# Chi exercise, then Divisors, Curves, R-R

### A nice exercise

Claim: Set . For , . The proof is by induction on n. For n = 1, its required to show . For m > -1, because there are m+1 monomials of degree m in two variables. For m = -1, all the cohomology vanishes so . For m < -1 there are no global sections, but there is . Serre duality gives so .

Also, for , the statement says which is correct because X only has constant global sections. So the statement holds for n=1 and all m, and for all n with m = 0. Now for the induction step, apply the equality to the s.e.s

Where are homogeneous coordinate rings of respectively. The binomial identity finishes the proof.

Next Hartshorne went on to prove the equivalent to exercise III.5.2 in Hartshorne (of which the above exercise is a special case). Unfortunately, I did not take very good notes for the next little bit that Ogus did (this was around the end of march). But here are some of the things he mentioned.

- Codimension 1 stuff, preparing for Weil divisors
- mentioned GAGA and used it to prove that for projective, that
- Blowing up as in II.7 in Hartshorne. And there was this nice picture

### Divisors etc.

Divisors on Curves and Riemann-Roch is a popular topic that is treated in many introductory texts on algebraic geometry. Here’s how Ogus does it.

Cartier divisors were introduced much the same way as in Hartshorne, but Ogus defined Weil divisors for arbitrary schemes. Namely, for , set . Then

= free monoid gen. by ht 1 pts.

= free group gen. by ht 1 pts.

Then Ogus went on to discuss all the great things that happen with Weil Divisors when X is integral, normal and separated (again, note the lack of Noetherian hypothesis); Ogus used ‘normal’ to replace the condition ‘regular in codimension 1.’ Then he proved the following thm:

Thm: Let be locally factorial, normal (e.g. regular) and separated. Then the group of Cartier divisors on X is isomorphic to the group of Weil divisors on X.

Now, some curve stuff. For Ogus, a curve is a scheme which satisfies

- X is quasi compact and finite type over field k
- for every closed pt.
- for every closed pt. (this is something about Cohen-Macaulay)
- X is separated

Here is some business about the degree of a Weil divisor. For any , set , then

Thm: For an effective Cartier = Weil divisor D, .

proof: Write with . Set ; it is a principal ideal. There is a s.e.s

Where the ideal sheaf . By the chinese remainder thm,

so , so I can reduce to the case . The result holds for . Also, for , which is also correct. Here’s the induction step. From the s.e.s of vector spaces

it suffices to prove that . Apparently this follows from smoothness and using the fact that the local rings are DVRs. QED.

With this result the Riemann-Roch thm can be proved for curves. The notation is that is the line bundle associated to the divisor ; in particular, .

Thm (R-R): For all , .

proof: First suppose is effective. Consider the s.e.s’

Note that is 0 dimensional so its Euler characteristic is just h^0, and since it has finite support, tensoring with a line bundle does not change the dimension of its global sections; that is , in fact . So the second s.e.s gives

using the fact that gives the result in this case. For the general case, write the divisor as the difference of two effective divisors: . Then . From which it follows that , so tensoring the s.e.s for with I get

from which I conclude that , using the first part I get

QED

Combining with Serre duality gives the usual statement.

Next Ogus spent some time on criterion for a morphism into projective space to be a closed immersion (it separates points and tangents); the treatment was not radically different than what is in Hartshorne. Also spent sometime classifying curves of low genus similar to Hartshorne and Miranda. Here’s a thm to end this post

Thm: Let be a nonconstant morphism of (smooth) curves. Then is

- surjective
- affine
- finite
- flat

proof: are both projective so is proper meaning is connected closed subset that is more than a point, hence all of . That the morphism is affine is the statement that a curve minus a finite number of points is affine. Here’s one way of doing this. Use global sections for k huge to get a closed immersion into projective space. The sections are nonconstant so they have poles at p, so the way this morphism works (think!) means that where is a function defined locally that vanishes at . The points is the affine intersected with gives another affine that as a point set is . For the general case, its another exercise to show that a one dimensional affine minus a finite number of points is still affine.

Another general result is that a proper affine map is finite (this is essentially exercise II.4.6 in Hartshorne). Finally is a loc. free module, which gives flatness, or also one could apply result III.9.7 in Hartshorne. It says is is integral and regular of dimension 1, then f is flat iff all the associated points of map to the generic point (these are points where consists of zero divisors). As is integral, only the generic point of X satisfies this criterion, so the morphism is flat. QED.

## About this entry

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- Published:
- August 13, 2009 / 9:27 am

- Category:
- alg. geo., Ogus Excerpts, Ogus/Hartshorne/Miranda

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