4 research outputs found
Strongly stable ideals and Hilbert polynomials
The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a
method to compute all saturated strongly stable ideals in a given polynomial
ring with a fixed Hilbert polynomial. A description of the main method and
auxiliary tools is given.Comment: Source code available as an ancillary file. Final versio
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
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
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