Revision as of 06:18, 23 February 2007 by imported>William Hart
In algebraic geometry the Riemann-Roch theorem states that if is a smooth algebraic curve, and is an invertible sheaf on then the the following properties hold:
- The Euler characteristic of is given by
- There is a canonical isomorphism
some examples
The examples we give arrise from considering complete linear systems on curves.
- Any curve of genus 0 is ismorphic to the projective line: Indeed if p is a point on the curve then ; hence the map is a degree 1 map, or an isomorphism.
- Any curve of genus 1 is a double cover of a projective line: Indeed if p is a point on the curve then ; hence the map is a degree 2 map,
- Any curve of genus 2 is a double cover of a projective line: Indeed the degree of the cannonical class is and therefor ; since the map is a degree 2 map,
Geometric Riemann-Roch
From the statement of the theorem one sees that an effective divisor of degree on a curve satisfies if and only if there is an effective divisor such that in . In this case there is a natural isomorphism
, where we identify with it's image in the dual canonical system .
As an example we consider effective divisors of degrees on a non hyperelliptic curve of genus 3. The degree of the canonical class is , whereas . Hence the canonical image of is a smooth plane quartic. We now idenitfy with it's image in the dual canonical system. Let be two points on then there are exactly two points
such that , where we intersect with multiplicities, and if we consider the tangent line instead of the line . Hence there is a natural isomorphism between and the unique point in representing the line . There is also a natural ismorphism between and the points in representing lines through the points .
Generalizations
Proofs
Using modern tools, the theorem is an immediate consequence of Serre's duality.