Arithmetically Cohen-Macaulay bundles and sets of points on general surfaces
Let \(S\subseteq\mathbb{P}^3\) be a general smooth surface of degree \(d\geq4\) with \(\text{Pic}(S)=\mathbb{Z}\). A vector bundle \(F\) on \(S\) is called \textit{initialized} if \(h^0(F)>h^0(F(-1))=0\). If \(F\) is a rank 2 initialized vector bundle, then it is stable if and only if \(c_1(F)>0\). A vector bundle \(F\) on \(S\) is said \textit{ACM} if it has no intermediate cohomology, that is if \(\mathrm{H}^1(S,F(t))=0\) for all \(t\in\mathbb{Z}\). The existence of rank 2 ACM bundles on \(S\) is related to the existence of arithmetically Gorenstein sets of points and also to the problem of pfaffian representation of forms. In the paper under review the authors give bounds for the first Chern class of rank \(2\) stable ACM bundles on \(S\). In particular they prove the following main results: -- if \(d\geq17\) and \(c\) is greater than a function of \(d\) growing like \(3\sqrt[3]{d}\), then a general surface of degree \(d\) contains no initialized, indecomposable ACM rank 2 vector bundles with first Chern class \(c\); -- for any \(2\leq c\leq 6\), there exists an initialized, indecomposable ACM rank 2 vector bundle with first Chern class \(c\) on a general surface of degree \(d\geq 2c\); -- for any integer \(c>0\), a general surface of degree \(d\gg c\) contains an initialized, indecomposable ACM rank 2 vector bundle with first Chern class \(c\).
scientific article; zbMATH DE number 6539244
scientific article