Riemann-Roch theorem: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>William Hart
imported>William Hart
Line 13: Line 13:
<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 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})}

Some examples

The examples we give arise from considering complete linear systems on curves.

  • Any 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 0 is isomorphic to the projective line: Indeed if p is a point on the curve then 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(p)-0=1-(0-1)=2} ; hence the map 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\to\mathbb{P}H^0(O_C(p))} 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 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(2p)-0=2-(1-1)=2} ; hence the map 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\to\mathbb{P}H^0(O_C(2p))} 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 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} 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 2g-2} and therefore 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)-h^0(O_C)=2-(2-1)=1} ; since 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(O_C)=1} the map 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\to\mathbb{P}H^0(K_C)} is a degree 2 map.

Geometric Riemann-Roch

From the statement 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} satisfies 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 canonical 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 canonical 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 2\mbox{genus}(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)=2\mbox{genus}(C)-2-(\mbox{genus}(C)-1)+h^0(0)=g} . Hence the canonical 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 idenitfy 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 canonical 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 points 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 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}} . Hence there is a natural isomorphism 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

Proofs

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

Retrieved from "https://citizendium.org/wiki/index.php?title=Riemann-Roch_theorem&oldid=149239"