On the Complexity of the Projective Classification of Surfaces.
On the curves through an general point of a smooth surface in IP3.
On the diffeomorphic type of the complement to a line arrangement in a projective plane
We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011,...
On the dimension of secant varieties
In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety under suitable regularity assumptions on , and we classify varieties for which the bound is attained.
On the dual variety in characteristic 2
On the enumerative geometry of real algebraic curves having many real branches.
On the existence of components of the Noether-Lefschetz Locus with given codiemsnion.
On the existence of curves with maximal rank in Pn.
On the extendability of elliptic surfaces of rank two and higher
We study threefolds having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings...
On the finiteness of rational curves on quintic threefolds
On the generation of certain bundles over P3.
On the genus of reducible surfaces and degenerations of surfaces
We deal with a reducible projective surface with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the -genus of , i.e. the dimension of the vector space of global sections of the dualizing sheaf . Then we prove that, when is smoothable, i.e. when is the central fibre of a flat family parametrized by a disc, with smooth general fibre, then the -genus of the fibres of is constant.
On the geometry of quadrics and their degenerations.
On the gonality of curves in
Here we study the gonality of several projective curves which arise in a natural way (e.gċurves with maximal genus in , curves with given degree and genus for all possible , if and with large for arbitrary ).
On the graded Betti numbers for large finite subsets of curves
We prove a recent conjecture of S. Lvovski concerning the periodicity behaviour of top Betti numbers of general finite subsets with large cardinality of an irreducible curve C ⊂ ℙⁿ.
On the group orders of elliptic curves over finite fields
On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces
Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of P n with one point as support and with high regularity index.
On the Hilbert Scheme of Curves of Maximal Genus in a Projective Space.
On the Hilbert scheme of Palatini threefolds.
On the homogeneous ideal of collinear punctual subschemes of P2.