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Pure and Applied Mathematics Quarterly
Volume 18 (2022)
Number 4
Special issue celebrating the work of Herb Clemens
Guest Editor: Ron Donagi
Singularities of normal quartic surfaces II (char=2)
Pages: 1379 – 1420
DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a5
Authors
Abstract
We show, in this second part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $14$, and that, if we have $14$ singularities, these are nodes and moreover the minimal resolution of $X$ is a supersingular K3 surface.
We produce an irreducible component, of dimension $24$, of the variety of quartics with $14$ nodes. We also exhibit easy examples of quartics with $7$ $A_3$-singularities.
Keywords
quartic surface, singularity, Gauss map, genus one fibration, supersingular K3 surface
2010 Mathematics Subject Classification
Primary 14J17, 14J28. Secondary 14J25, 14N05.
The first author acknowledges support of the ERC 2013 Advanced Research Grant – 340258 – TADMICAMT.
Received 6 October 2021
Received revised 3 March 2022
Accepted 23 May 2022
Published 25 October 2022