The geometry of A-graded algebras
arXiv:alg-geom/9410032
Abstract
We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only finitely many isomorphism types of such A-graded algebras, and in these cases I is an initial ideal (in the sense of Groebner bases) of a toric ideal. In this paper it is shown that Arnold's finiteness theorem does not extend to n = 4. Geometric conditions are given for I to be an initial ideal of a toric ideal. The varieties defined by A-graded algebras are characterized in terms of polyhedral subdivisions, and the distinct A-graded algebras are parametrized by a certain binomial scheme.
16 pages, plain TEX
Cited by in corpus (8)
- Non-connected toric Hilbert schemes
- Combinatorics of the toric Hilbert scheme
- Geometric bistellar flips. The setting, the context and a construction
- Multigraded Hilbert Schemes
- Non-coherent Components of the Toric Hilbert Scheme
- The chain property for the associated primes of A-graded ideals
- The toric Hilbert scheme of a rank two lattice is smooth and irreducible
- Algorithms for the Toric Hilbert Scheme