\(\mathbb{A}^1\) curves on log \(K3\) surfaces
type: Item
instance of: scholarly article
title: \(\mathbb{A}^1\) curves on log \(K3\) surfaces
zbMATH Open document ID: 1386.14135
author: Yi Zhu
author: Xi Chen
published in: Advances in Mathematics
publication date: 2017-05-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1602.02204
review text: The aim of the article under review is to study (the existence of) \(\mathbb{A}^1\) curves on genuine log \(K3\) surfaces of type II. A log \(K3\) surface of type II is by definition a pair \((X,\, D)\) of smooth projective rational surface \(X\) and a divisor \(D\) on \(X\) with logarithmic geometric genus \(1\), and logarithmic Kodaira dimension and logarithmic irregularity are zero. In the paper, it is assumed that \((X,\, D)\) is genuine: namely, the divisor \(D\) is numerically equivalent to the anticanonical divisor of \(X\), and the global sections of the sheaf of logarithmic \(1\)-forms on \(X\) are all trivial. In the first step, a classification of genuine log \(K3\) surfaces of type II is obtained by giving an explicit description of the divisor \(D\) such that a pair \((X,\, D)\) is a genuine log \(K3\) surface of type II. There are five classes C0 to C4 up to log isomorphism. With this classification, the main theorem is given that a genuine log \(K3\) surface of type II \((X,\, D)\) contains countably-many \(\mathbb{A}^1\) curves if and only if \((X,\, D)\) is log isomorphic to C0, C1, C2, or C3. Lastly, as an application, some examples of genuine log \(K3\) surfaces of type II that contain only finitely many \(\mathbb{A}^1\) curves are given. The reviewer wonders without the assumption ``genuine'', if it is possible to classify all of log \(K3\) surfaces of type II, and to study the existence of \(\mathbb{A}^1\) curves on them.
reviewed by: Makiko Mase
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 6720182
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: log varieties
zbMATH Keywords: rational curves
MaRDI profile type: Publication
OpenAlex ID: W2252450176
arXiv ID: 1602.02204
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DOI: 10.1016/J.AIM.2017.04.020
description: scientific article