2012/10/23 by Jenna Rajchgot, Rajchgot, Jenna
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1210.6305
openalex publication_date 2012/10/23 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Let k be an algebraically closed field of characteristic p>2. By a result of\nKumar and Thomsen, the standard Frobenius splitting of the affine plane induces\na Frobenius splitting of the Hilbert scheme of n points in the plane. In this\nthesis, we investigate the question, "what is the stratification of the Hilbert\nscheme of points in the plane by all compatibly Frobenius split subvarieties?"\n We provide the answer to this question when n is at most 4 and we give a\nconjectural answer when n=5. We prove that this conjectural answer is correct\nup to the possible inclusion of one particular one-dimensional subvariety of\nthe Hilbert scheme of 5 points, and we show that this particular\none-dimensional subvariety is not compatibly split for at least those primes p\nbetween 3 and 23.\n Next, we restrict the splitting of the Hilbert scheme of n points in the\nplane (now for arbitrary n) to the affine open patch U_<x,yn> and describe all\ncompatibly split subvarieties of this patch and their defining ideals. We find\ndegenerations of these subvarieties to Stanley-Reisner schemes, explicitly\ndescribe the associated simplicial complexes, and use these complexes to prove\nthat certain compatibly split subvarieties of U_<x,yn> are Cohen-Macaulay.\n