Suppose $ϕ$ is a wildly ramified cover of germs of curves defined over an algebraically closed field of characteristic p. We study unobstructed deformations of $ϕ$ in equal characteristic, which are equiramified in that the branch locus is constant and the ramification filtration is fixed. We show that the moduli space $M_ϕ$ parametrizing equiramified deformations of $ϕ$ is a subscheme of an explicitly constructed scheme. This allows us to give an explicit upper and lower bound for the Krull dimension $d_ϕ$ of $M_ϕ$. These bounds depend only on the ramification filtration of $ϕ$. When $ϕ$ is an abelian p-group cover, we use class field theory to show that the upper bound for $d_ϕ$ is realized.
The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus. As …
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…
In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 3$ for which the canonical map induces a triple cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$ or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ is again canonical and induces a triple cover. The extremal case when $Y$ is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. This is especially interesting as well, since it has n…
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. We then apply this result to obtain a positive-characteristic Brill-Noether-type result for ramified maps from general curves to the projective line, which even holds for wild ramification indices. Lastly, in the special case of rational functions on the projective line, we examine what we can say as a result about fami…
In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 2$, for which its canonical map induces a double cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$, or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ again is canonical and again induces a double cover. The second part of the article deals with the existence or non existence of canonical double structures on rational varieties. The negative result in this article has consequence…
By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties. In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group $G$ there exist infinitely many components $M$ of the moduli of varieties with ample canonical class such that the generic automorphism group $G_M$ is equal to $G$. In order to construct the examples, we use abelian covers, i.e. Galois cover whose Galois group is …
The configuration space of $n$ marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It is proved that this decomposition is a semi-algebraic topological stratification. It also forms a cell decomposition of the configuration space of $n$ marked points. Moreover, we prove that classical tools from deformation theory, ruled by a Maurer--Cartan equation, can be used only locally for Gauss-skizze. We prove that the deformation of the Gauss-skizze is governed by a Hamilton-…
Let M_m be the generic fibre of the formal deformation space of a one-dimensional formal module X of finite height together with level-m-structures. We show that it is defined over the mth Lubin-Tate extension of the ground field and that it is geometrically connected over this field. The action of the covering group of M_m over M_0 on the set of geomtrically connected components is calculated as well as the action of the action of the automorphism group of X.
This paper is concerned with a refinement of the Stein factorization, and with applications to the study of deformations of surjective morphisms. We show that every surjective morphism f:X->Y between normal projective varieties factors canonically via a finite cover of Y that is etale in codimension one. This "maximally etale factorization" is characterized in terms of positivity of the push-forward of the structure sheaf and satisfies a functorial property. It turns out that the maximally etale factorization is stable under deformations, and naturally decomposes an etale cover of the Hom-sche…
In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K_X^2 = 4p_g(X)-8$, for any even integer $p_g\geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $φ:X\to \mathbb{P}^N$, where $φ$ is Galois of degree 4. These canonical covers are classified in by the first two authors into four distinct families. We show that any deformation of $φ$ factors through a double cover of a ruled surface and, hence, is never birational. More interestin…
The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very ample complete linear series. Among other things, we prove that any deformation of the canonical morphism of such surfaces $X$ is again a morphism of degree 2. A priori, this is not at all obvious, for the invariants $(p_g(X),c_1^2(X))$ of most of these surfaces lie on or above the Castelnuovo line; thus, in principle, a deformation of such $X$ could have a…
We construct a two-level weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local Gromov-Witten theory of curves of Bryan-Pandharipande. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized projective line. The Frobenius Algebras we obtain are one parameter deformations of the class algebra of the symmetric group S_d. In certain special cases we are able to produce explicit closed formulas for such def…
We construct a 4-dimensional family of surfaces of general type with p_g=0 and K^2=3 and fundamental group Z/2xQ_8, where Q_8 is the quaternion group. The family constructed contains the Burniat surfaces with K^2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on (\PP^1)^4 and we also give an action of Z/2xQ_8 on (\PP^1)^4 lifting the natural action on the surfaces. The strategy is the following. We consider an étale (Z/2)^3-cover T of a surface with p_g=0 and K^2=3 and assume that it may be embedded in a Fano 3-fold V. We construct…
We construct a deformation family for each of the 34 Hilbert series of index 2 Fano 3-folds. In 18 cases we construct two different families, distinguished by the topology of their general members.
In this paper, we investigate the possibility of constructing isomonodromic deformations of logarithmic connections on curves by using ramified covers. We give new examples and prove a classification result.
Let X be a smooth projective curve over a field of characteristic p>0 and G a finite group of automorphism of X. Let n(X,G) be the characteristic of the versal equivariant deformation ring R(X,G) of (X,G). When the ramification is weak (i.e. all second ramification groups are trivial),we prove that n(X,G) is 0 or p and we compute R(X,G).
We study branched covers of curves with specified ramification points, under a notion of equivalence derived from linear series. In characteristic 0, no non-constant families of covers with fixed ramification points exist. In positive characteristic we formulate a necessary and sufficient condition for the existence of such a family. We unconditionally prove one direction of this conjecture, and by studying infinitesimal deformations show the other direction in characteristic 2 and 3.
A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and $H$ an ample divisor of square 2 for Beauville's quadratic form. If $n=2$ then according to the conjecture $X$ is a double cover of a (singular) sextic 4-fold in $\PP^5$. It follows from the conjecture that a deformation of $(K3)^{[n]}$ carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture…
Inspired by a string duality, we construct a deformation family for $G_2$-orbifolds given as total spaces of coassociative fibrations by ADE singularities over a closed and oriented smooth three-manifold $Q$. The deformations are parametrized by sections of a fiber bundle on $Q$ that can be interpreted as spectral/cameral covers associated to certain Riemannian analogs of Higgs bundles. The spectral cover picture is a unifying framework for several different approaches to coassociative fibrations appearing in the literature. Our construction generalizes, to the context of $G_2$-geometry, a wel…
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…
Using the developed deformation theory on moduli spaces of quadratic differentials we derive variational formulas for objects associated with generalized $SL(2)$ Hitchin's spectral covers: Prym matrix, Prym bidifferential, Hodge and Prym tau-functions. The resulting formulas are antisymmetric versions of Donagi-Markman residue formula. The second variation of the Prym matrix is a natural analogy to the expression previously derived for the period matrix of the $GL(n)$ spectral cover.
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…
Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is an irreducible symplectic 4-fold numerically equivalent to the Hilbert square of…
