Double covers and extensions

TL;DR

This paper investigates double covers of projective space related to the extension problem of varieties, especially canonical curves to K3 surfaces and Fano 3-folds, revealing new extension properties and classifying certain Fano varieties.

Contribution

It establishes conditions for the uniqueness of K3 surface extensions of certain curves and classifies associated Fano varieties with specific genera and dimensions.

Findings

Unique K3 surface extension for curves from degree ≥7 plane pullbacks.

Existence of singular Fano varieties with specific genera for degrees 4, 5, 6.

The Fano variety for k=6 is not further extendable, confirming known classifications.

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.