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On super curves and supervolumes

Ricardo Jesús Ramos Castillo

Abstract

We study the geometry of super curves with a chosen supervolume form. We consider the algebra of divergence free vector fields $S(1|N)$ associated to such curves. When $N=2$ its derived algebra, called $S(2)$, defines a special family of curves, named $S(2)$-super curves. We exhibit an involution on the moduli space of such curves that generalizes Deligne's involution for $N=1$ super curves. The fixed point set of this involution consists on Manin's $SUSY_2$-super curves. We describe the moduli spaces of these curves.

On super curves and supervolumes

Abstract

We study the geometry of super curves with a chosen supervolume form. We consider the algebra of divergence free vector fields associated to such curves. When its derived algebra, called , defines a special family of curves, named -super curves. We exhibit an involution on the moduli space of such curves that generalizes Deligne's involution for super curves. The fixed point set of this involution consists on Manin's -super curves. We describe the moduli spaces of these curves.
Paper Structure (21 sections, 21 theorems, 135 equations)

This paper contains 21 sections, 21 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Super algebra
  3. Super algebras
  4. Super modules
  5. Super derivations
  6. The group $\mathrm{Aut}_R(R[[ m|n]])$
  7. The group $\mathrm{Aut}_R^\delta(R[[1|n]])$
  8. The group $\mathrm{Aut}_R^\omega(R[[1|n]])$
  9. Super manifolds
  10. $S(2)$-super curves
  11. $SUSY$-super curves
  12. $S(2)$-super curves and $SUSY_4$-super curves
  13. Families of super curves
  14. A family of $S(2)$-super curves
  15. A family of $S(1|2)$-super curves
  16. ...and 6 more sections

Key Result

Theorem 1.4

There exists an involution $\mu$ of the moduli space $\mathcal{M}_{S(2)}$ of $S(2)$-super curves. The fixed point set of $\mu$ consists of the moduli space $\mathcal{M}_{K(1|2)}$ of orientable $SUSY_2$-super curves.

Theorems & Definitions (58)

  • Theorem 1.4: Theorem \ref{['teorema de automorfismo']}
  • Theorem 1.7: Theorem \ref{['split']}
  • Proposition
  • Proposition
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.6
  • Example 2.7
  • ...and 48 more