In this paper we study the possible Picard numbers ρ of an abelian variety A of dimension g. It is well known that this satisfies the inequality 1 ≤ ρ ≤ g 2 . We prove that the set R g of realizable Picard numbers of abelian varieties of dimension g is not complete for every g ≥ 3, namely that R g [1, g 2 ] ∩ N. Moreover, we study the structure of R g as g → +∞, and from that we deduce a structure theorem for abelian varieties of large Picard number. In contrast to the non-completeness of any of the sets R g for g ≥ 3, we also show that the Picard numbers of abelian varieties are asymptotically complete, i.e. lim g→+∞ #R g /g 2 = 1. As a byproduct, we deduce a structure theorem for abelian varieties of large Picard number. Finally we show that all realizable Picard numbers in R g can be obtained by an abelian variety defined over a number field.

In this note we study the local negativity for certain configurations of smooth rational curves in smooth surfaces with numerically trivial canonical class. We show that for such rational curves there is a bound for the so-called local Harbourne constants, which measure the local negativity phenomenon. Moreover, we provide explicit examples of interesting configurations of rational curves in some K3 and Enriques surfaces and compute their local Harbourne constants.

Dedicated to my father on the occasion of his 50th birthday.Abstract. Given an abelian surface, the number of its distinct decompositions into a product of elliptic curves has been described by Ma. Moreover, Ma himself classified the possible decompositions for abelian surfaces of Picard number 1 ≤ ρ ≤ 3. We explicitly find all such decompositions in the case of abelian surfaces of Picard number ρ = 4. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group on the factors of a given decomposition. We also provide an alternative and simpler proof of Ma's formula, and an application to singular K3 surfaces.

We give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections X ⊂ P n+2 C of multi-degree d = (d1, . . . , dn), and we provide a sharp and uniform bound on their H-constants, which only depends on d. P H(X; P),

We study the field of moduli of singular K3 surfaces. We discuss both the field of moduli over the CM field and over Q. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an explicit approach to the computation of the field of moduli.