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ABSTRACT:
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.

SUGGESTED CITATION:
Mark Haiman and B Sturmfels, "Multigraded Hilbert schemes" (2004). Journal of Algebraic Geometry. 13 (4), pp. 725-769. Postprint available free at: http://repositories.cdlib.org/postprints/1136

REQUIRED PUBLISHER STATEMENT:
First published in The Journal of Algebraic Geometry (2004), published by the American Mathematical Society.