Moduli of curves, Gr\"obner bases, and the Krichever map

TL;DR

This paper constructs and analyzes moduli spaces of algebraic curves with marked points and additional structures, using Gr"obner bases and the Krichever map, leading to new compactifications and explicit descriptions.

Contribution

It introduces explicit affine schemes for moduli of curves with certain properties and relates them to the Sato Grassmannian via the Krichever map, providing new modular compactifications.

Findings

Moduli spaces are affine schemes of finite type.

Krichever map identifies these spaces with Grassmannian quotients.

Some quotients give modular compactifications of ${ m M}_{g,n}$.

Abstract

We study moduli spaces of (possibly non-nodal) curves (C,p_1,\ldots,p_n) of arithmetic genus g with n smooth marked points, equipped with nonzero tangent vectors, such that ${\mathcal O}_C(p_1+\ldots+p_n)$ is ample and $H^1({\mathcal O}_C(a_1p_1+\ldots+a_np_n))=0$ for given weights ${\bf a}=(a_1,\ldots,a_n)$ such that $a_i\ge 0$ and $\sum a_i=g$. We show that each such moduli space $\widetilde{{\mathcal U}}^{ns}_{g,n}({\bf a})$ is an affine scheme of finite type, and the Krichever map identifies it with the quotient of an explicit locally closed subscheme of the Sato Grassmannian by the free action of the group of changes of formal parameters. We study the GIT quotients of $\widetilde{{\mathcal U}}^{ns}_{g,n}({\bf a})$ by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of ${\mathcal M}_{g,n}$ with projective coarse moduli spaces. More generally, using similar techniques, we construct moduli spaces of curves with chains of divisors supported at marked points, with prescribed number of sections, which in the case n=1 corresponds to specifying the Weierstrass gap sequence at the marked point.