Moduli spaces of vector bundles on a real nodal curve

Abstract

Let Y be a geometrically irreducible nodal projective algebraic curve of arithmetic genus $$g \ge 2$$ defined over $${{\mathbb {R}}}$$ . Let $$Y_{{\mathbb {C}}} = Y \times _{{\mathbb {R}}} {\mathbb {C}}$$ . Fix an $${\mathbb {R}}$$ -valued point $$\xi $$ of $$Pic^d(Y)$$ . For integers $$r \ge 2$$ and d, let $$M(r,\xi )$$ [respectively $$U(r,\xi )$$ ] be the moduli stack of vector bundles (respectively the moduli space of stable vector bundles) of rank r and determinant $$\xi $$ on Y. We determine the Picard group of $$M(r,\xi )$$ . We compute the Picard group and Brauer group of $$U(r,\xi )$$ .