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A compact moduli of orbifold projective curves

Tarig Abdelgadir, Daniel Chan, Shinnosuke Okawa, Kazushi Ueda

Abstract

We introduce the notion of stable orbifold projective curves, and show that the moduli stack of stable orbifold projective curves is isomorphic to the moduli stack of weighted pointed stable curves in the sense of Hassett with respect to the weights determined by the automorphism groups of the stacky points.

A compact moduli of orbifold projective curves

Abstract

We introduce the notion of stable orbifold projective curves, and show that the moduli stack of stable orbifold projective curves is isomorphic to the moduli stack of weighted pointed stable curves in the sense of Hassett with respect to the weights determined by the automorphism groups of the stacky points.
Paper Structure (10 sections, 10 theorems, 29 equations)

This paper contains 10 sections, 10 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Ample invertible sheaves on algebraic stacks
  3. Assumptions and notations
  4. Definition of ampleness
  5. Ampleness criterion after Abramovich and Hassett
  6. Ampleness criterion after Artin and Zhang
  7. Stable orbifold projective curves
  8. Weighted pointed stable curves
  9. Curves with weighted divisors
  10. The main theorem

Key Result

Lemma 2.3

If an invertible sheaf $L$ is coarsely ample, then the sheaf $\pi_* \operatorname{Sym}(L)$ of $\mathbb{N}$-graded $\mathcal{O}_B$-algebras is of finite type.

Theorems & Definitions (29)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6: MR2779465, cf. also MR2819757
  • Proposition 2.7
  • proof
  • Definition 2.8: MR1304753
  • Theorem 2.9
  • ...and 19 more