Desingularizations of the moduli space of rank 2 bundles over a curve

摘要

Let X be a smooth projective curve of genus g≥3 and M 0 be the moduli space of rank 2 semistable bundles over X with trivial determinant. There are three desingularizations of this singular moduli space constructed by Narasimhan-Ramanan [NR78], Seshadri [Ses77] and Kirwan [Kir86b] respectively. The relationship between them has not been understood so far. The purpose of this paper is to show that there is a morphism from Kirwan’s desingularization to Seshadri’s, which turns out to be the composition of two blow-downs. In doing so, we will show that the singularities of M 0 are terminal and the plurigenera are all trivial. As an application, we compute the Betti numbers of the cohomology of Seshadri’s desingularization in all degrees. This generalizes the result of [BS90] which computes the Betti numbers in low degrees. Another application is the computation of the stringy E-function (see [Bat98] for definition) of M 0 for any genus g≥3 which generalizes the result of [Kie03].