Surfaces containing a family of plane curves not forming a fibration

We complete the classification of smooth surfaces swept out by a 1-dimensional family of plane curves that do not form a fibration. As a consequence, we characterize manifolds swept out by a 1-dimensional family of hypersurfaces that do not form a fibration.Comment: Author's post-print, final version published online in Collect. Mat

Moduli of mathematical instanton vector bundles with odd c_2 on projective space

The problem of irreducibility of the moduli space I_n of rank-2 mathematical instanton vector bundles with arbitrary positive second Chern class n on the projective 3-space is considered. The irreducibility of I_n was known for small values of n: Barth 1977 (n=1), Hartshorne 1978 (n=2), Ellingsrud and Stromme 1981 (n=3), Barth 1981 (n=4), Coanda, Tikhomirov and Trautmann 2003 (n=5). In this paper we prove the irreducibility of I_n for an arbitrary odd n.Comment: 62 page

Weierstrass jump sequences and gonality

We prove some structure results on Weierstrass points on a non-hyperelliptc curve, which are total or almost total ramification points for the gonal covering. It turns out that the corresponding Weierstrass semigroup is strictly related to the splitting type of the gonal scroll containing the canonical model of the curve. We also give a description of the Weierstrass gap sequence in the case of a non ramification point for the gonal cover

Surfaces in P^4 with a family of plane curves

We prove that if is a smooth nondegenerate surface covered by a one-dimensional family D={Dx}x 08T of plane (nondegenerate) curves, not forming a fibration, and if the hypersurface given by the union of the planes \u3008Dx\u3009 spanned by such curves is not a cone, then for any general x 08T, the genus g(Dx) 641, and S is either: 1. the projected Veronese surface, and the plane curves are conics; 2. the rational normal cubic scroll, and the plane curves are conics; 3. a quintic elliptic scroll, and the plane curves are smooth cubics. Furthermore, if the number of curves of the family passing through a general point of S is m 653, only cases 1 and 2 may occur. The statement has been conjectured by Sierra and Tironi in [J. Sierra, A. Tironi, Some remarks on surfaces in containing a family of plane curves, J. Pure Appl. Algebra 209 (2) (2007) 361\u2013369., Conjecture 4.13

Canonical map of low codimensional subvarieties

Fix integers a and b.We prove that for certain projective varieties V 82 Pr (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties X of V such that the linear system |aK-bH| is empty. In particular, except for finitely many families of varieties, the canonical map of any irreducible, smooth, projective, low codimensional subvariety X of V, is birational