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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-113/3/d/solution