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Murphy's Law for Algebraic Stacks

Daniel Bragg, Max Lieblich

Abstract

We show that various natural algebro-geometric moduli stacks, including the stack of curves, have the property that every Deligne-Mumford gerbe over a field appears as the residual gerbe of one of their points. These gerbes are universal obstructions for objects of the stack to be defined over their fields of moduli, and for the corresponding coarse moduli space to be fine. Thus, our results show that many natural moduli stacks hold objects that are obstructed from being defined over their fields of moduli in every possible way, and have coarse spaces which fail to be fine moduli spaces in every possible way. A basic insight enabling our arguments is that many classical constructions in equivariant projective geometry generalize to the setting of relative geometry over an arbitrary Deligne-Mumford gerbe over a field.

Murphy's Law for Algebraic Stacks

Abstract

We show that various natural algebro-geometric moduli stacks, including the stack of curves, have the property that every Deligne-Mumford gerbe over a field appears as the residual gerbe of one of their points. These gerbes are universal obstructions for objects of the stack to be defined over their fields of moduli, and for the corresponding coarse moduli space to be fine. Thus, our results show that many natural moduli stacks hold objects that are obstructed from being defined over their fields of moduli in every possible way, and have coarse spaces which fail to be fine moduli spaces in every possible way. A basic insight enabling our arguments is that many classical constructions in equivariant projective geometry generalize to the setting of relative geometry over an arbitrary Deligne-Mumford gerbe over a field.
Paper Structure (35 sections, 81 theorems, 94 equations)

This paper contains 35 sections, 81 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Consequences of Murphy's Law
  4. Some explicit examples
  5. Organization of this paper
  6. Acknowledgments
  7. Gerbes over a field and the inertial action
  8. The inertial action
  9. Deligne--Mumford gerbes over a field
  10. The inertial action on a scheme over a gerbe
  11. Complete intersections over gerbes
  12. Complete intersections over a gerbe, over a finite field
  13. Curves over gerbes
  14. Pencils on curves over gerbes
  15. Finite morphisms of curves 1: condition $(\ast)$
  16. ...and 20 more sections

Key Result

Theorem 1.1.2

Let $K_0$ be a field. The following moduli stacks satisfy Stacky Murphy's Law.

Theorems & Definitions (229)

  • Definition 1.1.1
  • Theorem 1.1.2
  • Remark 1.1.3
  • Remark 1.1.4
  • Remark 1.1.5
  • Theorem 1.2.1
  • Corollary 1.2.2
  • Theorem 1.2.3
  • Example 1.2.4
  • Theorem 1.2.5
  • ...and 219 more