17 research outputs found
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
Three invariants of geometrically vertex decomposable ideals
We study three invariants of geometrically vertex decomposable ideals: the
Castelnuovo-Mumford regularity, the multiplicity, and the -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 -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 -invariant of a toric ideal of
a bipartite graph
Geometric vertex decomposition and liaison for toric ideals of graphs
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 of a finite simple graph 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 is a set of quadratic
binomials. We also prove that some other families of graphs have the property
that 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