David A. Cox introduced the total coordinate ring
S of a normal toric variety
X and the irrelevant ideal
B. The polynomial ring
S has one variable for each ray in the associated fan and a natural grading by the class group. The monomial ideal
B encodes the maximal cones. The following results of Cox indicate the significance of the pair
(S,B).
- The variety X is a good categorial quotient of Spec(S) - V(B) by a suitable group action.
- The category of coherent sheaves on X is equivalent to the quotient of the category of finitely generated graded S-modules by the full subcategory of B-torsion modules.
In particular, we may represent any coherent sheaf on
X by giving a finitely generated graded
S-module.
The following methods allow one to make and manipulate coherent sheaves on normal toric varieties.
Menu