A positive characteristic extension of a result of Del Busto on line bundles on an algebraic surface
In the last few years several striking results were proven (in characteristic 0) on the behaviour of line bundles on projective varieties using very strong and interesting refinements of Mori theory. Here we will show how to give a partial extension (see theorem 1) of the main result of a paper by \textit{G. Fernández del Busto} [J. Algebr. Geom. 5, No. 3, 513-520 (1996; Zbl 0871.14006)] to positive characteristic. In our opinion this extension is not straight-forward. Theorem 1. Let \(X\) be a smooth projective surface defined over an algebraically closed field \(\mathbb{K}\); set \(p:=\text{char} (\mathbb{K})\). Fix \(A\in \text{Pic} (X)\) with \(A\) ample. If \(X\) rational assume \(p>2\). If \(\kappa (X)=2\) assume \(X\) not uniruled. If \(\kappa(X)=0,1\), assume \(p>3\). Let \(k\) be an integer \(>0\) such that \(2k> [((A\cdot (K_X+4A) +1)/A^2) + (\kappa^2 +4\kappa-2)]\). Then \(kA\) is \(\kappa\)-jet ample.
scientific article; zbMATH DE number 1089789