# Riemann-Roch theorem

• May 9th 2012, 03:01 PM
Aliquantus
Riemann-Roch theorem
Hi all,

I have some questions regarding the divisors and the Riemann-Roch theorem.

1. What exactly is a hyperplane divisor? I cannot find the definition of it in any of my books or notes, and not even google was of assistance this time. The term first came up on a past paper, but I have never seen it before.

2. Riemann-Roch tells us that if $D$ is any divisor on a non-singular projective algebraic curve in $P_2$, and $\kappa$ a canonical divisor, then $l(D)-l(\kappa - D)=deg(D)+1-g$.

My lecture notes immediately after the statement of the theorem make the following point: If $deg(D) > 2g-2,$ then $deg(\kappa-D) <0,$ so.... etc.

I don't really see at all how this follows, any tips?

Thanks,
Aliquantus