Publication

Homology of the three flag Hilbert Scheme

Daniele Boccalini
2016
Theses

Résumé

We prove the existence of an affine paving for the three-step flag Hilbert scheme that parametrizes flag of three 0-dimensional subschemes of length, respectively, n, n+1 and n+2 that are supported at the origin of the affine plane. This is done by showing that the space stratifies in smooth subvarieties, the Hilbert-Samuel's strata, each of which has an affine paving with cells of known dimension, indexed by marked Young diagrams. The affine pavings of the Hilbert-Samuel's strata allow us to prove that the Poincaré polynomials for our spaces satisfy a generating function. In the process of proving the formula for the generating function we relate combinatorially the homology of our spaces with that of known smooth subspaces of another Hilbert scheme of flags, this time of length n and n+2. As a corollary we find an affine paving and a combinatorial formula for the Poincaré of these last ambient spaces.

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https://infoscience.epfl.ch/entities/publication/09770737-a222-4d29-aa5e-a37f6099c4d8

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Daniele Boccalini
2016
Theses

Résumé

Source officielle

https://infoscience.epfl.ch/entities/publication/09770737-a222-4d29-aa5e-a37f6099c4d8

À propos de ce résultat

Proximité ontologique

Mathématiques

Géométrie: Géométrie algébrique

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Variété projective

En géométrie algébrique, les variétés projectives forment une classe importante de variétés. Elles vérifient des propriétés de compacité et des propriétés de finitude. C'est l'objet central de la géométrie algébrique globale. Sur un corps algébriquement clos, les points d'une variété projective sont les points d'un ensemble algébrique projectif. On fixe un corps (commutatif) k. Algèbre homogène. Soit B le quotient de par un idéal homogène ( idéal engendré par des polynômes homogènes).

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In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.

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In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.

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