Counting point configurations in projective space
Alex Fink, Navid Nabijou, Rob Silversmith
TL;DR
The paper develops a higher-dimensional analogue of cross-ratio degrees by introducing projective configuration counts $d_{r,n}(\mathcal{I})$ for configurations of $n$ points in $\mathbb{P}^{r-1}$ subject to $k=n-r-1$ constraints of size $r+2$. It proves two main results: an explicit combinatorial upper bound $d_{r,n}(\mathcal{I})\le|T_{r-1}(\Gamma(\mathcal{I})\setminus S)|$ via weighted transversals of a pruned configuration graph, and a dimension-reduction principle $d_{r+1,n+1}(\mathcal{I}')=d_{r,n}(\mathcal{I})$ when a marking lies in every constraint. The authors deploy the Gelfand–MacPherson correspondence, Jacobi–Trudi and Thom–Porteous formulas, and a surplus/positivity framework from matching theory to connect these counts to intersection theory on moduli/Grassmannian spaces, and they discuss compactifications, potential tropical algorithms, and conjectural positivity criteria. They also present stochastic sample calculations illustrating the upper bound and its tightness in certain cases, while highlighting where naive bounds fail. Overall, the work lays a foundation for systematic enumeration of high-dimensional point configurations and suggests concrete avenues for algorithms and further structural understanding via matroid-like objects.
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.