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Moduli of $\mathbb{Q}$-Gorenstein pairs and applications

Stefano Filipazzi, Giovanni Inchiostro

Abstract

We develop a framework to construct moduli spaces of $\mathbb{Q}$-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of $\mathbb{Q}$-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than $\frac{1}{2}$. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.

Moduli of $\mathbb{Q}$-Gorenstein pairs and applications

Abstract

We develop a framework to construct moduli spaces of -Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of -stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than . Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.

Paper Structure

This paper contains 22 sections, 32 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Connections with other works
  4. Preliminaries
  5. Terminology and conventions
  6. Contractions
  7. Divisors
  8. Non-normal varieties and pairs
  9. Divisorial sheaves
  10. Boundedness
  11. Index of a set
  12. Stable pairs and $\mathbb{Q}$-stable pairs
  13. Families of pairs
  14. Families of $\mathbb{Q}$-pairs
  15. Hassett's example
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix an integer $n\in \mathbb{N}$ and a polynomial $p(t) \in \mathbb{Q}[t]$. Then, there is a proper Deligne--Mumford stack $\mathscr{F}_{n,p,1}$, with projective coarse moduli space, parametrizing $\mathbb{Q}$-stable ($\mathbb{Q}$-Gorenstein) pairs $(X;D)$ of dimension $n$ and with polynomial $p(t)$

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Corollary \ref{['cor proj moduli']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Remark 2.4
  • proof
  • Definition 2.5
  • Definition 2.6: HK04*§ 3
  • ...and 75 more