Double covers and extensions

Abstract: In this paper we consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to $K3$ surfaces and Fano 3-folds. In particular we consider $K3$ surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pull back of plane curves of degree $k\geq 7$ lies on a unique $K3$ surface. If $k\leq 6$ the general such curve is instead extendable to a higher dimensional variety. In the cases $k=4,5,6$, this gives the existence of singular index $k$ Fano varieties of dimensions 8, 5, 3, and genera 17, 26, 37 respectively. For $k = 6$ we recover the Fano variety $\mathbf{P}(3, 1, 1, 1)$, one of only two Fano threefolds with canonical Gorenstein singularities with the maximal genus 37, found by Prokhorov. We show that the latter variety is no further extendable. For $k=4$ and $5$ these Fano varieties have been identified by Totaro. We also study the extensions of smooth degree 2 sections of $K3$ surfaces of genus 3. In all these cases, we compute the co-rank of the Gauss--Wahl maps of the curves under consideration. Finally we observe that linear systems on double covers of the projective plane provide superabundant logarithmic Severi varieties.