Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms
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Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms Roberto Pignatelli1 Received: 30 January 2019 / Accepted: 25 November 2019 © Universidad Complutense de Madrid 2019
Abstract We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C × C)/G of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors. We study them through the further quotients (C × C)/G where G ⊃ G. As a first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp. The main result is a complete description of the Albanese morphism of S through a well determined further quotient (C ×C)/G that is an étale cover of the symmetric square of a curve. In particular, if the irregularity of S is at least 2, then S has maximal Albanese dimension. We apply our result to all the semi-isogenous mixed surfaces of maximal Albanese dimension constructed by Cancian and Frapporti, relating them with the other constructions appearing in the literature of surfaces of general type having the same invariants. Keywords Surfaces of general type · Albanese morphism · Automorphisms · Group actions · Quotients Mathematics Subject Classification Primary 14J29; Secondary 14J10 · 14J50 · 14K02 · 14L30
The author is grateful to Fabrizio Catanese for inviting him in Bayreuth with the ERC-2013-Advanced Grant - 340258- TADMICAMT and for the several enlightening conversations on this subject during his stay in May 2018 that allowed to considerably simplify several proofs. He is grateful to Nicola Cancian, Christian Gleißner, Pietro Pirola and Francesco Polizzi for several fruitful conversations on the subject of this paper, and to Davide Frapporti for his careful reading of the first draft of this paper. The author thanks the anonymous reviewers whose comments helped improve and clarify this manuscript. He was partially supported by the Project PRIN 2015 Geometria delle varietà algebriche. He is a member of GNSAGA-INdAM.
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Roberto Pignatelli [email protected] Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Trento, Italy
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R. Pignatelli
Contents Introduction . . . . . . . . . . . . . . . . Notation and conventions . . . . . . . 1 Mixed surfaces . . . . . . . . . . . . . 2 Subgroups of G 0 (2) containing G . . . 3 Further quotients . . . . . . . . . . . . 4 Albanese morphisms of mixed surfaces 5 Ramification of further quotients . . . 6 Applications . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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