Ciro Ciliberto

Linear systems of plane curves with base points of equal multiplicity

In this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m. A conjecture of Harbourne and Hirschowitz implies that when d > 3m, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple (-1)-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all m < 12.

Geometric Aspects of Polynomial Interpolation in More Variables and of Waring’s Problem

In this paper I treat the problem of determining the dimension of the vector space of homogeneous polynomials in a given number of variables vanishing with some of their derivatives at a finite set of general points in projective space. I will illustrate the geometric meaning of this problem and the main results and conjectures about it. I will finally point out its connection with the so-called Waring’s problem for forms, of which I will also indicate the geometric meaning.

PROJECTIVE DEGENERATIONS OF K3 SURFACES, GAUSSIAN MAPS, AND FANO THREEFOLDS

SummaryIn this article we exhibit certain projective degenerations of smoothK3 surfaces of degree 2g−2 in ℙg (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of suchK3 surfaces has a corank one Gaussian map, ifg=11 org≥13. We also prove that the general such hyperplane section lies on a uniqueK3 surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.

On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with pg > 3 and nonbirational bicanonical map. Our main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3, and therefore pg = q = 3 and K2 = 6. Furthermore we obtain some results towards the classification of minimal surfaces with pg = q = 3. Such surfaces have 6 < Kz < 9, and we show that Kz = 6 if and only if S is the symmetric product of a curve of genus 3. We also classify the minimal surfaces with pg = q = 3 with a pencil of curves of genus 2, proving in particular that for those one has Kz = 8.

Remarks on the bicanonical map for surfaces of general type

To the memory of our colleague and friend Mario Raimondo

On the Concept of k-Secant Order of a Variety

For a variety X of dimension n in

VARIETIES WITH ONE APPARENT DOUBLE POINT

{\mathbb P}^r,\ r\geq n(k+1)+k

Numerical Godeaux surfaces with an involution

, the k th secant order of X is the number

Factoriality Of Certain Hypersurfaces Of P4 With Ordinary Double Points

\mu_k(X)

The Chow motive of the Godeaux surface

of