Compatibly Frobenius split subschemes are rigid

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 $ϕ: F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $ϕ$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $ϕ$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.

Finiteness of the number of compatibly-split subvarieties

We show that for a Frobenius split variety, there are only finitely many closed subvarieties which are compatibly-split.

On the number of compatibly Frobenius split subvarieties, prime $F$-ideals, and log canonical centers

Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $θ: F_* Ø_X \to Ø_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $θ$. In particular, suppose $\sL$ is a sufficiently ample line bundle on $X$ (for example, if $\sL$ induces a projectively normal embedding) with $n = \dim H^0(X, \sL)$. We show that the number of $d$-dimensional irreducible subvarieties of $X$ that are compatibly split by $θ$ is less than or equal to ${n \choose d+1}$. This generalizes a well known result on the number of closed poin…

Frobenius splitting, point-counting, and degeneration

Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface {f=0} one may construct a number of other subschemes {Y} by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in \FF_p^n is not a multiple of p, then all these intersections in Å^n_{\FF_p} just described are reduced. (If this holds for infinitely many p, then it holds over \QQ as well.) More specifically, there is a_Frobenius splitting_ on Å^n_{\FF_p} compatibly splitting all these subschemes…

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-dime…

通过几何顶点分解提升弗罗贝尼乌斯分裂

Lifting Frobenius splittings through geometric vertex decomposition

研究通过几何顶点分解提升弗罗贝尼乌斯分裂。核心方法是利用其与戈伦斯坦联络关系，主要贡献是给出克努森结果部分逆命题，证明特定条件下可提升分裂，还讨论示例及证明李氏双行列式簇是弗罗贝尼乌斯分裂。