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On supercurves of genus 1 with an underlying odd spin structure

Alexander Polishchuk

Abstract

We study the standard family of supercurves of genus 1 with an underlying odd spin structures. We give a simple algebraic description of this family and of the compactified family of stable supercurves with one Neveu-Schwarz puncture. We also describe the Gauss-Manin connection on the first de Rham cohomology of this family and compute the superperiods of global differentials.

On supercurves of genus 1 with an underlying odd spin structure

Abstract

We study the standard family of supercurves of genus 1 with an underlying odd spin structures. We give a simple algebraic description of this family and of the compactified family of stable supercurves with one Neveu-Schwarz puncture. We also describe the Gauss-Manin connection on the first de Rham cohomology of this family and compute the superperiods of global differentials.
Paper Structure (11 sections, 11 theorems, 102 equations)

This paper contains 11 sections, 11 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Supercurves of genus $1$ with an underlying odd spin structure
  3. Weierstrass functions
  4. Standard family of cupercurves genus $1$ with an underlying odd spin structure
  5. Functions on the affine supercurve
  6. An algebraic model
  7. Kodaira-Spencer map and compactification
  8. Cohomology and Gauss-Manin connection
  9. Cohomology and differentials
  10. Gauss-Manin connection: generalities
  11. Gauss-Manin-connection for supercurves of genus $1$

Key Result

Proposition 2.2

The functions $\Psi_1$ and $\Psi_2$ are well defined holomorphic functions on $X-p$. For each $\tau$, the functions form a $\mathop{\mathrm{\mathbb C}}\nolimits[\varphi]$-basis of $H^0(X_\tau - p,{\mathcal{O}})$. One has

Theorems & Definitions (24)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • Theorem 2.6
  • ...and 14 more