PlanetMath: Riemann-Roch theorem for curves

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Riemann-Roch theorem for curves

(Theorem)

Let $C$ be a projective nonsingular curve over an algebraically closed field. If $D$ is a divisor on $C$ then

$$\ell(D) - \ell(K-D) = {\rm deg}(D) + 1 - g$$

where $g$ is the genus of the curve, and $K$ is the canonical divisor ($\ell(K)=g$ . Here $\ell(D)$ denotes the dimension of the space of functions associated to a divisor.

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Cross-references: dimension, canonical, genus, divisor, field, algebraically closed, curve, nonsingular
There are 3 references to this entry.

This is version 8 of Riemann-Roch theorem for curves, born on 2001-12-12, modified 2006-02-21.
Object id is 1098, canonical name is RiemannRochTheorem.
Accessed 6671 times total.

Classification:

AMS MSC:	14H99 (Algebraic geometry :: Curves :: Miscellaneous)
	19L10 ($K$-theory :: Topological $K$-theory :: Riemann-Roch theorems, Chern characters)

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