Birational contractions of $\overline{\mathrm{M}}_{0,n}$ and combinatorics of extremal assignments

TL;DR

This paper investigates the combinatorial structures of extremal assignments in the context of moduli spaces of pointed rational curves, revealing their atomic decomposition and connections to birational geometry.

Contribution

It demonstrates that extremal assignments can be decomposed into finite unions of atomic extremal assignments and links these to combinatorial objects and birational geometry.

Findings

Extremal assignments are finite unions of atomic extremal assignments.

Identified combinatorial objects corresponding to special classes of extremal assignments.

Established connections between extremal assignments and the birational geometry of moduli spaces.

Abstract

From Smyth's classification, modular compactifications of pointed smooth rational curves are indexed by combinatorial data, so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a finite union of atomic extremal assignments. We discuss a connection with the birational geometry of the moduli space of stable pointed curves. As applications, we study three special classes of extremal assignments: smooth, toric, and invariant with respect to the symmetric group action. We identify them with three combinatorial objects: simple intersecting families, complete multipartite graphs, and special families of integer partitions, respectively.