Surfaces with pg = q = 2, K2 = 6, and albanese map of degree 2

We classify minimal surfaces of general type with pg = q = 2 and K2 = 6 whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components MIa, MIb, M II of dimension 4, 4, 3, respectively. © Canadian Mathematical Society 2012

A new family of surfaces with pg = q = 2 and K2 = 6 whose Albanese map has degree 4

We construct a new family of minimal surfaces of general type with pg = q = 2 and K2 = 6, whose Albanese map is a quadruple cover of an abelian surface with polarization of type (1, 3). We also show that this family provides an irreducible component of the moduli space of surfaces with pg = q = 2 and K2 = 6. Finally, we prove that such a component is generically smooth of dimension 4 and that it contains the two-dimensional family of product-quotient examples previously constructed by the first author. The main tools we use are the Fourier-Mukai transform and the Schrödinger representation of the finite Heisenberg group H3

Calabi-Yau 4-folds of Borcea-Voisin type from F-theory

We apply Borcea-Voisin's construction and give new examples of Calabi- Yau 4-folds Y, which admit an elliptic fibration onto a smooth 3-fold V, whose singular fibers of type I5 lie above a del Pezzo surface dP 82 V. These are relevant models for F-theory according to Beasley et al. (2009a, 2009b). Moreover, we give the explicit equations of some of these Calabi-Yau 4-folds and their fibrations

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g'=0$. Here we find 6 families of Galois coverings, all with $g' = 1$ and $g=2,3,4$ and we show that these are the only families with $g'=1$ satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $\mathsf{A}_g$, while the other examples arise from certain Shimura subvarieties of $\mathsf{A}_g$ already obtained as families of Galois coverings of $\mathbb{P}^1$ in [FGP]. Finally we prove that if a family satisfies this sufficient condition with $g'\geq 1$, then $g \leq 6g'+1$.Comment: 18 pages, to appear in Geometriae Dedicat

A note on a family of surfaces with pg= q= 2 and K2= 7

We study a family of surfaces of general type with pg= q= 2 and K2= 7 , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus M in the moduli space of surfaces of general type. In particular we prove that M is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smoot

A note on surfaces with pg = q = 2 and an irrational fibration

We study several examples of surfaces with pg = q = 2 and maximal Albanese dimension that are endowed with an irrational fibration

On the cohomology of surfaces with PG = q = 2 and maximal Albanese dimension

In this paper we study the cohomology of smooth projective complex surfaces S of general type with invariants pg = q = 2 and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface X that we call the K3 partner of S. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner X and an algebraic correspondence in S 7X that relates the cohomology of S and X. Finally, we prove the Tate and Mumford\u2013Tate conjectures for those surfaces S that lie in connected components of the Gieseker moduli space that contain a product-quotient or a mixed surface

On zariski multiplets of branch curves from surfaces isogenous to a product

In this paper we give an asymptotic bound of the cardinality of Zariski multiples of particular irreducible plane singular curves. These curves have only nodes and cusps as singularities and are obtained as branched curves of ramified covering of the plane by surfaces isogenous to a product of curves with group (Z/2Z)k. The knowledge of the moduli space of these surfaces will enable us to produce Zariski multiplets whose number grows subexponentially in function of their degree

On surfaces with pg = q = 2, k2 = 5 and albanese map of degree 3

We construct a connected, irreducible component of the moduli space of minimal surfaces of general type with pg = q = 2 and K2 = 5, which contains both examples given by Chen-Hacon and the first author. This component is generically smooth of dimension 4, and all its points parametrize surfaces whose Albanese map is a generically finite triple cover