Angelo Felice Lopez

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.

Subvarieties of generic hypersurfaces in any variety

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in

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H^0(L^d)

Focal loci of families and the genus of curves on surfaces

a hypersurface of degree d which is generic among those given by sums of monomials from

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L

A linear bound on the Euler number of threefolds of Calabi-Yau and of general type

, and let

SURJECTIVITY OF GAUSSIAN MAPS FOR CURVES ON ENRIQUES SURFACES

f : Y \to X

On the irreducibility of secant cones, and an application to linear normality

be a generically finite map from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X in

Enumeration of surfaces containing an elliptic quartic curve

H^0(L^d)

Explicit Noether-Lefschetz for arbitrary threefolds

, f deforms in a family such that the corresponding deformations of