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Construction of the Moduli Space of Vector Bundles on an Orbifold Curve

Soumyadip Das, Souradeep Majumder

Abstract

Let $k$ be an algebraically closed field of any characteristic, and let $(X,P)$ be an orbifold curve over $k$. We construct the moduli space $\mathrm{M}_{(X,P)}^{\mathrm{ss}}(n, Δ)$ of $P$-semistable bundles on $(X,P)$ of rank $n$ and determinant $Δ$. In the characteristic zero case, this result is well known and follows from GIT techniques. Our construction follows a different approach inspired by a GIT-free construction of Faltings. We show that when the moduli space is non-empty, it is a finite disjoint union of irreducible projective varieties.

Construction of the Moduli Space of Vector Bundles on an Orbifold Curve

Abstract

Let be an algebraically closed field of any characteristic, and let be an orbifold curve over . We construct the moduli space of -semistable bundles on of rank and determinant . In the characteristic zero case, this result is well known and follows from GIT techniques. Our construction follows a different approach inspired by a GIT-free construction of Faltings. We show that when the moduli space is non-empty, it is a finite disjoint union of irreducible projective varieties.

Paper Structure

This paper contains 19 sections, 33 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moduli Space of Semistable Bundles on an Algebraic Curve
  4. Orbifold Curves
  5. Orbifold Sheaves of Modules
  6. Stability Conditions on Orbifold Bundles
  7. Descent and cohomologies of Orbifold Bundles
  8. Orbifold Semistable Bundles
  9. The Moduli Problem
  10. Construction of the Moduli Space
  11. Bounded Families
  12. Results on the non-emptiness
  13. The Unstable Locus
  14. Main Theorem
  15. Construction of the Moduli Space in the Classical Context
  16. ...and 4 more sections

Key Result

Lemma \oldthetheorem

Let $(X,P)$ be a connected orbifold curve. Then there is a geometric branch data $\tilde{P}$ on $X$ with $\tilde{P} \geq P$ such that the induced cover $\tau \colon (X,\tilde{P}) \longrightarrow (X,P)$ is tamely ramified.

Theorems & Definitions (63)

  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem: Brochard, SP
  • Corollary \oldthetheorem
  • ...and 53 more