Riemann-Roch theorem

In algebraic geometry the Riemann-Roch theorem states that if $C$ is a smooth algebraic curve, and ${\mathcal {L}}$ is an invertible sheaf on $C$ then the the following properties hold:

• The Euler characteristic of ${\mathcal {L}}$ is given by $h^{0}({\mathcal {L}})-h^{1}({\mathcal {L}})=deg({\mathcal {L}})-(g-1)$
• There is a canonical isomorphism $H^{0}(L)(K_{C}\otimes {\mathcal {L}}^{\vee })\cong H^{1}({\mathcal {L}})$

some examples

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

• Any curve $C$ of genus 0 is ismorphic to the projective line: Indeed if p is a point on the curve then $h^{0}(p)-0=1-(0-1)=2$; hence the map $C\to \mathbb {P} H^{0}(O_{C}(p))$ 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 $h^{0}(2p)-0=2-(1-1)=2$; hence the map $C\to \mathbb {P} H^{0}(O_{C}(2p))$ is a degree 2 map,
• Any curve of genus 2 is a double cover of a projective line: Indeed the degree of the cannonical class $K_{C}$ is $2g-2$ and therefor $h^{0}(K_{C})-h^{0}(O_{C})=2-(2-1)=1$; since $h^{0}(O_{C})=1$ the map $C\to \mathbb {P} H^{0}(K_{C})$ is a degree 2 map,

Geometric Riemann-Roch

From the statement of the theorem one sees that an effective divisor $D$ of degree $d$ on a curve $C$ satisfies $h^{0}(D)>d-(g-1)$ if and only if there is an effective divisor $D'$ such that $D+D'\sim K_{C}$ in $Pic(C)$. In this case there is a natural isomorphism $\{[H]\in |K_{C}|,H\cdot C=D'\}\cong \mathbb {P} H^{0}(D)$, where we identify $C$ with it's image in the dual canonical system $|K_{C}|^{*}$.

As an example we consider effective divisors of degrees $2,3$ on a non hyperelliptic curve $C$ of genus 3. The degree of the canonical class is $2genus(c)-2=4$, whereas $h^{0}(K_{C})=2genus(C)-2-(genus(C)-1)+h^{0}(0)=g$. Hence the canonical image of $C$ is a smooth plane quartic. We now idenitfy $C$ with it's image in the dual canonical system. Let $p,q$ be two points on $C$ then there are exactly two points $r,s$ 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 ${\overline {pq}}$. 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.