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NormalToricVarieties :: working with sheaves

working with sheaves -- information about coherent sheaves and total coordinate rings (a.k.a. Cox rings)

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.

Sheaf-theoretic methods

See also