2024/02/19 by Isidora Bailly-Hall, Christine Berkesch, Bailly-Hall, Isidora +9
Computer Science · Engineering · #14F06 #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Digital Image Processing Techniques #FOS: Mathematics #Primary: 13D02. Secondary: 14M25
paper · pdf · doi:10.48550/arxiv.2402.12495
openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For finite sets of points in ℙn × ℙm, we produce short virtual resolutions, as introduced by Berkesch--Erman--Smith. We first intersect with a sufficiently high power of one set of variables for points in ℙn × ℙm to produce a virtual resolution of length n+m. Then, we describe an explicit virtual resolution of length 3 for a set of points in sufficiently general position in ℙ1 × ℙ2, via a subcomplex of a free resolution. This first result generalizes to ℙn × ℙm work of Harada--Nowroozi--Van Tuyl, and the second partially generalizes work of Harada--Nowroozi--Van Tuyl and Booms-Peot, which were both for ℙ1 × ℙ1. Along the way, we also note an explicit relationship between Betti numbers and higher difference matrices of bigraded Hilbert functions for ℙn × ℙm.