On vector bundles whose general sections have all projectively equivalent zero-loci
By a well-known lemma of Serre, if a vector bundle \(E\) of rank \(r\) on a variety \(X\) is spanned by global sections, then there is a subspace \(W\) of \(H^ 0(X,E)\) with \(\dim W\leq r+\dim X\) such that \(W\) spans \(E\). The author studies here the case \(\dim W=4\), \(r=2\), \(X\) an integral projective surface. Let \(i\) be the morphism from \(X\) to the Grassmannian \(G(2,4)\subset \mathbb{P}^ 5\) associating to \(x\) in \(X\) the quotient \(E_ x\) of \(W\), \(E_ x=\) fibre of \(E\) at \(x\). Assume that (1) \(i\) is an embedding, \(i(X)\) is ordinary (i.e. reflexive with dual a hypersurface); (2) there is \(s\in W\) with zero scheme \(s_ 0\) union of \(C_ 2(E)-2\) smooth points of \(X\) (with reduced structure) and a length 2 scheme supported by another smooth point of \(X\); (3) all general \(s_ 0\) are reduced and projectively equivalent. The main result in the first part of the paper says that under these assumptions \(C_ 2(E)\leq 28\). The second part of the paper is devoted to plane curves \(C\) with all general tangent lines to \(C\) intersecting \(C\) in projectively equivalent subsets (under \(\Aut \mathbb{P}^ 1\)). It is shown that if \(\text{Card}(C\cap\ell)_{\text{red}})\geq 3\) for every line \(\ell\), then varying line \(\ell\) in \((\mathbb{P}^ 2)^*\) the projective equivalence classes of sets \(C\cap\ell\) vary in a two dimensional family.
scientific article; zbMATH DE number 492563
scientific article