Riemann-Roch theorem: Difference between revisions
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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 20:02, 22 February 2007
In algebraic geometry the Riemann-Roch theorem states that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a smooth algebraic curve, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}} is an invertible sheaf on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then the the following properties hold:
- The Euler characteristic of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(\mathcal{L})-h^1(\mathcal{L})=deg(\mathcal{L})-(g-1)}
- There is a canonical isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0(L)(K_C\otimes\mathcal{L}^\vee)\cong H^1(\mathcal{L})}
Geometric Riemann-Roch
From the statment of the theorem one sees that an effective divisor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} on a curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} satsifyies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(D)>d-(g-1)} if and only if there is an effective divisor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D'} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D+D'\sim K_C} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Pic(C)} . In this case there is a natural isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{[H]\in|K_C|, H\cdot C=D'\}\cong\mathbb{P}H^0(D)} , where we identify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with it's image in the dual cannonical system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |K_C|^*} .
As an example we consider effective divisors of degrees Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2,3} on a non hyperelliptic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} of genus 3. The degree of the cannonical class is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2genus(c)-2=4} , whereas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(K_C)=2genus(C)-2-(genus(C)-1)+h^0(0)=g} . Hence the cannonical image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a smooth plane quartic. We now idenityf Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with it's image in the dual cannonical system. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p,q} be two on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then there are exactly two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r,s} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\cap\overline{pq}=\{p,q,r,s\}} , where we intersect with multiplicities, and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=q} we consider the tangent line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p C} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{pq}} . Hence there is a natural ismorphism between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}h^0(O_C(p+q))} and the unique point in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |K_C|} representing the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{pq}} . There is also a natural ismorphism between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}(O_C(p+q+r))} and the points in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |K_C|} representing lines through the points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} .
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.