On an extension of Galligo's theorem concerning the Borel-fixed points on the Hilbert scheme

Sherman, Morgan

doi:10.1016/j.jalgebra.2007.02.045

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…As shown in [25], the set of monomials of a fixed degree of a Borel ideal i is a filter for the transitive closure of the partial ordering ≤ B induced by the relation (8.1) (X α X i−1 > B X α X i ). For each s, we can look for a term ordering ≺ s , obtained refining the partial order ≤ B , such that the ideal j s becomes a (6, ≺ s )-segment.…”

mentioning

confidence: 99%

Rational Components of Hilbert Schemes

Lella¹,

Roggero²

2011

Rend. Sem. Mat. Univ. Padova

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-The Gro Èbner stratum of a monomial ideal j is an affine variety that parameterizes the family of all ideals having j as initial ideal (with respect to a fixed term ordering). The Gro Èbner strata can be equipped in a natural way with a structure of homogeneous variety and are in a close connection with Hilbert schemes of subschemes in the projective space P n . Using properties of the Gro Èbner strata we prove some sufficient conditions for the rationality of components of rilb n p(z) . We show for instance that all the smooth, irreducible components in rilb n p(z) (or in its support) and the Reeves and Stillman component H RS are rational. We also obtain sufficient conditions for isomorphisms between strata corresponding to pairs of ideals defining a same subscheme, that can strongly improve an explicit computation of their equations.

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confidence: 99%

Rational Components of Hilbert Schemes

Lella¹,

Roggero²

2011

Rend. Sem. Mat. Univ. Padova

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“…As an application of our geometric study of the Gin decomposition of the Hilbert scheme, we give a geometric proof of the existence of generic initial ideals and their Borel-fixed properties. One of the key ingredients is that initial ideals can be thought of as flat limits with respect to a one-parameter subgroup action: Dave Bayer and Ian Morrison used this in their study of state polytopes of Hilbert points [BS87a], and more recently Morgan Sherman has also used it to prove that the one-parameter subgroup [She07] taking an ideal to its generic initial ideal is also Borel fixed.…”

Section: Primary and Secondary Generic Initial Idealsmentioning

confidence: 99%

Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

Hyeon

Park

2018

Proceedings of the Edinburgh Mathematical Society

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We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications: First, we give a completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Secondly, we prove that when a Hilbert scheme of nonconstant Hilbert polynomial is embedded by the Grothendieck-Plücker embedding of a high enough degree, it must be degenerate.

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“…Therefore, it is natural to investigate the structure of Hilb n p(t) using the Borel ideals and the action of the linear group. This will be our approach and was recently considered for instance by [23] to prove a first order infinitesimal version of Galligo's Theorem.…”

Section: 2mentioning

confidence: 99%

A Borel open cover of the Hilbert scheme

Bertone

Lella

Roggero

2013

Journal of Symbolic Computation

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Let p(t) be an admissible Hilbert polynomial in P n of degree d. The Hilbert scheme Hilb n p(t) can be realized as a closed subscheme of a suitable Grassmannian G, hence it could be globally defined by homogeneous equations in the Plücker coordinates of G and covered by open subsets given by the non-vanishing of a Plücker coordinate, each embedded as a closed subscheme of the affine space A D , D = dim(G). However, the number E of Plücker 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 E. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree ≤ d + 2 in their natural embedding in A D . Furthermore we find new embeddings in affine spaces of far lower dimension than D, and characterize those that are still defined by equations of degree ≤ d + 2. 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.

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On an extension of Galligo's theorem concerning the Borel-fixed points on the Hilbert scheme

Cited by 4 publications

References 11 publications

Rational Components of Hilbert Schemes

Rational Components of Hilbert Schemes

Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

A Borel open cover of the Hilbert scheme

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