How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

Question

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to solve it.

$4.2$. Let $k$ be a field, and let $F=x^n+y^n+z^n\in k[x,y,z]$, where $n\ge 1$ is prime to $\operatorname{char}(k)$. Determine $H^0(X,\Omega^1_{X/k})$ for $X=V_+(F)\subset \mathbb P^2_k$.

The preceding chapter was very theoretical, and I'm having trouble understanding how to apply the theory to do actual computations. In particular, it contains a lot of duality-like lemmas and I'm not sure which one to actually apply. To be honest, I don't understand it very well, and I was hoping doing something concrete with it would help.

I understand there are many different ways to compute cohomology groups. I would like to do this using the tools developed so far by Liu. In particular, he does not talk about Euler sequences like Hartshorne does. (It is possible they are in the book in disguise, but I do not see them. Please correct me if this is the case.) I would also appreciate any suggestions for elementary ways to use duality to compute this cohomology group. Thanks.

(The cohomology theory Liu uses is Čech cohomology.)

Edit. The answer below sketches a simple way to do this without any duality at all. My question is now this: What is this question doing in a chapter on duality? Does a duality argument make it even easier?

Edit 2. I think I see what was intended now. Once must use Corollary $4.14$ and proceed as in the following examples. The key observation here is that it suffices to compute the canonical sheaf $\omega_{X/k}$, since the canonical sheaf is isomorphic to $\Omega^1_{X/k}$ in this situation. Corollary $4.14$ then gives the tools necessary to compute $\omega_{X/k}$. Thinking in terms of duality was the wrong approach.

Answer 1

$H^0$ is just global sections, so he is askign you to describe the module of globally defined differential forms on your curve $X$.

• Find an open covering by affine open sets of the curve, and find the coordinate ring of those sets

• On each of them, find a presentation of the module of Kähler differentials.

• Finally, see which differentials on those open sets extend to the whole thing.

Answer 2

1) Forget about cohomology: $H^0(X,\Omega^1_{X/k})$ just means $\Omega^1_{X/k}(X)$, the vector space of global sections of the sheaf $\Omega^1_{X/k}$.

2) Forget about duality, which has nothing to do with the exercise.
The exercise is probably attached to section 6.4.2, devoted to the canonical sheaf.

3) The curve $X$ is smooth since $n$ is not divisible by $\text {char} (k)$.
Since it has degree $n$, its genus is $g=\frac {(n-1)(n-2)}{2}$.
And thus the dimension of the $k$-vector space $H^0(X,\Omega^1_{X/k})$ is $g=\frac {(n-1)(n-2)}{2}$, and this is a possible solution to the exercise.

Caveat
Since I have not read this great book systematically but only consulted it, I am not sure what solution the author had in mind (maybe he wants his reader to use theorem 4.9 (a), the adjunction formula, which would also provide a solution)
However I guarantee the correctness of the formula I described in point 3) .