If $C$ is a curve of genus $g$, I'm trying to prove the dimension of the divisor $(2g-1)P$ associated to this curve is $g$.
I'm using the Riemann-Roch theorem which says:
Let $W$ be a canonical divisor on $C$. Then for any divisor $D$: $$l(D)=deg(D)+1-g+l(W-D)$$
Using this theorem I get $l((2g-1)P)=g+l(W-(2g-1)P)$. So why $l(W-(2g-1)P)=0$?
Remark: I'm studying Fulton's Algebraic Curves book.
Thanks in advance