Supersingular quartic surfaces
type: Item
instance of: scholarly article
title: Supersingular quartic surfaces
zbMATH Open document ID: 1420.14085
author: Junmyeong Jang
published in: Journal of Pure and Applied Algebra
publication date: 2019-07-03T00:00:00Z
review text: The aim of the paper under review is to study supersingular \(K3\) surfaces over an algebraically-closed field \(k\) of odd characteristic which is not \(5\). The main result is to give a projective model of them by showing that any of these \(K3\) surfaces is isomorphic to a smooth quartic surface in \(\mathbb{P}^3_k\). In the proof, key lemmas allow it to be sufficient to prove the statement for a supersingular \(K3\) surface of Artin invariant \(1\), which uniquely exists. The main body of the proof is to show that a \(K3\) surface \(X_\lambda\) defined by a polynomial \(f_{\lambda}:=x^4+y^4+z^4+w^4+\lambda xyzw\) with \(\lambda^4\not=256\), which is either ordinary or supersingular, is supersingular of Artin invariant \(1\). By a criterion for \(X_\lambda\) not to be ordinary in terms of the Frobenius morphism, the condition in the main theorem is finally given. The author also makes a remark about the projective model of the \(K3\) surfaces defined over a finite field. The present result is resembled to a well-known fact that all complex \(K3\) surfaces are diffeomorphic in particular to a smooth quartic surface in \(\mathbb{P}^3_{\mathbb{C}}\).
reviewed by: Makiko Mase
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 7076014
zbMATH Keywords: supersingular \(K3\) surface
zbMATH Keywords: quartic polarization
Wikidata QID: Q112882061
MaRDI profile type: Publication
full work available at URL: j.jpaa.2019.02.012
OpenAlex ID: W2916757712
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DOI: 10.1016/J.JPAA.2019.02.012
description: scientific article; zbMATH DE number 7076014