Stability of the restriction of a rank 2 fibre bundle on an algebraic surface to a smooth curve
The aim of this work is to find a criterion for the stability of the restriction of a rank 2 vector bundle to a smooth curve embedded in an algebraic surface \(S\). If the bundle \(E\) is given as a nontrivial extension \[ 0 \to {\mathcal O}_S \to E \to L {\mathcal J}_Z \to 0, \tag{1} \] and if \(E |_C\) is unstable, with \(C\) satisfying certain numerical conditions, we obtain the existence of a curve \(F\) containing a subscheme of \(Z\) such that \(F^2\) and \(F \cdot C\) are bounded by two explicit quantities depending on \(E\). Then this result is applied to an abelian surface with \(\text{NS} (S) = \mathbb{Z} H\): Let \({\mathcal M}_m\) be the moduli space of rank 2 stable vector bundles over \(S\) with \(c_1 = 0\) and \(c_2 = m\). The restriction to a smooth curve in \(|nH |\) of a generic bundle from a component satisfying a property expressed in terms of the extension (1), is stable if \(n \geq 2 \sqrt {{m \over H^2}} + 5\).
stability of the restriction of a rank 2 vector bundle
scientific article; zbMATH DE number 840213