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Algorithms for the Toric Hilbert Scheme

Michael Stillman·Bernd Sturmfels·Rekha R. Thomas
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Abstract

The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component. However, it is unknown whether toric Hilbert schemes are always connected. In this chapter we illustrate the use of Macaulay 2 for exploring the structure of toric Hilbert schemes. In the process we will encounter algorithms from commutative algebra, algebraic geometry, polyhedral theory and geometric combinatorics.

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Michael Stillman, Bernd Sturmfels, Rekha R. Thomas. 2000-10-12. Algorithms for the Toric Hilbert Scheme. https://arxiv.org/abs/math/0010130

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