Brauer group of moduli spaces of PGL\((r)\)-bundles over a curve (Q1959661)

Let \(X\) be an irreducible smooth projective curve of genus \(g\) defined over an algebraically closed field \(k\). Let \(r\geq 2\) be an integer, assume that \(r>3\) if \(g=2\). Let \({\mathcal N}(r)\) (respectively \(N(r)\)) denote the moduli stack (respectively moduli space) of stable PGL\((r,k)\)-bundles on \(X\). The authors prove that the natural map \({\mathcal N}(r) \to N(r)\) is an isomorphism outside a closed subset of codimension three. This identifies the Brauer group of \({\mathcal N}(r)\) with the Brauer group \(\mathrm{Br}(N(r))\) of \(N(r)\). The moduli space \(N(r)\) has \(r\) irreducible components \(N(r)_i\) parametrized by \({\mathbb Z}/r{\mathbb Z}\). Let \(M(r,L)\) be the moduli space of vector bundles of rank \(r\) with determinant isomorphic to \(L\). Then \(N(r)_i\) is the quotient of \(M(r,L), d(L)=i\), by the \(r\)-torsion points of the Jacobian. Using the Leray spectral sequence of the quotient map, the authors show that there is a surjective homomorphism \(\mathrm{Br}(N(r)_i) \to \mathrm{Br}(M(r,L))\) and explicitly compute the kernel of this map. It is known that \(\mathrm{Br}(M(r,L)) \cong {\mathbb Z}/h{\mathbb Z}\), where \(h\) is the greatest common divisor of \(r\) and \(i\).