Brauer group of moduli of parabolic symplectic bundles
Indranil Biswas, Sujoy Chakraborty, Arijit Dey
TL;DR
This work computes the cohomological Brauer group of the smooth locus of the moduli space of parabolic symplectic bundles on a curve. By exploiting symmetry in the parabolic weights, the authors reduce to the classical twisted symplectic case and first treat concentrated weights, using the Brauer group of regularly stable symplectic bundles and a Leray-type sequence for the associated fibration to obtain explicit cyclic groups generated by the Brauer-Severi class of a Poincaré bundle. They then extend the result to arbitrary generic weights via Thaddeus wall-crossing, proving the Brauer group remains unchanged across walls. The main results express the Brauer group in terms of the degree of $L$ and the parabolic multiplicities $m_{p,i}$, yielding explicit formulas for both even and odd $\deg(L)$ cases and depending on the parity of $\frac{r}{2}$ when $\deg(L)$ is odd. This provides a precise algebraic invariant of parabolic symplectic moduli and connects to prior work on non-parabolic moduli via symmetry and GIT methods.
Abstract
Let $X$ be a smooth connected complex projective curve of genus $g$, with $g\,\geq\, 3$. Fix an integer $r\geq 2$, a finite subset $D\, \subset\, X$, and a line bundle $L$ on $X$. We compute the Brauer group of the smooth locus of the moduli space of parabolic symplectic stable bundles of rank $r$ on $X$ equipped with a symplectic form taking values in $L(D)$, where $L(D)$ is given the trivial parabolic structure.