Double covers and extensions

Abstract

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 threefolds. 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 pullback of plane curves of degree $k⩾7$ lies on a unique $K3$ surface. If $k⩽6$, then 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, genera 17, 26, 37, and indices 6, 3, 1, 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 not 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 corank 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.