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Multigraded Hilbert schemes. (English) Zbl 1072.14007

In his fundamental paper [“Techniques de construction et théorèmes d’existence en géométrie algébrique. IV: Les schemes de Hilbert”, Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], A. Grothendieck introduced the so called Hilbert scheme, which is an object which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial.
In the paper under review, the authors present a more general object, called the multigraded Hilbert scheme, parametrizing all homogeneous ideals with fixed Hilbert function in a graded polynomial ring \(S\).
As in the case of Hilbert schemes, the multigraded Hilbert scheme is a projective scheme (quasi-projective if the grading of \(S\) is not positive), and, when the ground ring is a field, its tangent space at a point corresponding to an ideal \(I\) has a simple description: it is canonically isomorphic to the degree \(0\) piece of \(\operatorname{Hom}(I,S/I)\).
The construction of the multigraded Hilbert scheme is obtained in a great generality, and it enables the authors to prove a conjecture from D. Bayer’s thesis [The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University (1982)] on equations defining the Hilbert scheme, and to construct a natural morphism from the toric Hilbert scheme to the toric Chow variety, resolving Problem 6.4 appearing in the paper of B. Sturmfels [The geometry of A-graded algebras, preprint, http://arXiv.org/abs/math.AG/9410032].

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 0236.14003
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References:

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