Related papers: Recent results on linear systems on generic K3 sur…
In this paper we introduce a technique to degenerate K3 surfaces and linear systems through fat points in general position on K3 surfaces. Using this degeneration we show that on generic K3 surfaces it is enough to prove that linear systems…
In this paper we prove the equivalence of two conjectures on linear systems through fat points on a generic K3 surface. The first conjecture is exactly as Segre conjecture on the projective plane. Whereas the second characterizes such…
We prove that any number of general fat points of any multiplicities impose the expected number of conditions on a linear system on a smooth projective surface, in several cases including primitive linear systems on very general K3 and…
In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.
In this paper we deal with linear systems of P^3 through fat points. We consider the behavior of these systems under a cubo-cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only…
We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a…
In this paper we develop techniques for determining the dimension of linear systems of divisors based at a collection of general fat points in P^n by partitioning the monomial basis for the vector space of global sections of O(d). The…
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
In this note we consider the behavior of linear systems of P^3 through fat points under a cubo-cubic Cremona transformation. This allows us to produce a class of special systems which we conjecture to be the only ones.
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are…
We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.
We proved the existence of rational curves in every linear system on a general K3 surface and that all rational curves in the hyperplane class are nodal on a general K3 surface of small genus.
We study the distribution of algebraic points on K3 surfaces.
We study arithmetic properties of derived equivalent K3 surfaces over the field of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.
This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good…
We report on our project to find explicit examples of $K3$ surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for…
We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.
We construct a lot of K3 surface automorphisms of positive entropy having rotation domains of ranks 1 and 2. To carry out this construction, we first lay theoretical foundations concerning equivariant linearization of nonlinear maps under…
In this work, we introduce $(r,i)$-regular fat linear sets, which are defined as linear sets containing exactly $r$ points of weight $i$ and all other points of weight one. This notion generalizes and unifies existing constructions;…
We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a…