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 A2k induces a Frobenius splitting of Hilbn( A2k ). In this thesis, we investigate the question, "what is the stratification of Hilbn( A2k ) 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( A2k ), 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( A2k ) (now for arbitrary n) to the affine open patch Ux,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 Ux,yn are Cohen-Macaulay.
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