# Finite # of points, open affines and Quotients

Here’s some random stuff. For an (abstract) variety, the condition ‘any finite set of points in contained in an open affine’ is equivalent to the variety being quasi projective. The details of the proof of this assertion are presented in pages 111 – 116 of JMiline’s algebraic geometry notes, here is a brief outline:

First prove the result when the variety is projective: a finite set of a projective variety, i.e. . Then there is a linear form which ‘misses’ all the points in , i.e. for all .

Now define a bijection with and the affine variety . For set , which makes sense since I’m picking so . Then

The map is a bijection with inverse

.

Next, prove an analogous result with the linear form replaced by a homogeneous polynomial , say of degree d. This uses the Veronese map or d-uple embedding (see Hartshorne ex. I.2.12). Basically this gives a map where where the hypersuface defined by maps isomorphically to a hyperplane.

Finally, deduce the general result: if is quasi projective then embed it as an open subset of projective space . Consider (Note is closed; look at its complement). Recall is a finite set which doesn’t meet . As is closed, , so means for all there is a homogeneous with . Then make all the have the same degree and argue that some linear combination of them don’t vanish at all points of . Then by the previous paragraph, is affine, contains all of and , so its an affine of .

### Quotienting out by a group

One particular place where the above comes into play is in talking about quotienting out a scheme by an action of the group scheme. For an affine scheme with an action by a finite group , the quotient is given by the ring of invariants: . For a general scheme , could be constructed by covering with affines and gluing together , this requires to be G-invariant. This can be guaranteed if is quasi projective. Indeed, given , the G-orbit is a finite set of points, so contained in an affine . Then

is invariant and affine, as is separated.

With a little more work, I can define where is a group scheme with an action of , i.e. a morphism which fits into some nice commutative diagrams. When is affine the task boils down to defining the ring of invariants. In this case if . Then . As above, for the general case, cover with affines and glue together the quotients.

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- Published:
- June 23, 2009 / 3:43 pm

- Tags:
- group schemes, quasi projective

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