Generation of jets on \(K3\) surfaces
type: Item
instance of: scholarly article
title: Generation of jets on \(K3\) surfaces
zbMATH Open document ID: 0956.14002
DOI: 10.1016/S0022-4049(98)00098-X
author: Thomas Bauer
author: Tomasz Szemberg
author: Sandra Di Rocco
published in: Journal of Pure and Applied Algebra
publication date: 2001-03-12T00:00:00Z
review text: Consider a K3 surface \(X\) and an ample line bundle \(L\) on \(X\). \textit{B. Saint-Donat} [Am. J. Math. 96, 602-639 (1974; Zbl 0301.14011)] established that \({\mathcal O}_X(3L)\) is globally generated and \({\mathcal O}_X(2L)\) is very ample. The purpose of this note is to see how these basic facts generalize to the generation of jets and to jet ampleness. The main result gives the optimal bounds and the sharp bound for jet ampleness as follow. Let \(X\) be a K3 surface, \(L\) an ample line bundle on \(X\) and \(k\) a non-negative integer. Then either (a) \({\mathcal O}_X(nL)\) is \(k\)-jet ample for \(n\geq k+2\), or (b) \(L\) is of the form \(L={\mathcal O}_X(aE +\Gamma)\), where \(E\subset X\) is an elliptic curve, \(\Gamma\subset X\) is a \((-2)\)-curve with \(E\cdot\Gamma=1\) and \(a\geq 3\).
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1398844
zbMATH Keywords: K3 surface
zbMATH Keywords: ample line bundle
zbMATH Keywords: jet ampleness
reviewed by: Shou Lin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1398844