2021/04/29 by Soohyun Park, Park, Soohyun
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Codimension #Combinatorics #Commutative Algebra and Its Applications #Computer science #FOS: Mathematics #Fano plane #Field (mathematics) #Finite field #Hypersurface #Identity (music) #Mathematical analysis #Mathematics #Measure (data warehouse) #Number Theory (math.NT) #Physics #Polynomial and algebraic computation #Pure mathematics #Ring (chemistry) #Stability (learning theory) #Statistics #Subspace topology #Variety (cybernetics) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.2104.14381
published in arXiv (Cornell University) (Cornell University) · Final version with edits made in response to referee comments; 50 pages, 7 figures
openalex publication_date 2021/04/29 · arxiv created 2022/04/24 · arxiv updated 2022/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the probability that an (n - m)-dimensional linear subspace in ℙn or a collection of points spanning such a linear subspace is contained in an m-dimensional variety Y ⊂ ℙn. This involves a strategy used by Galkin--Shinder to connect properties of a cubic hypersurface to its Fano variety of lines via cut and paste relations in the Grothendieck ring of varieties. Generalizing this idea to varieties of higher codimension and degree, we can measure growth rates of weighted probabilities of k-planes contained in a sequence of varieties with varying initial parameters over a finite field. In the course of doing this, we move an identity motivated by rationality problems involving cubic hypersurfaces to a motivic statistics setting associated with cohomological stability.