On semi‐isogenous mixed surfaces

Abstract

Let be a smooth projective curve and be a finite subgroup of Aut( ) 2 ⋊ ℤ 2 whose action is mixed, i.e. there are elements in exchanging the two isotrivial fibrations of × . Let 0 ⊲ be the index two subgroup ∩ Aut( ) 2 . If 0 acts freely, then ∶= ( × )∕ is smooth and we call it semi-isogenous mixed surface. In this paper we give an algorithm to determine semi-isogenous mixed surfaces with given geometric genus, irregularity and self-intersection of the canonical class. As an application we classify irregular semi-isogenous mixed surfaces with 2 > 0 and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with = 1. We provide an example of a minimal surface of general type with 2 = 7 and = = 2. K E Y W O R D SSurfaces of general type, finite group actions M S C ( 2 0 1 0 ) 14J29, 14L30, 14Q10

A family of surfaces with and Albanese map of degree 3

“…Since S does not contain any rational curve and K 2 S > 0, we deduce that S is a minimal surface of general type with ample canonical class. Now, as K B = π * K A + E = E, the Riemann-Hurwitz formula yields 7) and this allows us to compute Z 2 . In fact, using (2.6) and (2.7), we can write…”

“…The existence of surfaces S was first established in [7], using a computer-aided construction based on Magma computations. The present paper provides the first computer-free description of them.…”

“…The paper [7] provides a detailed study of semi-isogenous mixed surfaces, showing, among other things, that they are smooth and how to compute their invariants. For instance, [7, Proposition 2.6] allows us to prove the equality q(S) = 2 without exploiting the classification of surfaces with p g = q ≥ 3.…”

“…As the title suggests, in this paper we consider a family M of minimal surfaces of general type with p g = q = 2 and K 2 = 7. The existence of such surfaces was originally established in [7] with the help of the Computer Algebra System MAGMA [6]; the present work provides an alternative, computer-free construction of them, that allows us to describe their Albanese map and their moduli space.…”

A Family of Threefolds of General Type with Canonical Map of High Degree

“…In recent years the intensive work on product-quotient varieties has produced several interesting examples, e.g. the mentioned example of a surface of general type with canonical map of degree 32; new topological types for surface of general type, in particular a family of surfaces of general type with K 2 = 7, p g = q = 2 (see [CF18] and the references therein); and recently the first examples of rigid but not infinitesimally rigid compact complex manifolds ( [BP18]).…”