Cox Rings
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:
Remark.
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.
Finite Generation
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.
Cubic Surface
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
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.
Back to the
main index
for Rational and integral points on higher dimensional varieties.