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 $C\cap {\overline {pq}}=\{p,q,r,s\}$ , where we intersect with multiplicities, and if $p=q$ we consider the tangent line $T_{p}C$ instead of the line ${\overline {pq}}$ . Hence there is a natural isomorphism between $\mathbb {P} h^{0}(O_{C}(p+q))$ and the unique point in $|K_{C}|$ representing the line ${\overline {pq}}$ . There is also a natural ismorphism between $\mathbb {P} (O_{C}(p+q+r))$ and the points in $|K_{C}|$ representing lines through the points $s$ .