# Mixed quasi-Ã©tale surfaces and new surfaces of general type

Frapporti, Davide (2012) Mixed quasi-Ã©tale surfaces and new surfaces of general type. PhD thesis, University of Trento.

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## Abstract

In this thesis we define and study the mixed quasi-Ã©tale surfaces. In particularwe classify all the mixed quasi-Ã©tale surfaces whose minimal resolution of the singularities is a regular surface with p_g=0 and K^2>0. It is a well known fact that each Riemann surface with p_g=0 is isomorphic to P^1. At the end of XIX century M. Noether conjectured that an analogous statement holds for the surfaces: in modern words, he conjectured that every smooth projective surface with p_g=q=0 be rational. The first counterexample to this conjecture is due to F. Enriques (1869). He constructed the so called Enriques surfaces. The Enriques-Kodaira classification divides compact complex surfaces in four main classes according to their Kodaira dimension k: -oo, 0, 1, 2. A surface is said to be of general type if k=2. Nowadays this class is much less understood than the other three. The Enriques surfaces have k=0. The first examples of surfaces of general type with p_g=0 have been constructed in the 30's by L. Campedelli e L. Godeaux. The idea of Godeaux to construct surfaces was to consider the quotient of simpler surfaces by the free action of a finite group. In this spirit, Beauville proposed a simple construction of surfaces of general type, considering the quotient of a product of two curves C_1 and C_2 by the free action of a finite group G. Moreover he gave an explicit example with p_g=q=0 considering the quotient of two Fermat curves of degree 5 in P^2. There is no hope at the moment to achieve a classification of the whole class of the surfaces of general type. Since for a surface in this class the Euler characteristic of the structure sheaf \chi is strictly positive, one could hope that a classification of the boundary case \chi=1 is more affordable. Some progresses in this direction have been done in the last years through the work of many authors, but this (a priori small) case has proved to be very challenging, and we are still very far from a classification of it. At the same time, this class of surfaces, and in particular the subclass of the surfaces with p_g=0 contains some of the most interesting surfaces of general type. If S is a surface of general type with \chi=1, which means p_g=q, then p_g = q < 5, and if p_g=q=4, then S is birational to the product of curves of genus 2. The surfaces with p_g = q = 3 are completely classified. The cases p_g = q < 3 are still far from being classified. Generalizing the Beauville example, we can consider the quotient (C_1 x C_2)/G, where the C_i are Riemann surfaces of genus at least two, and G is a finite group. There are two cases: the mixed case where the action of G exchanges the two factors (and then C_1 = C_2); and the unmixed case where G acts diagonally. Many authors studied the surfaces birational to a quotient of a product of two curves, mainly in the case of surfaces of general type with \chi=1. In all these works the authors work either in the unmixed case or in the mixed case under the assumption that the group acts freely. The main purpose of this thesis is to extend the results and the strategies of the above mentioned papers in the non free mixed case. Let C be a Riemann surface of genus at least 2, let G be a finite group that acts on C x C with a mixed action, i.e. there exists an element in G that exchanges the two factors. Let G^0 be the index two subgroup of the elements that do not exchange the factors. We say that X=(C x C)/G is a mixed quasi-Ã©tale surface if the quotient map C x C -> (C x C)/G has finite branch locus. We present an algorithm to construct regular surfaces as the minimal resolution of the singularities of mixed quasi-Ã©tale surfaces. We give a complete classification of the regular surfaces with p_g=0 and K^2>0 that arise in this way. Moreover we show a way to compute the fundamental group of these surfaces and we apply it to the surfaces we construct. Some of our construction are more interesting than others. We have constructed two numerical Campedelli surfaces (K^2 = 2) with topological fundamental group Z/4Z. Two of our constructions realize surfaces whose topological type was not present in the literature before. We also have three examples of Q-homology projective planes, and two of them realize new examples of Q-homology projective planes.

Item Type: Doctoral Thesis (PhD) Mathematics XXIV Area 01 - Scienze matematiche e informatiche > MAT/03 GEOMETRIA 01 Mar 2012 09:33

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