Proof of the genus-degree formula using long exact sequence of cohomology.

Question

Let $i: C \subset \mathbb{P}^2$ be a nonsingular curve of degree $d$ and genus $g$. There is a conormal sequence $0 \to i^* \mathcal{O}(-d) \to i^* \Omega^1_{\mathbb{P}^2} \to \Omega^1_{C} \to 0.$ Thus, we have $$0 \to H^0 (C, i^* \mathcal{O}(-d)) \to0\to H^0(C, i^* \Omega^1_{\mathbb{P}^2}) \to H^0(C, \Omega^1_{\mathbb{P}^2}) \to H^1(C, i^* \mathcal{O}(-d)) \to H^1(C, i^* \Omega^1_{\mathbb{P}^2}) \to H^1(C, \Omega^1_{C}) \to 0.$$

We know that $H^1(C, \Omega^1_{C})$ is one-dimensional by the Serre duality. Let's compute $H^1(C, i^* \Omega^1_{\mathbb{P}^2})$. Using the Euler sequence we get $ 0 \to i^* \Omega^1_{\mathbb{P}^2} \to i^* \mathcal{O}(-1)^{\oplus 3} \to \mathcal{O}_C \to 0$ and writing a long exact sequence we get $0 \to k \to H^1(C, i^* \Omega^1_{\mathbb{P}^2}) \to H^1(C, i^* \mathcal{O}(-1)^{\oplus 3}) \to 0$. Here I'm not sure how to compute first cohomology of $i^* \mathcal{O}(-1)$ even though I suspect it's zero. The same for $i^* \mathcal{O}(-d)$; I don't see how to compute its cohomology. How can I do this?

And a general question: suppose I have a hypersurface $j: Y \to X$ and a sheaf $F$ on $X$. Is is possible to compute cohomology of $j^* F$ knowing cohomology of $F$?

Answer 1

The basic idea to compute the cohomology of $i^\ast \mathscr{O}_{\mathbb{P}^2}(-1)$ is the following. Recall that we have the closed subscheme exact sequence $$ 0 \to \mathscr{O}_{\mathbb{P}^2}(-d) \to \mathscr{O}_{\mathbb{P}^2} \to i_\ast \mathscr{O}_C \to 0. $$ We can tensor by $\mathscr{O}_{\mathbb{P}^2}(-1)$ to get $$ 0 \to \mathscr{O}_{\mathbb{P}^2}(-d-1) \to \mathscr{O}_{\mathbb{P}^2}(-1) \to i_\ast \mathscr{O}_C(-1) \to 0. $$ This is a bit subtle the first time. Tensoring by $\mathscr{O}_{\mathbb{P}^2}(-1)$ is exact as it is a locally free sheaf. Also, $\mathscr{O}_C(-1)$ is (by definition) $i^\ast \mathscr{O}_{\mathbb{P}^2}(-1)$, and we used $$ i_\ast \mathscr{O}_C \otimes \mathscr{O}_{\mathbb{P}^2}(-1) \cong i_\ast (\mathscr{O}_C \otimes i^\ast \mathscr{O}_{\mathbb{P}^2}(-1)) = i_\ast i^\ast\mathscr{O}_{\mathbb{P}^2}(-1)\qquad (= i_\ast \mathscr{O}_C(-1) )$$ by the projection formula.

This way, taking cohomology, and using that $H^i(\mathbb{P}^2, i_\ast \mathscr{F})=H^i(C, \mathscr{F})$ for any coherent sheaf $\mathscr{F}$ yields the long exact sequence $$ 0 \to H^0(\mathbb{P}^2, \mathscr{O}_{\mathbb{P}^2}(-d-1)) \to H^0(\mathbb{P}^2, \mathscr{O}_{\mathbb{P}^2}) \to H^0(C, \mathscr{O}_C(-1)) \to H^1(\mathbb{P}^2, \mathscr{O}_{\mathbb{P}^2}(-d-1) \to \cdots. $$ This is enough to compute the cohomology, which I'll omit for now.

Of course, this works in general for locally free sheaves $\mathscr{F}$, by taking the closed subscheme exact sequence, tensoring by $\mathscr{F}$ and taking cohomology.