Moduli of vector bundles on $μ_n$-gerbes over genus 2 curves and the period-index problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909999598403584 |
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| author | Gong, Ting |
| author_facet | Gong, Ting |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moduli of vector bundles on $μ_n$-gerbes over genus 2 curves and the period-index problem Gong, Ting Algebraic Geometry Number Theory 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. |
| title | Moduli of vector bundles on $μ_n$-gerbes over genus 2 curves and the period-index problem |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2512.03417 |