o Flamini

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p − 2, for 2 ≤ g = p − δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ℳ g that associates to X the isomorphism class [C] of its normalization.

On the branch curve of a general projection of a surface to a plane

In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps. This is a basic result in surface theory, extremely useful in various applications. However, its proof, in this general setting, was so far lacking. Our approach substantially uses a powerful tool from projective differential geometry, i.e., the concept of focal schemes.

SINGULAR CURVES ON A K3 SURFACE AND LINEAR SERIES ON THEIR NORMALIZATIONS

We study the Brill–Noether theory of the normalizations of singular, irreducible curves on a K3 surface. We introduce a singular Brill–Noether number ρsing and show that if Pic(K3) = ℤ[L], there are no s on the normalizations of irreducible curves in |L|, provided that ρsing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for ρsing < 0, without any assumption on the Picard group.

DEGENERATIONS OF SCROLLS TO UNIONS OF PLANES

In this paper we study degenerations of scrolls to union of planes, a problem already considered by G. Zappa in (23) and (24). We prove, using techniques different from the ones of Zappa, a degeneration result to union of planes with the mildest possible singularities, for linearly normal scrolls of genus g and of degree d ≥ 2g + 4 in P d 2g+1 . We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally we review Zappas original approach.

SOME RESULTS OF REGULARITY FOR SEVERI VARIETIES OF PROJECTIVE SURFACES

For a linear system |C| on a smooth projective surface S, whose general member is a smooth, irreducible curve, the Severi variety V |C|,δ is the locally closed subscheme of |C| which parametrizes curves with only δ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper bounds on δ, ensuring that the corresponding Severi variety is smooth of codimension δ, Our result generalizes what is proven in [7] and [10]. We also consider examples of smooth Severi varieties on surfaces of general type in P 3 which contain a line. The author is a member of GNSAGA-CNR.

Moduli of nodal curves on smooth surfaces of general type

In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by V_d(|D|)), which parametrize universal families of irreducible, d -nodal curves in a complete linear system |D|, on a smooth projective surface S of general type. We determine geometrical and numerical conditions on D and numerical conditions on d ensuring that such a number coincides with the dimension of such a variety. As related facts, we also determine some sharp results concerning the geometry of some Severi varieties.

BRILL-NOETHER LOCI OF STABLE RANK-TWO VECTOR BUNDLES ON A GENERAL CURVE

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill-Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to Hilbert scheme of scrolls.

Genera of Curves on a Very General Surface in ℙ3

In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface

Hilbert schemes of some threefold scrolls over F_e

S

Families of nodal curves on projective threefolds and their regularity via postulation of nodes

of degree