Riemann-Roch theorem: Difference between revisions
imported>David Lehavi (added examples) |
imported>David Lehavi m (typos) |
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The examples we give arrise from considering complete [[linear systems]] on curves. | The examples we give arrise from considering complete [[linear systems]] on curves. | ||
* Any curve <math>C</math> of genus 0 is ismorphic to the projective line: Indeed if p is a point on the curve then <math>h^0(p)-0=1-(0-1)=2</math>; hence the map <math>C\to\mathbb{P}H^0(O_C(p))</math> is a degree 1 map, or an isomorphism. | * Any curve <math>C</math> of genus 0 is ismorphic to the projective line: Indeed if p is a point on the curve then <math>h^0(p)-0=1-(0-1)=2</math>; hence the map <math>C\to\mathbb{P}H^0(O_C(p))</math> 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 <math>h^0( | * Any curve of genus 1 is a double cover of a projective line: Indeed if p is a point on the curve then <math>h^0(2p)-0=2-(1-1)=2</math>; hence the map <math>C\to\mathbb{P}H^0(O_C(2p))</math> 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]] <math>K_C</math> is <math>2g-2</math> and therefor <math>h^0(K_C)-h^0(O_C | * Any curve of genus 2 is a double cover of a projective line: Indeed the degree of the [[cannonical class]] <math>K_C</math> is <math>2g-2</math> and therefor <math>h^0(K_C)-h^0(O_C)=2-(2-1)=1</math>; since <math>h^0(O_C)=1</math> the map <math>C\to\mathbb{P}H^0(K_C)</math> is a degree 2 map, | ||
=== Geometric Riemann-Roch === | === Geometric Riemann-Roch === | ||
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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 idenitfy <math>C</math> with it's image in the dual cannonical 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 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 idenitfy <math>C</math> with it's image in the dual cannonical 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 <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>. | <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 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 == | == Generalizations == |
Revision as of 22:53, 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
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 the line . 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
- Cliford's theorem
- Riemann-Roch for surfaces and Noether's formula
- Hirzebruch-Riemann-Roch theorem
- Grothendieck-Riemann-Roch theorem
- Atiya-Singer index theorem
Proofs
Using modern tools, the theorem is an immediate consequence of Serre's duality.