This paper considers some algebraic surfaces that can deform to planar Zappatic stable surfaces with a unique singularity of type En. We prove that the Galois covers of these surfaces are all simply connected of general type, for n >= 4, and we give a formula for Chern numbers of such Galois covers. As an application, we prove that such surfaces do not exist for n>30. Furthermore, Kollar improves the result to n>9 in Appendix 5.
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…
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.
In this paper we consider the Prym variety $P(\widetilde{C}/C)$ associated to a Galois coverings of curves $f:\widetilde{C}\to C$ branched at $r$ points. We discuss some properties and equivalent definitions and then consider the Prym map $\mathcal{P}=\mathcal{P}(G,g,r):R(G,g,r)\to A_{p,δ}$ with $δ$ the type of the polarization. For Galois coverings whose Galois group is abelian and metabelian (non-abelian) we show that the differential of this map at certain points is injective. We also consider the Abel-Prym map $u:\widetilde{C}\to P(\widetilde{C}/C)$ and prove some results for its injectivi…
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.
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…
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.
In this paper we study the Galois group of the Galois cover of the composition of a $q$-cyclic étale cover and a cyclic $p$-gonal cover for any odd prime $p$. Furthermore, we give properties of isogenous decompositions of certain Prym and Jacobian varieties associated to intermediate subcovers given by subgroups.
This paper computes the Galois group of the Galois cover of the composition of an étale double cover of a cyclic $p$-gonal cover for any prime $p$. Moreover a relation between some of its Prym varieties and the Jacobian of a subcover is given. In a sense this generalizes the trigonal construction.
We study $p$-group Galois covers $X \rightarrow \mathbb{P}^1$ with only one fully ramified point. These covers are important because of the Katz-Gabber compactification of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$--rank: we focus on maximal curves and curves that admit a big action. Moreover the Galois module structure of polydifferentials is…
In this work we study the monodromy group of covers $φ\circ ψ$ of curves \linebreak $\mathcal{Y}\xrightarrow {\quad ψ} \mathcal{X} \xrightarrow {\quad φ} \mathbb{P}^{1}$, where $ψ$ is a $q$-fold cyclic étale cover and $φ$ is a totally ramified $p$-fold cover, with $p$ and $q$ different prime numbers with $p$ odd. We show that the Galois group $\mathcal{G}$ of the Galois closure $\mathcal{Z}$ of $φ\circ ψ$ is of the form $ \mathcal{G} = \mathbb{Z}_q^s \rtimes \mathcal{U}$, where $0 \leq s \leq p-1$ and $\mathcal{U}$ is a simple transitive permutation group of degree $p$. Since the simple transi…
We study triple covers of K3 surfaces, following Miranda's theory of triple covers. We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vec…
The aim of this paper is to construct "special" isogenies between K3 surfaces, which are not Galois covers between K3 surfaces, but are obtained by composing cyclic Galois covers, induced by quotients by symplectic automorphisms. We determine the families of K3 surfaces for which this construction is possible. To this purpose we will prove that there are infinitely many big families of K3 surfaces which both admit a finite symplectic automorphism and are (desingularizations of) quotients of other K3 surfaces by a symplectic automorphism. In the case of involutions, for any $n\in\mathbb{N}_{>0}…
In this thesis, we study the Berkovich skeleton of an algebraic curve over a discretely valued field $K$. We do this using coverings $C\rightarrow{\mathbb{P}^{1}}$ of the projective line. To study these coverings, we take the Galois closure of the corresponding injection of function fields $K(\mathbb{P}^{1})\rightarrow{K(C)}$, giving a Galois morphism $\overline{C}\rightarrow{\mathbb{P}^{1}}$. A theorem by Liu and Lorenzini tells us how to associate to this morphism a Galois morphism of semistable models $\mathcal{C}\rightarrow{\mathcal{D}}$. That is, we make the branch locus disjoint in the s…
