An algorithm for computing compatibly Frobenius split subvarieties

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $\phi$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $\phi$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.Comment: 15 pages, many statements clarified and numerous other substantial improvements to the exposition (thanks to the referees). To appear in the Journal of Symbolic Computatio

An algorithm for constructing certain differential operators in positive characteristic

Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some specific families of polynomials, we also study the level of such a differential operator $\delta$, i.e., the least integer $e$ such that $\delta$ is $R^{p^e}$-linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator.Comment: 23 pages. Comments are welcom

Strong F-regularity and generating morphisms of local cohomology modules

We establish a criterion for the strong F-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least 2, containing a perfect field of prime characteristic p. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an n×(n−1) matrix X of indeterminates. For p≥5, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of X is strongly F-regular

D-module and F-module length of local cohomology modules

Let R be a polynomial or power series ring over a field k. We study the length of local cohomology modules HjI (R) in the category of D-modules and Fmodules. We show that the D-module length of HjI (R) is bounded by a polynomial in the degree of the generators of I. In characteristic p > 0 we obtain upper and lower bounds on the F-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of R/I. The obtained upper bound is sharp if R/I is an isolated singularity, and the lower bound is sharp when R/I is Gorenstein and F-pure. We also give an example of a local cohomology module that has different D-module and F-module lengths

Compatibly split subvarieties of the Hilbert scheme of points in the plane

Let k be an algebraically closed field of characteristic p>2. By a result of Kumar and Thomsen, the standard Frobenius splitting of the affine plane induces a Frobenius splitting of the Hilbert scheme of n points in the plane. In this thesis, we investigate the question, "what is the stratification of the Hilbert scheme of points in the plane by all compatibly Frobenius split subvarieties?" We provide the answer to this question when n is at most 4 and we 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 the Hilbert scheme of 5 points, and we show that this particular one-dimensional subvariety is not compatibly split for at least those primes p between 3 and 23. Next, we restrict the splitting of the Hilbert scheme of n points in the plane (now for arbitrary n) to the affine open patch U_ 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_ are Cohen-Macaulay.Comment: Graduate thesi