  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.
