Riemann-Roch theorem: Difference between revisions

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imported>David Lehavi
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* There is a [[canonical isomorphism]] <math>H^0(L)(K_C\otimes\mathcal{L}^\vee)\cong H^1(\mathcal{L})</math>
* There is a [[canonical isomorphism]] <math>H^0(L)(K_C\otimes\mathcal{L}^\vee)\cong H^1(\mathcal{L})</math>


= Generalizations =
=== Geometric Riemann-Roch ===
From the statment of the theorem one sees that an [[effective divisor]] <math>D</math> of degree <math>d</math> on a curve <math>C</math> satsifyies <math>h^0(D)>d-(g-1)</math> if and only if there is an effective divisor <math>D'</math> such that <math>D+D'\sim K_C</math> in <math>Pic(C)</math>. In this case there is a natural isomorphism
<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 [[cannonical 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 cannonical 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 cannonical image of <math>C</math> is a smooth plane quartic. We now idenityf <math>C</math> with it's image in the dual cannonical system. Let <math>p,q</math> be two 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 <math>\overline{pq}</math>. Hence there is a natural ismorphism 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>.
 
== Generalizations ==
* [[Cliford's theorem]]
* [[Riemann-Roch for surfaces]] and [[Noether's formula]]
* [[Riemann-Roch for surfaces]] and [[Noether's formula]]
* [[Hirzebruch-Riemann-Roch theorem]]
* [[Hirzebruch-Riemann-Roch theorem]]
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* [[Atiya-Singer index theorem]]
* [[Atiya-Singer index theorem]]


= Proofs=
== Proofs==
Using modern tools, the theorem is an immediate consequence of [[Serre's duality]].
Using modern tools, the theorem is an immediate consequence of [[Serre's duality]].


[[Category:Mathematics Workgroup]]
[[Category:Mathematics Workgroup]]
[[Category:CZ Live]]
[[Category:CZ Live]]

Revision as of 21:02, 22 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

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 idenityf with it's image in the dual cannonical system. Let be two 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.