Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds

Abstract

In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in ${\mathbb{P}}^g$ (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 such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\ge 13$. We also prove that the general such hyperplane section lies on a unique $K3$ 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.

24 pages, AMS-LaTeX 1.1

Cited by in corpus (33)

• Enumeration of $n$-fold tangent hyperplanes to a surface
• Twisted connected sums and special Riemannian holonomy
• Wahl maps and extensions of canonical curves and K3 surfaces
• On the Corank of Gaussian Maps for General Embedded K3 Surfaces
• On the extendability of projective surfaces and a genus bound for Enriques-Fano threefolds
• Syzygies of Abelian and Bielliptic Surfaces in P^4
• On the geometric genus of reducible surfaces and degenerations of surfaces to unions of planes
• Seshadri constants of $K3$ surfaces of degrees 6 and 8
• Half Nikulin surfaces and moduli of Prym curves
• On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds
• Geometry of the moduli space of $n$-pointed K3 surfaces of genus 11
• Stable maps and singular curves on K3 surfaces
• On degenerations of surfaces
• On the existence of curves with a triple point on a K3 surface
• Hilbert schemes of some threefold scrolls over F_e
• On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds (first version, superseded by arXiv:1204.4838)
• Density of Rational Curves on K3 Surfaces
• On the Mumford - Narasimhan problem
• Surjectivity of Gaussian maps for curves on Enriques surfaces
• Non-special scrolls with general moduli
• Moduli of curves on Enriques surfaces
• On the extendability of elliptic surfaces of rank two and higher
• Rational curves on \bar{M}_g and K3 surfaces
• Degenerations of scrolls to unions of planes
• Fundamental group of Galois covers of degree $6$ surfaces
• Braid monodromy factorization for a non-prime $K3$ surface branch curve
• Singular curves on a K3 surface and linear series on their normalizations
• Fundamental group of Galois covers of degree 5 surfaces
• Gaussian maps, Gieseker-Petri loci and large theta-characteristics
• Qualitative aspects of counting real rational curves on real K3 surfaces
• Extending Nirenberg-Spencer's question on holomorphic embeddings to families of holomorphic embeddings
• On universal Severi varieties of low genus K3 surfaces
• Degeneration of differentials and moduli of nodal curves on $K3$ surfaces