# Equations of $\,\overline{{M}}_{0,n}$

## TL;DR

This paper provides explicit polynomial equations in the Cox ring that define the moduli space of stable n-pointed rational curves, , and proves their correctness for small n, advancing understanding of its algebraic structure.

## Contribution

It introduces explicit equations in the Cox ring for , with proofs for n=5 to 8 and a cohomological embedding for n.

## Key findings

- Equations generate the ideal for n=5,6,7,8.
-  is embedded in projective space via the log canonical system.
- Cohomological proof confirms the embedding for n.

## Abstract

Following work of Keel and Tevelev, we give explicit polynomials in the Cox ring of $\mathbb{P}^1\times\cdots\times\mathbb{P}^{n-3}$ that, conjecturally, determine $\overline{M}_{0,n}$ as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for $n=5, 6, 7, 8$. For $n \leq 6$ we give a cohomological proof that these polynomials realize $\overline{M}_{0,n}$ as a projective variety, embedded in $\mathbb{P}^{(n-2)!-1}$ by the complete log canonical linear system.