Riemann-Roch theorem

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Revision as of 22:51, 22 February 2007 by imported>David Lehavi (added examples)
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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 statment of the theorem one sees that an effective divisor of degree on a curve satsifyies 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 cannonical system .

As an example we consider effective divisors of degrees on a non hyperelliptic curve of genus 3. The degree of the cannonical class is , whereas . Hence the cannonical image of is a smooth plane quartic. We now idenitfy with it's image in the dual cannonical 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 . Hence there is a natural ismorphism 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.