Generation of jets on \(K3\) surfaces
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 \(a\geq 3\).
scientific article; zbMATH DE number 1398844