8,401 research outputs found
Segments and Hilbert schemes of points
Using results obtained from the study of homogeneous ideals sharing the same
initial ideal with respect to some term order, we prove the singularity of the
point corresponding to a segment ideal with respect to the revlex term order in
the Hilbert scheme of points in . In this context, we look inside
properties of several types of "segment" ideals that we define and compare.
This study led us to focus our attention also to connections between the shape
of generators of Borel ideals and the related Hilbert polynomial, providing an
algorithm for computing all saturated Borel ideals with the given Hilbert
polynomial.Comment: 19 pages, 2 figures. Comments and suggestions are welcome
Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial
Let be an algebraically closed field of null characteristic and a
Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity
of closed subschemes of projective spaces over with Hilbert
polynomial . Experimental evidences led us to consider the idea that
could be achieved by schemes having a suitable minimal Hilbert
function. We give a constructive proof of this fact. Moreover, we are able to
compute the minimal Castelnuovo-Mumford regularity of
schemes with Hilbert polynomial and given regularity of the
Hilbert function, and also the minimal Castelnuovo-Mumford regularity of
schemes with Hilbert function . These results find applications in the study
of Hilbert schemes. They are obtained by means of minimal Hilbert functions and
of two new constructive methods which are based on the notion of
growth-height-lexicographic Borel set and called ideal graft and extended
lifting.Comment: 21 pages. Comments are welcome. More concise version with a slight
change in the title. A further revised version has been accepted for
publication in Experimental Mathematic
Discrete denoising of heterogenous two-dimensional data
We consider discrete denoising of two-dimensional data with characteristics
that may be varying abruptly between regions.
Using a quadtree decomposition technique and space-filling curves, we extend
the recently developed S-DUDE (Shifting Discrete Universal DEnoiser), which was
tailored to one-dimensional data, to the two-dimensional case. Our scheme
competes with a genie that has access, in addition to the noisy data, also to
the underlying noiseless data, and can employ different two-dimensional
sliding window denoisers along distinct regions obtained by a quadtree
decomposition with leaves, in a way that minimizes the overall loss. We
show that, regardless of what the underlying noiseless data may be, the
two-dimensional S-DUDE performs essentially as well as this genie, provided
that the number of distinct regions satisfies , where is the total
size of the data. The resulting algorithm complexity is still linear in both
and , as in the one-dimensional case. Our experimental results show that
the two-dimensional S-DUDE can be effective when the characteristics of the
underlying clean image vary across different regions in the data.Comment: 16 pages, submitted to IEEE Transactions on Information Theor
A survey of Heisenberg categorification via graphical calculus
In this expository paper we present an overview of various graphical
categorifications of the Heisenberg algebra and its Fock space representation.
We begin with a discussion of "weak" categorifications via modules for Hecke
algebras and "geometrizations" in terms of the cohomology of the Hilbert
scheme. We then turn our attention to more recent "strong" categorifications
involving planar diagrammatics and derived categories of coherent sheaves on
Hilbert schemes.Comment: 23 pages; v2: Some typos corrected and other minor improvements made;
v3: Some small errors corrected; v4: Code corrected to fix problem with
missing arrows on some diagram
Ideals with an assigned initial ideal
The stratum St(J,<) (the homogeneous stratum Sth(J,<) respectively) of a
monomial ideal J in a polynomial ring R is the family of all (homogeneous)
ideals of R whose initial ideal with respect to the term order < is J. St(J,<)
and Sth(J,<) have a natural structure of affine schemes. Moreover they are
homogeneous w.r.t. a non-standard grading called level. This property allows us
to draw consequences that are interesting from both a theoretical and a
computational point of view. For instance a smooth stratum is always isomorphic
to an affine space (Corollary 3.6). As applications, in Sec. 5 we prove that
strata and homogeneous strata w.r.t. any term ordering < of every saturated
Lex-segment ideal J are smooth. For Sth(J,Lex) we also give a formula for the
dimension. In the same way in Sec. 6 we consider any ideal R in k[x0,..., xn]
generated by a saturated RevLex-segment ideal in k[x,y,z]. We also prove that
Sth(R,RevLex) is smooth and give a formula for its dimension.Comment: 14 pages, improved version, some more example
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