On the cohomological strata of families of vector bundles on algebraic surfaces (Q1175687)

This paper belongs to a series of papers on rank two vector bundles on algebraic surfaces by the author (and many others). Under the assumption \(P_ g(X)=0\), the author defines families (called sheets) of rank two vector bundles on an algebraic surface \(X\) by the following cohomological conditions. Let \(M\) be a (fixed) line bundle on \(X\). Define \(U(M,c_ 1,c_ 2)\) to be the set of isomorphism classes of rank two vector bundles on \(X\) with Chern classes \(c_ 1, c_ 2\) such that \(E(M)\) has a section \(s\) with its zero locus of dimension zero and \( h^ 0(E(M\otimes K))=0\). Sufficiency conditions for \(U(M,c_ 1,c_ 2)\) to be nonempty are studied in the paper. Note that if \(P_ g(X)>0\), then \(U(M,c_ 1,c_ 2)=0\) for all \(M\). In good cases, these sheets turn out to be irreducible and give a stratification of the moduli space. The examples studied here in detail are \(\mathbb{P}^ 2\) and rational ruled surfaces. For these varieties the properties of the general bundle in a sheet are studied and the existence of bundles with exceptional behaviour in each sheet is proved. Finally the author considers the `recognition problem', i.e. given a bundle defined using extensions, how to recognise it is isomorphic to a certain bundle \(E\)? The problem is solved in some cases when \(X=\mathbb{P}^ 2\), \( \mathbb{P}^ 1\times \mathbb{P}^ 1\) or \(\mathbb{P}^ 3\). The most interesting case is \(X=\mathbb{P}^ 3\), \( E=T\mathbb{P}^ 3(t)\), a twist of the tangent bundle on \(\mathbb{P}^ 3\). It is related to the study of codimension 1 meromorphic foliations with singularities on \(\mathbb{P}^ 3\).