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Uniruledness of Some Moduli Spaces of Pointed Spin Curves

Bogdan-Petru Carasca

TL;DR

The paper addresses whether certain moduli spaces $\mathcal{S}_{g,2n}$ of pointed spin curves are uniruled. It extends the Nikulin-surface framework, particularly non-standard Nikulin surfaces, to construct families of rational curves via pencils in linear systems associated to ramified double covers, leveraging dominant maps to the ramified-cover moduli $\mathcal{R}_{g,2n}$. It proves uniruledness for nine specific pairs $(g,2n)$: $(2,4),(2,6),(3,2),(3,4),(3,6),(4,2),(4,4),(5,2),(5,4)$. This work enhances the understanding of the birational geometry of pointed spin moduli and connects spin-curve geometry with Nikulin surface theory and ramified double covers, complementing existing results on the Kodaira classification of spin-moduli spaces.

Abstract

The moduli space $\mathcal{S}_{g, 2n}$ parametrizes pointed curves with spin structure. We prove that $\mathcal{S}_{g, 2}$, $\mathcal{S}_{g, 4}$ and $\mathcal{S}_{g, 6}$ are uniruled for particular values of $g$.

Uniruledness of Some Moduli Spaces of Pointed Spin Curves

TL;DR

The paper addresses whether certain moduli spaces of pointed spin curves are uniruled. It extends the Nikulin-surface framework, particularly non-standard Nikulin surfaces, to construct families of rational curves via pencils in linear systems associated to ramified double covers, leveraging dominant maps to the ramified-cover moduli . It proves uniruledness for nine specific pairs : . This work enhances the understanding of the birational geometry of pointed spin moduli and connects spin-curve geometry with Nikulin surface theory and ramified double covers, complementing existing results on the Kodaira classification of spin-moduli spaces.

Abstract

The moduli space parametrizes pointed curves with spin structure. We prove that , and are uniruled for particular values of .

Paper Structure

This paper contains 4 sections, 3 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Irreducibility of
  4. Proof of the main theorem

Key Result

Theorem 1

The moduli spaces $\mathcal{S}_{2, 4}$, $\mathcal{S}_{2, 6}$, $\mathcal{S}_{3, 2}$, $\mathcal{S}_{3, 4}$, $\mathcal{S}_{3, 6}$, $\mathcal{S}_{4, 2}$, $\mathcal{S}_{4, 4}$, $\mathcal{S}_{5, 2}$ and $\mathcal{S}_{5, 4}$ are uniruled.

Theorems & Definitions (6)

  • Theorem 1
  • Definition 2
  • Theorem 3: cf. lelli2023uni
  • Proposition 4
  • proof
  • proof : Proof of Theorem 1