). The second main theorem states that the field resulting by adjoining to the field of moduli all…

Semi-direct Galois covers of the affine line

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $ψ:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.

Shimura subvarieties in the Prym locus of ramified Galois coverings

We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

Decomposing Jacobians via Galois covers

Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genu…

Classification of quadruple Galois canonical covers I

In this article we classify quadruple Galois canonical covers of smooth surfaces of minimal degree. The classification shows that they are either non-simple cyclic covers or bi-double covers. If they are bi-double then they are all fiber products of double covers. We construct examples to show that all the possibilities in the classification do exist. There are implications of this classification that include the existence of families with unbounded geometric genus and families with unbounded irregularity, in sharp contrast with the case of double and triple canonical covers. Together with the…

Tannakian Fundamental Groups Associated to Galois Groups

This should be the final version of this paper. Numberous minor improvements have been made to the manuscript, one argument has been corrected, and an appendix has been added.

Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P^1 - {0,1,infinity}

This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number $ł$. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-$ł$ completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted …

$p$-group Galois covers of curves in characteristic $p$

We study cohomologies of a curve with an action of a finite $p$-group over a field of characteristic $p$. Assuming the existence of a certain 'magical element' in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic $p$-group cover has a 'magical element'. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.

Galois structure of Zariski cohomology for weakly ramified covers of curves

We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani's computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case.

算术概型的伽罗瓦共变式与基本群

Galois coinvariants and fundamental groups of arithmetic schemes

该研究针对有限生成域上的几何平展基本群与平展上同调，建立特定伽罗瓦扩张下共变式的有限性性质，进而得到数域上算术概型平展基本群的有限性结果。

Field of moduli and field of definition of Galois covers

In this paper we investigate the cohomological obstruction for the field of moduli of a $G$-cover to be a field of definition, in the case of local fields and covers with tame admissible reduction. This applies in particular to $p$-adic fields where $p$ does not divide the order of the group $G$. We give examples of $G$-covers with field of moduli $\QQ_p$ that cannot be defined over $\QQ_p$, for all primes $p>5$.

Classification of Fundamental Groups of Galois Covers of Surfaces of Small Degree Degenerating to Nice Plane Arrangements

Let $X$ be a surface of degree $n$, projected onto $\mathbb{CP}^2$. The surface has a natural Galois cover with Galois group $S_n.$ It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of $X.$ In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree $n \leq 4$, that degenerate to a nice plane arrangement, namely a union of $n$ planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree $4$ em…

Nodal degenerations of plane curves and Galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois coverings.

Galois覆盖的特化的算术下降

Arithmetic descent of specializations of Galois covers

本文研究数域上Galois覆盖的特化能否算术下降到Q的问题，证明了允许基变换时在G正则可实现条件下答案通常为肯定，不允许基变换时循环群情形为肯定，并给出一个无反例的显式构造。