The geometry of A-graded algebras

TL;DR

This paper explores the structure and classification of A-graded algebras, revealing that finiteness results for low dimensions do not extend to higher dimensions and providing geometric characterizations.

Contribution

It demonstrates that Arnold's finiteness theorem fails for n=4, offers geometric conditions for initial ideals of toric ideals, and characterizes varieties via polyhedral subdivisions.

Findings

Finiteness of A-graded algebras does not hold for n=4.

Geometric conditions for initial ideals of toric ideals are established.

Varieties are characterized by polyhedral subdivisions and parametrized by a binomial scheme.

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.