February 2024 Double covers and extensions
Ciro Ciliberto, Thomas Dedieu
Author Affiliations +
Kyoto J. Math. 64(1): 75-94 (February 2024). DOI: 10.1215/21562261-2023-0012


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 k7 lies on a unique K3 surface. If k6, 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 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.


Download Citation

Ciro Ciliberto. Thomas Dedieu. "Double covers and extensions." Kyoto J. Math. 64 (1) 75 - 94, February 2024. https://doi.org/10.1215/21562261-2023-0012


Received: 25 March 2021; Revised: 29 January 2022; Accepted: 10 May 2022; Published: February 2024
First available in Project Euclid: 12 December 2023

MathSciNet: MR4677748
Digital Object Identifier: 10.1215/21562261-2023-0012

Primary: 14E20
Secondary: 14H10 , 14J28 , 14N05

Keywords: double covers , extension problem and Gaussian maps , K3 surfaces and canoical curves

Rights: Copyright © 2023 by Kyoto University


This article is only available to subscribers.
It is not available for individual sale.

Vol.64 • No. 1 • February 2024
Back to Top