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The homogeneous coordinate ring of a toric variety (1992)
| Citations: | 473 - 7 self |
BibTeX
@MISC{Cox92thehomogeneous,
author = {David A. Cox},
title = {The homogeneous coordinate ring of a toric variety },
year = {1992}
}
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Abstract
This paper will introduce the homogeneous coordinate ring S of a toric variety X. The ring S is a polynomial ring with one variable for each one-dimensional cone in the fan ∆ determining X, and S has a natural grading determined by the monoid of effective divisor classes in the Chow group An−1(X) of X (where n = dim X). Using this graded ring, we will show that X behaves like projective space in many ways. The paper is organized into four sections as follows. In §1, we define the homogeneous coordinate ring S of X and compute its graded pieces in terms of global sections of certain coherent sheaves on X. We also define a monomial ideal B ⊂ S that describes the combinatorial structure of the fan ∆. In the case of projective space, the ring S is just the usual homogeneous coordinate ring C[x0,..., xn], and the ideal B is the “irrelevant ” ideal 〈x0,..., xn〉. Projective space P n can be constructed as the quotient (C n+1 −{0})/C ∗. In §2, we will see that there is a similar construction for any toric variety X. In this case, the algebraic group G = HomZ(An−1(X), C ∗ ) acts on an affine space C ∆(1) such that the categorical quotient (C ∆(1) − Z)/G exists and is isomorphic to X. The exceptional set Z is the zero
Keyphrases
toric variety homogeneous coordinate ring projective space homogeneous coordinate graded ring combinatorial structure affine space usual homogeneous coordinate graded piece chow group certain coherent sheaf exceptional set natural grading many way irrelevant ideal x0 monomial ideal similar construction polynomial ring algebraic group homz effective divisor class global section one-dimensional cone categorical quotient
