Jenna Rajchgot

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 A_k² induces a Frobenius splitting of Hilbⁿ(A_k²). In this thesis, we investigate the question, “what is the stratification of Hilbⁿ(A_k²) by all compatibly Frobenius split subvarieties?”

We provide the answer to this question when n ≤ 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 one-dimensional subvariety of Hilb⁵(A_k²), and we show that this particular one-dimensional subvariety is not compatibly split for at least those primes p satisfying 2 < p ≤ 23.

Next, we restrict the splitting of Hilbⁿ(A_k²) (now for arbitrary n) to the affine open patch U_〈x,yⁿ〉 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,yⁿ〉are Cohen-Macaulay.