Counting point configurations in projective space

TL;DR

This paper develops a combinatorial framework for counting specific point configurations in projective space, generalizing cross-ratio degrees and providing bounds and recursive formulas for these counts.

Contribution

It introduces projective configuration counts, establishes an upper bound via bipartite graph transversals, and derives a recursion relating counts across different dimensions.

Findings

Provides a combinatorial upper bound for configuration counts.

Establishes a recursion linking counts in different projective dimensions.

Connects geometric enumeration with bipartite graph matching theory.

Abstract

We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed relative positions. The $\mathbb{P}^1$ case recovers cross-ratio degrees, which arise naturally in numerous contexts. We establish two main results. The first is a combinatorial upper bound given by the number of weighted transversals of a bipartite graph. The second is a recursion that relates counts associated to projective spaces of different dimensions, by projecting away from a given point. Key inputs include the Gelfand-MacPherson correspondence, the Jacobi-Trudi and Thom-Porteous formulae, and the notion of surplus from matching theory of bipartite graphs.