This follows an exposition due to Hu and Keel (Yi Hu and Seán Keel, Mori dream spaces and GIT, Michigan Math. J., 48 (2000), 331-348). Let , and a projective smooth variety over . Let be line bundles on . If , then .
We have a multiplication map . We have a ring , which is often not finitely generated.
If are (semi)ample (a semiample bundle is a pullback of an ample bundle), then is finitely generated.
Definition. Let be smooth and projective, and assume (e.g. a Fano variety). The Cox ring is where:
The Hilbert function if has no higher cohomology, so is a `polynomial in '. For example, if ample cone, Kodaira vanishing implies that .
If is nef ( is nef if for every curve ) and big ( is big if is in the interior of the effective cone), for example if is Fano, then for all nef and big. Our basic strategy: use knowledge of the Hilbert function to read off the structure of . There is hope that the ring will be finitely generated from this polynomial expression.
What are necessary conditions for to be finitely generated? (Part of a theorem of Hu, Keel which give necessary and sufficient conditions.) We must have:
There are also sufficient conditions:
Remark. The universal torsor as an explicitly defined open subset, if is finitely generated.
This is joint work with Tschinkel. The cubic surface is defined by the equation embedded in ; it contains a unique line , and a unique singularity .
Analysis of the singularity: In affine coordinates, we have . We rewrite this as , and up to analytic equivalence, this is , which is the normal form of an singularity.
The resolution has six exceptional curves in an -diagram, and has intersection form
The inverse of this matrix is given by
The inverse proves that: Proposition. The effective cone of is generated by ; the nef cone is generated by .
We have , but this is not surjective. We need additional generators. We let ; since is semiample, we have by Riemann-Roch, and is a conic bundle. Then
Fact. We have a surjection .
Now we look for relations among these generators. By the Hilbert function, we know that , but we have elements of degree in the polynomial ring:
Since , , , after renormalization, so the original equation gives the relation
This method should also work for other very singular cubic surfaces.
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