the dimension of the zero coomology space of normal bundle

Question

Suppose that $S$ is a surface of general type canonically embedded by the canonical divisor in the space $\mathbb{P}^n$ for some $n$. Take the curve $C=H\cap S$ with $H$ the divisor of a hyperplane in $\mathbb{P}^n$. We can define the normal bundle $N_{C/S}$. How can i compute the dimension of the space $H^0(C,N_{C/S})$.
i don't know if it is usefull but, using the adjunction formula, we get that $N_{C/S}\equiv H_{|C}$.

Answer 1

you need $C^2$ this number is the degree of S in $\mathbb{P}^n$. It depends on the embedding. Then applying adjunction formula(for both S and C in $\mathbb{P}^n$) and Riemann-Roch.