Multigraded Hilbert schemes

By Mark D. Haiman, Bernd Sturmfels. 2002

193 citations recorded by the source corpus.

We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer's conjecture on equations defining Grothendieck's classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bc16fe74ca2deca21765afc0aeb3325bc6ef9104

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Citations in the corpus

Higher-dimensional analogues of stable curves (2006, background): “We note that there exists another moduli space closely related to our moduli space M of stable toric varieties: it is the toric Hilbert scheme HilbTPn [46], [24] parameterizing subschemesX ⊂ P corresponding to the multiplicity-free multigraded algebras.”
Hilbert schemes of 8 points (2008, background): “Multigraded Hilbert schemes, introduced in [Haiman and Sturmfels 2004], parametrize homogeneous ideals with a fixed Hilbert function.”
Rational components of Hilbert schemes (2009, methodology): “ For the meaning of p(z), r, t, M, t 1, M 1 and i∈Hilbn p(z) we refer to Deﬁnition 5.1 and Notation 5.2. First of all we recall how equations deﬁning Hilbn p(z) are usually obtained (see for instance [1, 11]). Every ideal i ∈Hilbn p(z) is generated by the t-dimensional vector space ir. On the other ”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, background): “From the construction [20], Hilb is a sub-scheme of a certain projective space P, and the action of Gξ extends to P(V ).”
Multigraded Castelnuovo-Mumford Regularity (2003, unknown): “bτ ]. By combining Corollary 2.13 in [ BbS ] and Theorem 6.2 in [Mil], we deduce that Hi”
Mapping stacks and categorical notions of properness (2014, background): “a of certain quotient stacks exhibiting properties analogous to proper schemes has appeared a few places in the literature. It is implicit in the work on equivariant Hilbert schemes for torus actions [HS] and more general reductive groups [AB]. In addition, a recent preprint by Geraschenko and Zurei”
Integral bounds on curvature and Gromov–Hausdorff limits (2012, background): “maller, we may assume by Proposition 2.12 that Φ′ is generically one-to-one and so by Lemma 3.17 Φ′ is an embedding. In particular, C(Y′) also deﬁnes an element [C(Y′)] in Hilb. From the construction [20], Hilb is a sub-scheme of a certain projective space P, and the action of Gξ extends to P(V). It”
Uniform bounds on multigraded regularity (2003, methodology): “The proof of Proposition 3.2 of [ HS ] establishes that this algorithm terminates.”
Compactification of the moduli space of hyperplane arrangements (2005, methodology): “M is a multigraded Hilbert scheme as defined in [ HS04 ].”