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Gluing for stable families of surfaces in mixed characteristic

Quentin Posva

Abstract

We study the normalizations of non-normal stable families of slc surfaces over an excellent DVR. In mixed characteristic, we establish a gluing statement that is relevant for the properness of the moduli space of such surfaces. We also study the fibers of such glued families, in mixed and equi-characteristic, and provide an essential result for slc adjunction in residue characteristic >5.

Gluing for stable families of surfaces in mixed characteristic

Abstract

We study the normalizations of non-normal stable families of slc surfaces over an excellent DVR. In mixed characteristic, we establish a gluing statement that is relevant for the properness of the moduli space of such surfaces. We also study the fibers of such glued families, in mixed and equi-characteristic, and provide an essential result for slc adjunction in residue characteristic >5.

Paper Structure

This paper contains 16 sections, 37 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Conventions and notations
  5. Quotients by finite equivalence relations
  6. Demi-normal varieties and slc pairs
  7. Semi-normality, weak normality and demi-normality
  8. Slc compactification for surfaces
  9. Gluing for families of surfaces
  10. Existence of the quotient
  11. Descent of the log canonical sheaf
  12. Proof of the gluing theorem
  13. Fibers of the quotient
  14. Commutativity of fibers and quotients
  15. The case of locally stable families.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $R$ be an excellent DVR of mixed characteristic with maximal ideal $\pi R$. Then normalization gives a bijection (On the right-hand side, $\bar{D}$ is understood to be a Weil divisor, possibly reducible, whose every coefficient is equal to $1$.)

Theorems & Definitions (97)

  • Theorem 1: see \ref{['proposition:gluing_exists']} and \ref{['proposition:canonical_sheaf_descends']}
  • Theorem 2: \ref{['thm:commutativity_in_slc_case']}
  • Theorem 3: \ref{['proposition:intersection_of_lc_centers']}
  • Theorem 4
  • Theorem 5
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • ...and 87 more