Families of \(K3\) surfaces and curves of \((2,3)\)-torus type
type: Item
instance of: scholarly article
title: Families of \(K3\) surfaces and curves of \((2,3)\)-torus type
zbMATH Open document ID: 1436.14068
author: Makiko Mase
published in: Kodai Mathematical Journal
publication date: 2020-02-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1902.02195
full work available at URL: https://projecteuclid.org/euclid.kmj/1572487224
review text: Curves of \((2,3)\)-torus type are plane sextic curves defined by a polynomial of the form \(F=F_2^3+F3^2\) with \(F_i\) homogeneous of degree \(i\), and have at most simple singularities. This note studies families of \(K3\) surfaces that are double covers of the projective plane curves of \((2,3)\)-torus type, in particular, their Picard lattices. Let \(X\to\mathbb{P}^2\) be a non-Galois triple cover branched along a curve of \((2,3)\)-torus type \(B\), and let \(\hat{X}\) be the Galois closure. If \(Z\) is a minimal model of the double cover \(D(X/\mathbb{P}^2)\) of \(\mathbb{P}^2\) branching along \(B\), then \(\hat{X}\) is obtained as the cyclic triple cover of \(Z\). In this case, a cyclic triple cover \(\hat{X}\to Z\) is obtained where \(Z\) is a \(K3\) surface, and \(\hat{X}\) is an abelian surface (with singularities), or a Gorenstein \(K3\) surface (a \(K3\) surface with at most simple singularities). A Gorenstein \(K3\) surface \(D(X/\mathbb{P}^2)\) has the defining equation \(W^2-F(X,Y,Z)=0\) in the weighted projective space \(\mathbb{WP}(1,1,1,3)\). These two cases are distinguished by the invariant \(\delta\) which takes values \(9\) in the first case, and \(6\) in the latter. Indeed, \(\delta=9\) if and only if \(\text{Sing}(B)=9A_2\), and \(\delta=6\) if \(\text{Sing}(B)\) is one of the 19 cases. This paper is focused in the latter case, namely \(\delta=6\). A Gorenstein \(K3\) surface \(D(X/\mathbb{P}^2)\) has the defining equation \(W^2-F(X,Y,Z)=0\) in the weighted projective space \(\mathbb{WP}(1,1,1,3)\), where \(F\) is the equation for the plance curve \(B\). This gives rise to families of Gorenstein \(K3\) surfaces. In particular, the main result is concerned with the structure of the Picard lattices of such families. Three families \(\mathcal{F}_i,\,i=1,2,3\) of \(K3\) surfaces are constructed by toric method corresponding to reflexive polytopes \(\Delta_i,\,i=1,2,3\), respectively, which are parametrized by the complete anticannonical linear …
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 7174410
zbMATH Keywords: families of Gorenstein \(K3\) surfaces
zbMATH Keywords: double cover of the projective plane
zbMATH Keywords: curves of (2,3)-torus type
zbMATH Keywords: Picard lattice
MaRDI profile type: Publication
OpenAlex ID: W2985097794
arXiv ID: 1902.02195
DOI: 10.2996/KMJ/1572487224
description: scientific article