Algebra of global sections of $\psi$-bundles on $\bar{M}_{0,n}$

TL;DR

This paper studies the algebra of global sections of line bundles on the moduli space of stable genus 0 curves, providing a simple presentation and proving geometric properties of a related moduli space.

Contribution

It offers a new presentation of the algebra of sections and establishes Cohen-Macaulayness and normality of the moduli space of ψ-stable curves.

Findings

Presented a quadratic relations-based algebraic description.

Proved the moduli space is Cohen-Macaulay and normal.

Showed the natural map is a rational resolution.

Abstract

We consider the ${\mathbb Z}^n$-graded algebra of global sections of line bundles generated by the standard line bundles $L_1,\ldots,L_n$ on $\bar{M}_{0,n}$. We find a simple presentation of this algebra by generators and quadratic relations. As an application we prove that the moduli space $\bar{M}_{0,n}[\psi]$ of $\psi$-stable curves of genus $0$ is Cohen-Macaulay and normal, and the natural map $\bar{M}_{0,n}\to \bar{M}_{0,n}[\psi]$ is a rational resolution.