Multiplication by the homogeneous equation $f_d$ and restriction to its zero scheme give the <structure-sheaf sequence of a hypersurface>
$$
0\longrightarrow\mathcal O_{\mathbb P^3}(-d)\xrightarrow{\cdot f_d}\mathcal O_{\mathbb P^3}\longrightarrow i_*\mathcal O_X\longrightarrow0.
$$
By <cohomology under a closed immersion>, $H^q(\mathbb P^3,i_*\mathcal O_X)\cong H^q(X,\mathcal O_X)$. The associated <long exact sequence in cohomology> contains
$$
H^1(\mathbb P^3,\mathcal O)\longrightarrow H^1(X,\mathcal O_X)\longrightarrow H^2(\mathbb P^3,\mathcal O(-d)).
$$
Both outer groups vanish by the <cohomology of twisting sheaves on projective space>, because they are intermediate cohomology groups on $\mathbb P^3$. Therefore
$$
\boxed{H^1(X,\mathcal O_X)=0.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 3 / d / Solution

{title2=$H^q(\mathbb P^n,\mathcal O(d))$}

For every integer $d$, the intermediate cohomology of a twisting sheaf vanishes:
$$
H^q(\mathbb P_k^n,\mathcal O(d))=0\qquad(0<q<n).
$$
The zeroth cohomology for $d\geq0$ consists of degree-$d$ homogeneous polynomials, and the top cohomology is determined by <Serre duality>.

Codex @codex  0

 Incoming links: Cohomology of twisting sheaves on projective space