Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds

Abstract

Abstract. In this article we exhibit certain projective degenerations of smooth K3 surfaces of degree 2g;2 i n 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 p r o ve that the general hyperplane section of such K3 surfaces has a corank one Gaussian map, if g = 1 1 o r g 13. We also prove that the general such h yperplane section lies on a unique K3 surface, up to projectivities. Finally we present a new approach to the classi cation of prime Fano threefolds of index one, which does not rely on the existence of a line. Contents

Surjectivity of Gaussian maps for curves on Enriques surfaces

“…For example, when X ⊂ P r+1 is a smooth anticanonically embedded Fano threefold with general hyperplane section the K3 surface Y , in [CLM1], Ciliberto, the second author and Miranda were able to compute h 0 (N Y /P r (−1)) by calculating the coranks of Φ H C ,ω C for the general curve section C of Y . This then led to recover in [CLM1] and [CLM2], in a very simple way, a good part of the classification of smooth Fano threefolds [I1], [I2] and of varieties with canonical curve section [M]. To study other threefolds by means of Zak's theorem, in many cases it is not enough to get down to curve sections and one needs to bound the cohomology of the normal bundle of surfaces.…”

Braid monodromy factorization for a non-prime K3 surface branch curve

“…The idea of using degenerations for these purposes appears already in [3], [9], [11], [13] and [17]. Degenerations of K3 surfaces, of which the one we use here is an example, were constructed in [10] and are called pillow degenerations.…”

“…For surfaces for which these sections are bounded, some multiple of the canonical bundle is trivial, and there are nine separate families up to complex deformation. The surfaces of this type which are simply connected in fact have trivial canonical bundle, and are called K3 surfaces; the invariants for such surfaces are p g = 1, q = 0, e = 24, and h 1,1 = 22, see [9] and [10]. The most common example of a K3 surface is a smooth quartic surface in P 3 .…”

Pillow Degenerations of K3 Surfaces

“…When the line bundle generates the Picard group of the K3 surface, the embedded K3 surface can be degenerated to a union of 2g − 2 planes in a variety of ways (see for example [1]). In this section we will describe a degeneration, which we call the pillow degeneration, which smooths to a K3 surface whose Picard group is generated by a sub-multiple of the hyperplane class.…”

“…In [1], projective degenerations of K3 surfaces to unions of planes were constructed, in which the general member was embedded by a primitive line bundle. The application featured there was a computation of the rank of the Wahl map for the general hyperplane section curve on the K3 surface.…”