Dynkin diagrams of rank 20 on supersingular \(K3\) surfaces
type: Item
instance of: scholarly article
title: Dynkin diagrams of rank 20 on supersingular \(K3\) surfaces
zbMATH Open document ID: 1316.14071
author: Ichiro Shimada
author: De-Qi Zhang
published in: Science China. Mathematics
publication date: 2015-04-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.5474
review text: A \textit{Dynkin type} is a finite formal sum of \(A_l,\, l\geq 1,\, D_m, \, m\geq 4,\, E_6,\, E_7,\) and \(E_8\) with non-negative integer coefficients corresponding to an even negative-definite lattice \(L(R)\) of discriminant denoted by \(\mathrm{disc} (R)\). Let \(R\) be a Dynkin type of rank \(20\). A prime integer \(p\) is called a \textit{\(R\)-supersingular \(K3\) prime} if \(p\) is odd not dividing \(\mathrm{disc}(R)\), and there exists a normal \(K3\) surface over an algebraically-closed field \(k\) of characterstic \(p\) whose singularity is of type \(R\). The minimal resolution of such a normal \(K3\) surface is the supersingular \(K3\) surface of Artin invariant \(1\) over \(k\). The question asked in the paper under review is whether or not there exist \(R\)-supersingular \(K3\) primes for a given Dynkin type \(R\) of rank \(20\). In other words, for a given \(R\), the existence of normal \(K3\) surfaces with singularity being of type \(R\) is the problem. This question in pure geometry is reduced to be an existence problem of certain lattices in Theorem 2.1. Denote by \(D_L\) the discriminant group, and \(q_L\) the discriminant form of a lattice \(L\). The theorem asserts that that an odd prime integer \(p\) being a \(R\)-supersingular \(K3\) prime when it does not divide \(\mathrm{disc} (R)\) is equivalent to that there existing an overlattice \(\tilde{L}\) of \(L(R)\) whose set of roots is the same as that of \(L(R)\), and a lattice \(T'\) of rank \(2\) and signature \((1,1)\) such that \((D_{T'}, pq_{T'})\simeq (D_{\tilde{L}}, q_{\tilde{L}})\). Besides, a tuple \(\tau(p):=(\tau_l(p))_l\) defined in sections 3 and 4, where \(l\) runs all odd prime divisors of \(\mathrm{disc}(R)\), characterizes whether or not this criterion holds. In conclusion, a list of all \(R\)-supersingular \(K3\) primes is given as a subset of the multiplicative group of units of a cyclic group.
reviewed by: Makiko Mase
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6430225
zbMATH Keywords: supersingulari \(K3\) surface
zbMATH Keywords: normal \(K3\) surface
MaRDI profile type: Publication
OpenAlex ID: W2035414451
arXiv ID: 1408.5474
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DOI: 10.1007/S11425-014-4902-3
description: scientific article