arXiv:1302.3717v2 [math.AG] 19 Nov 2013 MIXED QUASI- ´ ETALE QUOTIENTS WITH ARBITRARY SINGULARITIES DAVIDE FRAPPORTI, ROBERTO PIGNATELLI Abstract. A mixed quasi-´ etale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-´ etale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-´ etale sur- faces with given geometric genus, irregularity, and self-intersection of the canonical class. We prove that all irregular mixed quasi-´ etale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi ´ etale surfaces of general type with genus equal to the irregularity, and all the regular ones with K 2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with p g = q = 1 and Albanese fibre of genus bigger than K 2 . Introduction In the last decade, after the seminal paper [Cat00], there has been growing interest in those surfaces birational to the quotient of the prod- uct of two curves of genus at least 2 by the action of a subgroup of its automorphism group. These have shown to be a very productive source of examples, espe- cially in the very interesting and still mysterious case of the surfaces of general type with χ(S ) = 1 (equivalently p g (S )= q (S )). Here and in the following we use the standard notation of the theory of the complex surfaces, as in [Bea78, BHPV04]. For motivation and for the state of the art (few years ago) of the research on the surfaces of general type with p g = q = 0 we suggest to the reader the survey [BCP11], while some information on the more general case χ(S ) = 1 can be found in Date : November 20, 2013. 2000 Mathematics Subject Classification. 14J29, 58E40, 14Q10 . Key words and phrases. Surfaces of general type, finite group actions. Some of the results in this paper were developed in fall 2012, when the first author was supported by the Department of Mathematics of the University of Trento. The second author was partially supported by the FCT Project PTDC/MAT/111332/2009 Moduli Spaces in Algebraic Geometry, by the PRIN 2010-2011 Project Geometria delle variet` a algebriche and by the FIRB 2012 Project Spazi di moduli e applicazioni. Both authors are members of G.N.S.A.G.A. of I.N.d.A.M. 1