A polygonal presentation of $Pic(\overline{\mathfrak{M}}_{0,n})$

TL;DR

This paper provides a new, simplified polygonal presentation of the Picard group of the moduli space of stable n-pointed rational curves, improving understanding of its structure with minimal relations.

Contribution

It introduces an explicit, simple expression for boundary divisors in terms of a known basis, offering a minimal relation presentation of the Picard group.

Findings

Explicit formula for boundary divisors in terms of a basis

New minimal relation presentation of Picard group

Simplified polygonal representation of the moduli space

Abstract

In the first section of this article, we recall Keel's well-known presentation of $Pic(\overline{\mathfrak{M}}_{0,n})$ using irreducible boundary divisors of $\overline{\mathfrak{M}}_{0,n}$ as generators, and describe a basis for $Pic(\overline{\mathfrak{M}}_{0,n})$ recently discovered by Keel and Gibney. In the second section, we present a theorem which gives an explicit and very simple expression for every boundary divisor in terms of this basis, thereby yielding a new presentation with a minimal set of relations.