arXiv subjects

Ingrid Bauer

Publications and source records attributed to Ingrid Bauer.

34 records · Page 2Linked to original sources

arXiv2011

The classification of minimal product-quotient surfaces with $p_g=0$

A product-quotient surface is the minimal resolution of the singularities of the quotient of a product of two curves by the action of a finite group acting separately on the two factors. We classify all minimal product-quotient surfaces of general type with geometric genus 0: they form 72 families. We show that there is exactly one product-quotient surface of general type with big canonical class which is not minimal, and describe its (-1) curves. For all these surfaces the Bloch conjecture holds.

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arXiv2011

The moduli space of Keum-Naie surfaces

Using a new description of Keum Naie surfaces and their fundamental group, we prove the following main result: Let S be a smooth complex projective surface which is homotopically equivalent to a Keum - Naie surface. Then S is a Keum - Naie surface. The connected component of the Gieseker moduli space corresponding to Keum - Naie surfaces is irreducible, normal, unirational of dimension 6.

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arXiv2010

Burniat surfaces III: deformations of automorphisms and extended Burniat surfaces

We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with K_S^2 = 4, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with K_S^2 = 4, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=4 form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with K_S^2=3 form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms G. In fact, we show that for the minimal model S of a nodal Burniat surface (G = (\ZZ/2 \ZZ)^2) we have Def(S,G) \neq Def(S), whereas for the canonical model X it holds Def(X,G) = Def(X). All deformations of S have a G-action, but there are different deformation types for the pairs (S,G) of the minimal models S together with the G-action, while the pairs (X,G) have a unique deformation type.

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arXiv2010

Surfaces of general type with geometric genus zero: a survey

In the last years there have been several new constructions of surfaces of general type with $p_g=0$, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with $p_g=0$, and gives an updated list of the existing surfaces, in the case where $K^2= 1,...,7$. It also focuses on certain important aspects of this classification.

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arXiv2009

Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space

We prove in one go that each of the 4 families of Burniat surfaces with K^2 = 6,5,4, is a connected component of the moduli space of surfaces of general type. We prove also the rationality of each component. In the nodal case (one of the two families for K^2_S = 4) a very surprising and new phenomenon occurs. Both the moduli space for the minimal models and the Gieseker moduli space for canonical models are everywhere non reduced. But the nilpotence order is higher for the first.

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arXiv2009

Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups

The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.

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arXiv2009

Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces

This is the first in a series of articles on the moduli spaces of Burniat surfaces. We prove that Burniat surfaces are Inoue surfaces and we calculate their fundamental groups. As a main result of this paper we prove that every smooth projective surface which is homotopically equivalent to a primary Burniat surface (i.e., a Burniat surface with K^2 = 6) is indeed a primary Burniat surface.

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arXiv2009

The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then follows from Dolgachev's theorem that M is rational.

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arXiv2008

The rationality of certain moduli spaces of curves of genus 3

We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, η) where C has genus 3 and ηis a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <η>) as above and where <η> is the cyclic subgroup of order 3 in Pic_0(C) generated by η.

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arXiv2007

Surfaces with K^2=8, p_g=4 and canonical involution

In this paper we classify completely all regular minimal surfaces with K^2=8, p_g=4 whose canonical map is composed with an involution. We obtain six unirational families of respective dimensions 28,28,32,33,38,34. The last two are irreducible components of the moduli space of minimal surfaces with K^2=8, p_g=4. These families hit three different topological types.

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arXiv2007

The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type

We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $σ\in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^σ$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting preprint by Robert Easton and Ravi Vakil (\cite{e-v}), where it is proven, with a completely different type of examples, that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of irreducible components of the moduli spaces of surfaces of general type. We also give other simpler examples of surfaces with nonisomorphic fundamental groups which are Galois conjugate, hence have isomorphic algebraic fundamental groups.

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arXiv2006

The classification of surfaces with p_g = q = 0 isogenous to a product of curves

We classify all the surfaces with p_g = q = 0 which admit an unramified covering which is isomorphic to a product of curves. Beyond the trivial case \PP^1 x \PP^1 we find 17 families which we explicitly describe. We reduce the problem to a combinatorial description of certain generating systems for finite groups which we solve using also MAGMA's library of groups of small order.

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arXiv2006

A volume maximizing canonical surface in 3-space

Answering a question posed by Enriques, we construct a minimal smooth algebraic surface $S$ of general type over the complex numbers with $K^2 = 45$ and $p_g = 4$, and with birational canonical map. Our surface is a regular (q=0) ball quotient which is an etale quotient of a Hirzebruch covering of the plane. The canonical system $|K_S|$ has a fixed part and the degree of the canonical image is 19.

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arXiv2006

Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory

We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck's program of 'Dessins d' enfants', aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups.

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arXiv2006

Complex surfaces of general type: some recent progress

Chapters : Old and new inequalities; Surfaces with $χ=1$ and the bicanonical map; Surfaces with $p_g=4$; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF and other equivalence relations.

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arXiv2004

Beauville surfaces without real structures, I

Inspired by a construction by Arnaud Beauville of a surface of general type with $K^2 = 8, p_g =0$, the second author defined the Beauville surfaces as the surfaces which are rigid, i.e., they have no nontrivial deformation, and admit un unramified covering which is isomorphic to a product of curves of genus at least 2. In this case the moduli space of surfaces homeomorphic to the given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces. It may also happen that a Beauville surface is biholomorphic to its complex conjugate surface, neverless it fails to admit a real structure. First aim of this note is to provide series of concrete examples of the second situation, respectively of the third. Second aim is to introduce a wider audience, especially group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them.

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