This is Exercise I.3 (P.23).
(X is a complex smooth projective surface.) Let $C$ be a smooth rational curve on $X$. For $n>0$, find $\dim H^0(nC;\mathcal{O}_{nC})$ and $p_a(nC)$ in terms of $C^2$. What can you say if instead $g(C)>0$?
I come up with the following. From the SES (equation (1.10) in the book) $$0\to \mathcal{O}_C(-(n-1)C)\to\mathcal{O}_{nC}\to\mathcal{O}_{(n-1)C}\to0$$ we have $$\chi(\mathcal{O}_{nC})-\chi(\mathcal{O}_{(n-1)C})=\chi(\mathcal{O}_C(-(n-1)C)).$$ By Riemann-Roch on $C$ $$\chi(\mathcal{O}_C(-(n-1)C)=-(n-1)\deg(\mathcal{O}_C(C))+1-g(C)=-(n-1)C^2+1.$$ Then inductively we have $$\chi(\mathcal{O}_{nC})=\chi(\mathcal{O}_C)-\binom{n}{2}C^2+(n-1)=-\binom{n}{2}C^2+n. $$ By $p_a(nC)=1-\chi(\mathcal{O}_{nC})$, we have $$p_a(nC)=\binom{n}{2}C^2-n+1. $$
However, I cannot get $h^0(\mathcal{O}_{nC})$. For $n=1$, I see $h^1(\mathcal{O}_C)=0$ and then $h^0(\mathcal{O}_C)=\chi(\mathcal{O}_C)$. For $n>1$, I want the vanishing of $h^1(\mathcal{O}_C(-(n-1)C))$, but I can not get it.