# Jenna Rajchgot

### First Position

Research member at the Mathematical Sciences Research Institute (fall 2012); postdoc at the University of Michigan### Dissertation

*Compatibly Split Subvarieties of the Hilbert Scheme of Points in the Plane*

### Advisor

### Research Area

### 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 A_{k}^{2} induces a Frobenius splitting of Hilb^{n}(A_{k}^{2}). In this thesis, we investigate the question, “what is the stratification of Hilb^{n}(A_{k}^{2}) 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 Hilb^{5}(A_{k}^{2}), 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 Hilb^{n}(A_{k}^{2}) (now for arbitrary *n*) to the affine open patch *U*_〈*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.