Lines on \(K3\) quartic surfaces in characteristic \(3\)
type: Item
instance of: scholarly article
title: Lines on \(K3\) quartic surfaces in characteristic \(3\)
zbMATH Open document ID: 1486.14056
author: Davide Cesare Veniani
published in: Manuscripta Mathematica
publication date: 2022-03-15T00:00:00Z
review text: I the last decade or so, the classical problem of estimating the number of lines (conics, etc.)\ on a polarized \(K3\)-surface, most notably quartic \(X\subset\mathbb{P}^3\), has become quite popular. The author addresses the number of lines on a quartic \(X\) with simple singular points defined over an algebraically closed field of characteristic \(3\). Recall that, in the smooth case, the number is \(112\) (and then \(X\) is the Fermat quartic) or at most \(60\); if \(X\) is known to be supersingular, the number is \(112\), \(58\), or at most \(52\). In the presence of singularities, the author proves, by mostly geometric arguments, the bound of at most \(67\) lines (at most \(58\) under an extra geometric assumption on the configuration of lines); on the other hand, he constructs an example with \(8\) singular points and \(48\) lines. The sharp upper bound in the presence of singularities remains open. Besides, the author finds explicit defining equations of the three \(1\)-parameter families of (necessarily supersingular, with the Artin invariant \(\sigma=2\)) smooth quartics with \(58\) lines; these families were recently discovered by abstract purely lattice theoretical means based on the global Torelli theorem.
reviewed by: Alex Degtyarev
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14N25
zbMATH DE Number: 7489552
zbMATH Keywords: spatial quartic
zbMATH Keywords: \(K3\)-surface
zbMATH Keywords: line count
MaRDI profile type: Publication
full work available at URL: s00229-021-01284-9
OpenAlex ID: W3127433236
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DOI: 10.1007/S00229-021-01284-9
description: scientific article; zbMATH DE number 7489552