I have recently started working with Cech cohomology and there is a lot of results that I am not common with, so I have come up with the following question. I will take the notation that Perrin uses in chapter VII of his book "Algebraic Geometry: An Introduction".
Let $F,G\in k[X,Y,T]$ be two homogeneous polynomials of degree $s$ and $t$ respectively such that $F$ and $G$ have no factors in common and let $Z:=V(F,G)$. The thing is that once we take the exact sequences $$0\rightarrow\mathcal{O}_{\mathbb{P}^2}(-s-t)\rightarrow\mathcal{O}_{\mathbb{P}^2}(-s)\oplus\mathcal{O}_{\mathbb{P}^2}(-t)\rightarrow\mathcal{J}_Z\rightarrow0$$ and $$0\rightarrow\mathcal{J}_Z\rightarrow\mathcal{O}_{\mathbb{P}^2}\rightarrow\mathcal{O}_Z\rightarrow 0,$$ I do not understand why $h^0(\mathcal{O}_Z)=1+h^1(\mathcal{J}_Z)$, $h^2(\mathcal{J}_Z)=h^2(\mathcal{O}_{\mathbb{P}^2})=0$ and $h^1(\mathcal{J}_Z)=h^2\mathcal{O}_{\mathbb{P}^2}(-s-t)-h^2\mathcal{O}_{\mathbb{P}^2}(-s)-h^2\mathcal{O}_{\mathbb{P}^2}(-t)$. Which results does he use in order to obtain those relations? Does he use the long exact sequence of Cech cohomology? How?
Thanks.