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The line bundles on the moduli stack of principal bundles on families of curves

Roberto Fringuelli, Filippo Viviani

TL;DR

This work determines the Picard group of the moduli stack Bun_G^δ(C/S) of principal G-bundles over a family of curves, for G reductive, by decoupling abelian and nonabelian contributions through tautological line bundles, determinant of cohomology, and Deligne pairing. Central tools include the transgression map τ_G^δ, the abelianization map ab_sharp, and a detailed analysis of the weight map wt on line bundles after rigidification by the center Z(G). The authors establish exact sequences and push-out diagrams that describe when the full Picard is generated by tautological classes, and they treat genus zero and positive genus separately, with a comprehensive account of when the Picard is controlled by the torus part, the derived subgroup, and the center. The paper also develops a framework for the rigidified stack Bun_G^δ by Z(G), relating its Picard group to NS(Bun_G^δ) and the obstructions given by wt, enabling future study of Brauer groups and related invariants. Overall, the results give a precise, functorial description of the relative Picard groups across families of curves, extending prior work of Faltings, Melo–Viviani, Fringuelli, and FV1–FV2 in a broad, stack-theoretic setting.

Abstract

Given a connected reductive algebraic group G, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.

The line bundles on the moduli stack of principal bundles on families of curves

TL;DR

This work determines the Picard group of the moduli stack Bun_G^δ(C/S) of principal G-bundles over a family of curves, for G reductive, by decoupling abelian and nonabelian contributions through tautological line bundles, determinant of cohomology, and Deligne pairing. Central tools include the transgression map τ_G^δ, the abelianization map ab_sharp, and a detailed analysis of the weight map wt on line bundles after rigidification by the center Z(G). The authors establish exact sequences and push-out diagrams that describe when the full Picard is generated by tautological classes, and they treat genus zero and positive genus separately, with a comprehensive account of when the Picard is controlled by the torus part, the derived subgroup, and the center. The paper also develops a framework for the rigidified stack Bun_G^δ by Z(G), relating its Picard group to NS(Bun_G^δ) and the obstructions given by wt, enabling future study of Brauer groups and related invariants. Overall, the results give a precise, functorial description of the relative Picard groups across families of curves, extending prior work of Faltings, Melo–Viviani, Fringuelli, and FV1–FV2 in a broad, stack-theoretic setting.

Abstract

Given a connected reductive algebraic group G, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.
Paper Structure (10 sections, 23 theorems, 130 equations)

This paper contains 10 sections, 23 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Family of curves
  4. Reductive groups
  5. The stack of $G$-bundles on a family of curves
  6. The Picard group of $\mathop{\mathrm{\mathrm{Bun}}}\nolimits^{\delta}_G(C/S)$
  7. Recollection on $\mathop{\mathrm{Pic}}\nolimits \mathop{\mathrm{\mathrm{Bun}}}\nolimits^{\delta}_G(C/S)$
  8. The Picard group of $\mathop{\mathrm{\mathrm{Bun}}}\nolimits^d_T(C/S)$ for a torus $T$
  9. The Picard group of $\mathop{\mathrm{\mathrm{Bun}}}\nolimits^{\delta}_G(C/S)$ for $G$ reductive
  10. The rigidification of $\mathop{\mathrm{\mathrm{Bun}}}\nolimits_{G}^{\delta}(C/S)$ by the center

Key Result

Theorem 1

(see Theorems T:PicBunT>0, T:all-taut) Let $C\to S$ be a family of curves of genus $g>0$ on a regular and integral quotient stack. Let $T$ be a torus and fix $d\in \Lambda(T)=\pi_1(T)$.

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 45 more