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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.

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    $\begingroup$ Cohomology of the projective spaces is known. For example, in the above, use the facts $h^0(O)=1, h^1(O)=h^2(O)=0$ where $O=\mathcal{O}_{\mathbb{P}^2}$. Also, your exact sequence is correct only if you assume $F,G$ have no common factors, in which case $Z$ is zero dimensional and thus $h^1(\mathcal{O}_Z)=0$. $\endgroup$
    – Mohan
    May 5, 2017 at 13:37
  • $\begingroup$ @Mohan Thanks. Why is $Z$ zero dimensional? $\endgroup$
    – adrisala
    May 5, 2017 at 13:53
  • $\begingroup$ Also, I had to use the long exact sequence, and thanks to your observation we have an exact sequence $$0\rightarrow\Gamma(Z,\mathcal{J}_Z)\rightarrow\cdots\rightarrow H^1(Z,\mathcal{J}_Z)\rightarrow 0,$$ right? Why is it that $h^0(\mathcal{J}_Z)=0$? $\endgroup$
    – adrisala
    May 5, 2017 at 14:00
  • $\begingroup$ $Z$ is zero dimensional, since having a one dimensional component implies $F,G$ have a common factor. $h^0(J_Z)=0$, since $h^0$ of the middle term is zero (in the first sequence) and $h^1$ of the left term is also zero. As I said, you must learn the chomologies of $\mathcal{O}_{\mathbb{P}^n}(k)$ which are well understood. $\endgroup$
    – Mohan
    May 5, 2017 at 14:41

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