Ideal-sheaf sequence of a projective hypersurface
= Ideal-sheaf sequence of a projective hypersurface
Multiplication by the defining polynomial of a degree-$d$ <projective hypersurface> gives the short exact sequence
$$
0\longrightarrow\mathcal O_{\mathbb P^n}(-d)
\xrightarrow{\cdot F}\mathcal O_{\mathbb P^n}
\longrightarrow i_*\mathcal O_{X_d}\longrightarrow0.
$$
Thus the ideal sheaf of $X_d$ is $\mathcal O_{\mathbb P^n}(-d)$.