A remark about the rigidity of curves on \(K3\) surfaces
type: Item
instance of: scholarly article
title: A remark about the rigidity of curves on \(K3\) surfaces
zbMATH Open document ID: 1230.14054
DOI: 10.1007/BF03191236
author: Mihai Halic
published in: Collectanea Mathematica
publication date: 2011-08-16T00:00:00Z
full work available at URL: http://www.collectanea.ub.edu/index.php/Collectanea/article/view/5348
review text: A classical result due to Bogomolov and Mumford states that on any projective \(K3\) surface there are rational curves and 1-dimensional families of elliptic curves. Moreover, \textit{Xi Chen} [J. Algebr. Geom. 8, 245--278 (1999; Zbl 0940.14024)] has shown that if \(S \subset \mathbb{P}^n\) is any \(K3\) surface of Picard number 1, then the linear system \(|\mathcal{O}_S(d)|\) contains a \((p(d)-\delta)\)-dimensional family of irreducible nodal curves whose arithmetic genus is \(p(d)=d^2(n-1)+1\) and which possess exactly \(\delta\) nodes, for \(\delta\geq 0\). In this paper, the author gives another proof of this result but with certain restrictions on the number of nodes \(\delta\).
reviewed by: Nikolaos Tziolas
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 5939723
zbMATH Keywords: K3 surfaces
zbMATH Keywords: rational curves
zbMATH Keywords: stable bunbles
MaRDI profile type: Publication
cites work: Q4699499
cites work: PSEUDO-EFFECTIVE LINE BUNDLES ON COMPACT KÄHLER MANIFOLDS
cites work: A bound on the dimensions of complete subvarieties of \({\mathcal M}_ g\).
cites work: Restrictions of semistable bundles on projective varieties
cites work: Notes on stable maps and quantum cohomology
cites work: Q3128810
cites work: Semistable sheaves on projective varieties and their restriction to curves
cites work: Dimension-Theoretic Characterization of Maximal Irreducible Algebraic Systems of Plane Nodal Curves of a Given Order n and with a Given Number d of Nodes
description: scientific article; zbMATH DE number 5939723