Moduli spaces of vector bundles on higher-dimensional varieties.
Let \(X\) be a \(\mathbb{P}^d\)-bundle over a smooth projective curve \(C\). The authors study the behaviour of moduli of \(\mu\)-stable vector bundles of rank two on \(X\) with increasing second Chern class. In contrast to the case of surfaces, where large \(c_2\) guarantees irreducibility and generical smoothness of such moduli spaces, the situation is usually much more complicated in higher dimensions, involving certain relations between the polarization and the Chern classes. However, for projective bundles there exist polarizations closely related to the second Chern class, providing sufficient assumptions to prove the desired results. More precisely, for second Chern classes fulfilling a certain condition, and a suitably chosen polarization on \(X\), a general point in the moduli space is an extension of two line bundles. By using this and the technically important result that for a suitable twist of a rank two vector bundle on \(X\) there exists a section with zero locus of codimension two, the authors prove the irreducibility and smoothness of the moduli spaces for such invariants, and additionally the rationality of these spaces if \(C\) is rational. In the last section, an example for a change of the polarization is discussed.
scientific article; zbMATH DE number 1742564