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The supermoduli space of genus zero SUSY curves with Ramond punctures

Nadia Ott, Alexander A. Voronov

Abstract

We give an explicit quotient construction of the supermoduli space $\mathfrak{M}_{0, n_R}$ of genus zero super Riemann surfaces with $n_R \ge 4$ Ramond punctures and prove it is a Deligne-Mumford superstack. We also make an explicit quotient construction of the moduli space of genus zero supercurves without a SUSY structure and thereby prove it is an algebraic superstack.

The supermoduli space of genus zero SUSY curves with Ramond punctures

Abstract

We give an explicit quotient construction of the supermoduli space of genus zero super Riemann surfaces with Ramond punctures and prove it is a Deligne-Mumford superstack. We also make an explicit quotient construction of the moduli space of genus zero supercurves without a SUSY structure and thereby prove it is an algebraic superstack.

Paper Structure

This paper contains 10 sections, 32 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Algebraic Supergeometry
  4. Supercurves
  5. Weighted Projective Superspace
  6. Super Riemann Surfaces
  7. The Moduli Problem
  8. Deformations of $\mathbb{W} \mathbb{P}$
  9. The moduli superstack $M(1-n_R/2)$ of supercurves
  10. Construction of the moduli superstack $\mathfrak{M}_{0,n_R}$ of SUSY curves

Key Result

Lemma 4.2

Let $X$ be a genus-zero supercurve over $k$. Then there exists a unique $m \in \mathbb{Z}$ such that $X \cong \mathbb{W}\mathbb{P}^{1|1}(1,1\, |\, m)$.

Theorems & Definitions (89)

  • Definition 3.1
  • Definition 3.2: Superscheme
  • Remark 3.3
  • Example 3.4: Affine Superscheme
  • Example 3.5: Affine Superspace
  • Example 3.6: Projective Superspace
  • Example 3.7: Weighted Projective Superspace
  • Definition 4.1: Relative Supercurve
  • Lemma 4.2
  • proof
  • ...and 79 more