84 research outputs found
A Borel open cover of the Hilbert scheme
Let be an admissible Hilbert polynomial in \PP^n of degree . The
Hilbert scheme \hilb^n_p(t) can be realized as a closed subscheme of a
suitable Grassmannian , hence it could be globally defined by
homogeneous equations in the Plucker coordinates of and covered by
open subsets given by the non-vanishing of a Plucker coordinate, each embedded
as a closed subscheme of the affine space , . However,
the number of Plucker coordinates is so large that effective computations
in this setting are practically impossible. In this paper, taking advantage of
the symmetries of \hilb^n_p(t), we exhibit a new open cover, consisting of
marked schemes over Borel-fixed ideals, whose number is significantly smaller
than . Exploiting the properties of marked schemes, we prove that these open
subsets are defined by equations of degree in their natural
embedding in \Af^D. Furthermore we find new embeddings in affine spaces of
far lower dimension than , and characterize those that are still defined by
equations of degree . The proofs are constructive and use a
polynomial reduction process, similar to the one for Grobner bases, but are
term order free. In this new setting, we can achieve explicit computations in
many non-trivial cases.Comment: 17 pages. This version contains and extends the first part of version
2 (arXiv:0909.2184v2[math.AG]). A new extended version of the second part,
with some new results, is posed at arxiv:1110.0698v3[math.AC]. The title is
slightly changed. Final version accepted for publicatio
Segre Class Computation and Practical Applications
Let be closed (possibly singular) subschemes of a smooth
projective toric variety . We show how to compute the Segre class
as a class in the Chow group of . Building on this, we give effective
methods to compute intersection products in projective varieties, to determine
algebraic multiplicity without working in local rings, and to test pairwise
containment of subvarieties of . Our methods may be implemented without
using Groebner bases; in particular any algorithm to compute the number of
solutions of a zero-dimensional polynomial system may be used
Semi-inverted linear spaces and an analogue of the broken circuit complex
The image of a linear space under inversion of some coordinates is an affine
variety whose structure is governed by an underlying hyperplane arrangement. In
this paper, we generalize work by Proudfoot and Speyer to show that circuit
polynomials form a universal Groebner basis for the ideal of polynomials
vanishing on this variety. The proof relies on degenerations to the
Stanley-Reisner ideal of a simplicial complex determined by the underlying
matroid. If the linear space is real, then the semi-inverted linear space is
also an example of a hyperbolic variety, meaning that all of its intersection
points with a large family of linear spaces are real.Comment: 16 pages, 1 figure, minor revisions and added connections to the
external activity complex of a matroi
A Borel open cover of the Hilbert scheme
Let be an admissible Hilbert polynomial in of degree . The Hilbert scheme can be realized as a closed subscheme of a suitable Grassmannian , hence it could be globally defined by homogeneous equations in the Plücker coordinates of and covered by open subsets given by the non-vanishing of a Plücker coordinate, each embedded as a closed subscheme of the affine space , . However, the number of Plücker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of , we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than . Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree in their natural embedding in . Furthermore we find new embeddings in affine spaces of far lower dimension than , and characterize those that are still defined by equations of degree . The proofs are constructive and use a polynomial reduction process, similar to the one for Gröbner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases
On the Rapoport-Zink space for over a ramified prime
In this work, we study the supersingular locus of the Shimura variety
associated to the unitary group over a ramified prime. We
show that the associated Rapoport-Zink space is flat, and we give an explicit
description of the irreducible components of the reduction modulo of the
basic locus. In particular, we show that these are universally homeomorphic to
either a generalized Deligne-Lusztig variety for a symplectic group or to the
closure of a vector bundle over a classical Deligne-Lusztig variety for an
orthogonal group. Our results are confirmed in the group-theoretical setting by
the reduction method \`a la Deligne and Lusztig and the study of the admissible
set
Quasi-Splines and their moduli
We study what we call quasi-spline sheaves over locally Noetherian schemes.
This is done with the intention of considering splines from the point of view
of moduli theory. In other words, we study the way in which certain objects
that arise in the theory of splines can be made to depend on parameters. In
addition to quasi-spline sheaves, we treat ideal difference-conditions, and
individual quasi- splines. Under certain hypotheses each of these types of
objects admits a fine moduli scheme. The moduli of quasi-spline sheaves is
proper, and there is a natural compactification of the moduli of ideal
difference-conditions. We include some speculation on the uses of these moduli
in the theory of splines and topology, and an appendix with a treatment of the
Billera-Rose homogenization in scheme theoretic language
A counterexample to the parity conjecture
Let be a zero-dimensional subscheme of the
affine three-dimensional complex space of length . Okounkov and
Pandharipande have conjectured that the dimension of the tangent space of
at and have have the same parity. The
conjecture was proven by Maulik, Nekrasov, Okounkov and Pandharipande for
points defined by monomial ideals and very recently by Ramkumar and
Sammartano for homogeneous ideals. In this paper we exhibit a family of
zero-dimensional schemes in , which disproves the
conjecture in the general non-homogeneous case.Comment: 11 pages. Comments are welcom
Polynomial Equations: Theory and Practice
Solving polynomial equations is a subtask of polynomial optimization. This
article introduces systems of such equations and the main approaches for
solving them. We discuss critical point equations, algebraic varieties, and
solution counts. The theory is illustrated by many examples using different
software packages.Comment: This article will appear as a chapter of a forthcoming book
presenting research acitivies conducted in the European Network POEMA. It
discusses polynomial equations, with optimization as point of entry. 24
pages, 7 figure
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