Moduli of vector bundles on $μ_n$-gerbes over genus 2 curves and the period-index problem
Ting Gong
TL;DR
The paper develops a framework for the moduli of vector bundles on μ_n-gerbes over genus 2 curves using twisted sheaf theory and determinantal line bundles, yielding explicit moduli descriptions and Theta-divisor interpretations. It demonstrates period–index phenomena by proving the existence of genus-2 Brauer classes with per=ind and, over C1-fields, that every 2-torsion class satisfies per=ind; it also constructs higher-dimensional fiber-product varieties with 2-torsion Brauer classes obeying the period–index relation. The results combine explicit genus-2 computations (twisted rank-2 and rank-3 cases), obstruction theory for descent of determinantal data, and applications to iterative constructions, strengthening the evidence for the period–index conjecture in geometric settings. Overall, the work links twisted moduli, Brauer–Severi geometry, and period–index behavior to produce concrete arithmetic and geometric consequences for curves and their higher-dimensional analogues.
Abstract
We develop a framework for describing vector bundles on $μ_n$-gerbes over curves and illustrate the construction through two detailed examples. Using the interpretation of Brauer classes as obstructions to descending determinantal line bundles from the algebraic closure, together with a geometric analysis of the moduli space of twisted sheaves, we prove that for genus $2$ curves there exist Brauer classes over the base field whose period equals their index. Over $C_1$-fields, we further show that every $2$-torsion class in the Brauer group of a genus $2$ curve satisfies the period-index problem. As an application, we construct higher-dimensional varieties obtained as fibre products of genus $2$ curves over $C_1$-fields whose $2$-torsion algebraic Brauer classes also satisfy the period-index problem, providing new evidence toward the period-index conjecture.