TITLE: Genus of intersection of two surfaces in $\mathbb{P}^3$
QUESTION [6 upvotes]: Let $F_1$ and $F_2$ be two (smooth) surfaces in $\mathbb{P}^3$, of degrees $d_1$ and $d_2$ respectively. Let $C$ denote curve given as their intersection. How one can compute arithmetical genus of the curve $C$?
Perhaps I need to add some assumptions on $F$ and $G$ to exclude degenerate cases.

REPLY [4 votes]: The cohomology of the intersection of two surfaces can also be computed by writing resolutions. First of all, we have the ideal sheaf sequence $$ 0 \to \mathcal I \to \mathcal O_\mathbb P \to \mathcal O_C \to 0.$$
We also have the exact sequence
$$
0 \to \mathcal O(-d-e) \to \mathcal O(-d) \oplus \mathcal O(-e) \to \mathcal O \to \mathcal I \to 0.
$$
Then you can use arguments with long exact sequences to compute the cohomology of $C$. First of all, the cohomology of the ideal sheaf is completely determined by the second exact sequence together with results on the cohomology of projective space. Then combine this with the first sequence to determine the  cohomology of $\mathcal O_C$. 
(this is more bothersome than Asal Beag Dubh's method however...)