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Context: Let $k$ be an algebraically closed field of characteristic 3. Let $H\hookrightarrow \mathbb{P}^2_k$ be a closed inmersion, where $H$ a hypersurface of degree 3, then we have the short exact sequence $$0\longrightarrow \mathcal{O}_{\mathbb{P}^2_k}(-3)\longrightarrow \mathcal{O}_{\mathbb{P}^2_k} \longrightarrow i_*\mathcal{O}_E\longrightarrow 0 . $$ Taking cohomology we get a long exact sequence and I am interested in calculating the term $H^2(\mathbb{P}^2_k, \mathcal{O}_{\mathbb{P}^2_k}(-3))$.

(*) Up to the Theorem 5.1, III, pag 225 in Hartshorne it seems that this term is $k$-vector space, $k$, of dimension 1, since it corresponds to homogeneous polynomials of dregree 3 and since there is unique posible combination of the exponents in the mononial expression.

To be honest, I am really guessing (*), since I do not understand the details of the process. Probably it is really evident but I do not see it.

Question: How to proof in detail that this is really a $k$-vector space of dimension 1?

My guess: this has to be with the fact that in general, the dimension of homogeneous polynomials in degree $d$ is $ \binom{n+d}{d}$ but I do not see how this fits in the calculations, since I would get $\binom{5}{3}=10$. Moreover, $(k[x_0,x_1,x_2])_3$ is a free $k$-module of rank $ \binom{2+3}{3}$.

Can someone please help me to understand in detail this calculation? Any help will be greatly appreciated.

I also saw the proof of the Theorem 5.1 I mentioned above (not just in Hartshorne but in other references and also in other posts like Compute the cohomology of projective schemes) but I can not understand how to conclude properly the dimension of this vector space.

Ps.: I am new in stackexachange, I am learning how to formulate properly my questions. Please if you think I need to be more precise, let me know and I will.

Pedro
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1 Answers1

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For me, the notes by Vakil (the July 2024 version, pp. 494-495, 504-507) really helped when learning this stuff.

It is a bit tedious to work through, but what you end up with is that the vector space $H^2(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(-3))$ may be identified with the space of Laurent Polynomials of degree $-3$, in which each variable occurs with degree at most $-1$ (essentially 18.3.2 + 18.3.4 in the notes). But since we have only three variables, this means that we have the (unique up to scaling) basis consisting of the element $x_0^{-1}x_1^{-1}x_2^{-1}$.

Remark: This approach also yields results for more general cohomology groups.

ClemensB
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