17 research outputs found

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

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    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

    Three invariants of geometrically vertex decomposable ideals

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    We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the aa-invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process. As an application, we prove that the aa-invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the aa-invariant of a toric ideal of a bipartite graph

    Geometric vertex decomposition and liaison for toric ideals of graphs

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    The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci). In this paper, we initiate an investigation into when the toric ideal IGI_G of a finite simple graph GG is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gr\"obner basis of IGI_G is a set of quadratic binomials. We also prove that some other families of graphs have the property that IGI_G is glicci.Comment: 37 pages; in this revised version, Section 7 has been removed due to an error in the example found in previous version
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