Moduli spaces of nonspecial pointed curves of arithmetic genus 1

TL;DR

This paper provides an explicit description of GIT semistable loci for moduli stacks of genus 1 curves with marked points, connecting these to known modular compactifications and advancing the understanding of their geometric structure.

Contribution

It offers a detailed characterization of GIT semistable loci in the moduli stack of genus 1 curves with nonspecial marked points, linking GIT quotients to Smyth's modular compactifications.

Findings

Explicit description of GIT semistable loci in the moduli stack.

Identification of some GIT quotients with Smyth's modular compactifications.

Enhanced understanding of the geometric structure of genus 1 curve moduli.

Abstract

In this paper we study the moduli stack ${\mathcal U}_{1,n}^{ns}$ of curves of arithmetic genus 1 with n marked points, forming a nonspecial divisor. In arXiv:1511.03797 this stack was realized as the quotient of an explicit scheme $\widetilde{\mathcal U}_{1,n}^{ns}$, affine of finite type over ${\Bbb P}^{n-1}$, by the action of ${\Bbb G}_m^n$ . Our main result is an explicit description of the corresponding GIT semistable loci in $\widetilde{\mathcal U}_{1,n}^{ns}$. This allows us to identify some of the GIT quotients with some of the modular compactifications of ${\mathcal M}_{1,n}$ defined by Smyth in arXiv:0902.3690 and arXiv:0808.0177.