Counting point configurations in projective space
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917253841158144 |
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| author | Fink, Alex Nabijou, Navid Silversmith, Rob |
| author_facet | Fink, Alex Nabijou, Navid Silversmith, Rob |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15421 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting point configurations in projective space Fink, Alex Nabijou, Navid Silversmith, Rob Algebraic Geometry Combinatorics 14N10, 14N20, 14M15, 14C17, 05D15 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. |
| title | Counting point configurations in projective space |
| topic | Algebraic Geometry Combinatorics 14N10, 14N20, 14M15, 14C17, 05D15 |
| url | https://arxiv.org/abs/2601.15421 |