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Distinguishing Algebraic Spaces from Schemes

Andres Fernandez Herrero, Dario Weißmann, Xucheng Zhang

Abstract

We introduce local invariants of algebraic spaces and stacks which measure how far they are from being a scheme. Using these invariants, we develop mostly topological criteria to determine when the moduli space of a stack is a scheme. As an application we study moduli of principal bundles on a smooth projective curve.

Distinguishing Algebraic Spaces from Schemes

Abstract

We introduce local invariants of algebraic spaces and stacks which measure how far they are from being a scheme. Using these invariants, we develop mostly topological criteria to determine when the moduli space of a stack is a scheme. As an application we study moduli of principal bundles on a smooth projective curve.

Paper Structure

This paper contains 22 sections, 37 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. The most unschematic points
  3. Uniform base points
  4. Schematically trivial points
  5. Descending schematic triviality
  6. Local uniform base points versus schematically trivial points
  7. The locally factorial case
  8. The case of algebraic spaces with mild singularities
  9. Local geometry of stacks via maps to schemes
  10. Schematic dimension and schematic fibers
  11. Computing schematic fibers
  12. Nonschematic points of algebraic spaces
  13. Schematic points of weak topological moduli spaces
  14. Applications to moduli of bundles
  15. Moduli of principal bundles on a curve
  16. ...and 7 more sections

Key Result

Theorem 1.1

If $X$ is a locally factorial irreducible algebraic space proper over an algebraically closed field $k$, then a $k$-point $x$ of $X$ is a local uniform base point if and only if $x$ is schematically trivial, i.e., every Zariski open neighborhood of $x$ has $\mathop{\mathrm{Spec}}(k)$ as its universa

Theorems & Definitions (100)

  • Theorem 1.1: \ref{['thm: schematically trivial = local ubp for proper locally factorial']}
  • Theorem 1.2: \ref{['thm: reason why algebraic space at a point local']}
  • Theorem 1.3: \ref{['prop: schematic points for adequate moduli spaces']}
  • Theorem 1.4: \ref{['thm: semistable and schematic']}, \ref{['thm: open substacks of semistable G-bundles']}
  • Definition 1.5: Schematic point
  • Definition 1.6: Local factoriality
  • Definition 2.1: Uniform base locus and uniform base points
  • Example 2.2
  • Example 2.3
  • Example 2.4: kollar-non-quasiprojectic-moduli
  • ...and 90 more