Picard groups of the moduli spaces of vector bundles (Q1298172)

In the paper under review, the author studies the Picard group of the moduli spaces of certain vector bundles over nodal curves. Let \(Y\) be an irreducible reduced curve over the field of complex numbers such that \(Y\) has at most ordinary nodes as singularities. Fix a line bundle \(\mathcal L\) on \(Y\). Let \(U'_{\mathcal L}(n, d)\) (respectively, \({U'}_{\mathcal L}^{s}(n, d)\)) be the moduli space of semistable (respectively, stable) vector bundles of rank-\(n\) with fixed determinant \(\mathcal L\) on \(Y\). Let \(g\) (respectively, \(g_Y\)) be the geometric (respectively, arithmetic) genus of \(Y\). The main result in the paper says that if \(g \geq 2\), then \(\text{Pic }{U'}_{\mathcal L}(n, d) \cong \text{Pic }{U'}_{\mathcal L}^{s}(n, d) \cong \mathbb{Z}\) and \({U'}_{\mathcal L}(n, d)\) is locally factorial except possibly in case \(g =n=2\) and \(2|d\). Moreover, if \(g_Y=n=2\), then \(\text{Pic }{U'}_{\mathcal L}(n, d) \cong \mathbb{Z}\). These generalize the well-known results when the curve \(Y\) is non-singular. The main technique in the proof is to carry out various codimension computations among the relevant moduli spaces. Other interesting results concerning semistable sheaves over singular curves are also obtained.