Ideal sheaf of intersection of two surfaces in $\mathbb P^3$

Question

Let X be an intersection of 2 surfaces of degree $d_1,d_2$ in $\mathbb P^3$. Is it true that there is a short exact sequence $$ 0\to\mathcal{O}_{\mathbb{P}^3}(-d_1-d_2)\to\mathcal{O}_{\mathbb{P}^3}(-d_1)\oplus\mathcal{O}_{\mathbb{P}^3}(-d_2)\to\mathcal{I}_X\to0, $$ where $\mathcal{I}_X$ is ideal sheaf of $X$. And if it is true, how it could be proven?

Answer 1

If $X$ is a so-called complete intersection, meaning that the ideal sheaf of $X$ is generated by the generators, say $f$ and $g$, of the ideals of the two surfaces, and the two surfaces have no components in common, then yes, it pops out very naturally: Anything in the ideal of $X$ is locally a linear combination of the two generators, hence the surjection on the right, given by $(a,b)\mapsto af+gb$. The kernel is as advertised, because since the surfaces have no common components, $f$ and $g$ are relatively prime, so the relations they satisfy are generated by the obvious one $gf-fg$ and the map on the left is accordingly given by $c \mapsto (cg,-cf)$.