# line bundles, finite morphisms between curves

These posts are for my benefit and to save me time I’m going to start writing instead of .

Let be an integral scheme. The long exact sequence associated to

gives that is exact, on the other hand, the last group is the higher cohomology of a flasque sheaf on an irreducible space, hence the group is zero, and the first map is a surjection. Recalling the that I conclude any line bundle is of the form for some Cartier divisor .

Recall a sheaf is a generated by global sections if the images of the global sections generate the stalk at every point. For a line bundle, this says for every point p there is a global section that doesn’t vanish at p. Also recall that consist of the global sections that vanish at p. Putting this all together, I get is generated by global sections iff for all p, . Here’s a related thm:

Thm: If is a curve of genus then the canonical sheaf is generated by global sections.

proof: Say there exists a point p such that . Then applying R-R to the line bundle , I get

This implies which is a contradiction. QED

Next little bit of lecture concerned the following setup. is a finite morphism between smooth curves ( I think it might be enough for X to be smooth as long as Y is at least integral). If is a closed point, then

where the are determined via

Recall is principle. Define .

Prop: With the above setup, .

proof: The degree of a finite morphisms is defined to be . Now using a result in Hartshorne like or something similar, plus using cohomology with base change I can conclude is constant so in fact is locally free on Y, and so

where is some closed point. Continuing to compute,

The commutativity of f lower star and quotient can be checked for every open set (i.e. they give the same group for every open set, so they are the same). Continue computing:

where ranges over a subset of the open sets in that contain all the points mapping to y. Using the chinese remainder theorem, this can be expressed as

.

QED.

Here something else using the set up as above. Let be a Cartier divisor on Y. Consider the associated closed subscheme of codimesion 1 and define considered as a closed subscheme in . Then as the fibered product of and the inclusion . Let be the inclusion map. Using flat base change it follows that . Now, . Putting this all together (and using that tensoring with is exact) and starting with the s.e.s I get that

is exact. That is, is the ideal sheaf of . Here’s another construction: . On points it is . It can be check that .

Here’s another theorem.

Thm: For a curve , is not very ample iff there exists a line bundle of degree 2 with iff there exists a degree 2 map to .

proof: Using the criterion for a line bundle to define a closed immersion, I conclude that is not very ample iff there exists of degree 2 s.t. so R-R says . But this implies there is a nonconstant function with poles only at two points (assuming g(X) is not zero) which gives a degree 2 map, and this argument can be reversed. QED.

Ogus also talked about wild and tame ramifications and hyperelliptic curves and Hurwitz thm and Clifford’s theorem, but I didn’t have notes on this stuff, but it does appear in Hartshorne.

## About this entry

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- Published:
- August 18, 2009 / 11:50 am

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

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