Compatibly Split Subvarieties Of The Hilbert Scheme Of Points In The Plane

Abstract

Let k be an algebraically closed field of characteristic p > 2. By a result of Kumar and Thomsen (see [KT01]), the standard Frobenius splitting of A2 induces a Frobenius splitting of Hilbn (A2 ). In k k this thesis, we investigate the question, "what is the stratification of Hilbn (A2 ) by all compatibly k Frobenius split subvarieties?" We provide the answer to this question when n [LESS-THAN OR EQUAL TO] 4 and give a conjectural answer when n = 5. We prove that this conjectural answer is correct up to the possible inclusion of one particular onedimensional subvariety of Hilb5 (A2 ), and we show that this particular one-dimensional subvariety k is not compatibly split for at least those primes p satisfying 2 < p [LESS-THAN OR EQUAL TO] 23. Next, we restrict the splitting of Hilbn (A2 ) (now for arbitrary n) to the affine open patch U k x,y n and describe all compatibly split subvarieties of this patch and their defining ideals. We find degenerations of these subvarieties to Stanley-Reisner schemes, explicitly describe the associated simplicial complexes, and use these complexes to prove that certain compatibly split subvarieties of U x,y n are Cohen-Macaulay.