Riemann-Roch theorem: Difference between revisions

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imported>William Hart
imported>William Hart
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<math>\{[H]\in|K_C|, H\cdot C=D'\}\cong\mathbb{P}H^0(D)</math>, where we identify <math>C</math> with it's image in the dual [[canonical system]] <math>|K_C|^*</math>.
<math>\{[H]\in|K_C|, H\cdot C=D'\}\cong\mathbb{P}H^0(D)</math>, where we identify <math>C</math> with it's image in the dual [[canonical system]] <math>|K_C|^*</math>.


As an example we consider effective divisors of degrees <math>2,3</math> on a non hyperelliptic curve <math>C</math> of genus 3. The degree of the canonical class is <math>2genus(c)-2=4</math>, whereas <math>h^0(K_C)=2genus(C)-2-(genus(C)-1)+h^0(0)=g</math>. Hence the canonical image of <math>C</math> is a smooth plane quartic. We now idenitfy <math>C</math> with it's image in the dual canonical system. Let <math>p,q</math> be two points on <math>C</math> then there are exactly two points
As an example we consider effective divisors of degrees <math>2,3</math> on a non hyperelliptic curve <math>C</math> of genus 3. The degree of the canonical class is <math>2\mbox{genus}(c)-2=4</math>, whereas <math>h^0(K_C)=2\mbox{genus}(C)-2-(\mbox{genus}(C)-1)+h^0(0)=g</math>. Hence the canonical image of <math>C</math> is a smooth plane quartic. We now idenitfy <math>C</math> with it's image in the dual canonical system. Let <math>p,q</math> be two points on <math>C</math> then there are exactly two points
<math>r,s</math> such that <math>C\cap\overline{pq}=\{p,q,r,s\}</math>, where we intersect with multiplicities, and if <math>p=q</math> we consider the tangent line <math>T_p C</math> instead of the line <math>\overline{pq}</math>. Hence there is a natural isomorphism between <math>\mathbb{P}h^0(O_C(p+q))</math> and the unique point in <math>|K_C|</math> representing the line <math>\overline{pq}</math>. There is also a natural ismorphism between <math>\mathbb{P}(O_C(p+q+r))</math> and the points in <math>|K_C|</math> representing lines through the points <math>s</math>.
<math>r,s</math> such that <math>C\cap\overline{pq}=\{p,q,r,s\}</math>, where we intersect with multiplicities, and if <math>p=q</math> we consider the tangent line <math>T_p C</math> instead of the line <math>\overline{pq}</math>. Hence there is a natural isomorphism between <math>\mathbb{P}h^0(O_C(p+q))</math> and the unique point in <math>|K_C|</math> representing the line <math>\overline{pq}</math>. There is also a natural ismorphism between <math>\mathbb{P}(O_C(p+q+r))</math> and the points in <math>|K_C|</math> representing lines through the points <math>s</math>.



Revision as of 06:22, 23 February 2007

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 arise from considering complete linear systems on curves.

  • Any curve of genus 0 is isomorphic 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 canonical class is and therefore ; 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.