[proofplan]
The proof is the adjunction computation for a smooth divisor in projective space. First we compute the canonical bundle of $\mathbb{P}^n_k$ from the Euler sequence. Then we use the conormal exact sequence for the smooth hypersurface $X_d \subset \mathbb{P}^n_k$ and take determinants. Since a degree-$d$ hypersurface has conormal bundle $\mathcal{O}_{X_d}(-d)$, the determinant formula gives $K_{X_d} \simeq i^*K_{\mathbb{P}^n_k} \otimes \mathcal{O}_{X_d}(d)$, and substituting $K_{\mathbb{P}^n_k} \simeq \mathcal{O}_{\mathbb{P}^n_k}(-n-1)$ gives the result.
[/proofplan]
[step:Compute the canonical bundle of projective space from the Euler sequence]
Let $P := \mathbb{P}^n_k$. The Euler exact sequence on $P$ is
\begin{align*}
0 \longrightarrow \Omega^1_P \longrightarrow \mathcal{O}_P(-1)^{\oplus(n+1)} \longrightarrow \mathcal{O}_P \longrightarrow 0.
\end{align*}
Here $\Omega^1_P$ is the sheaf of Kähler differentials of $P$ over $k$. The Euler sequence is a standard exact sequence for projective space (citing a result not yet in the wiki: Euler sequence for projective space).
Because $P$ is smooth of dimension $n$, its canonical bundle is
\begin{align*}
K_P := \det \Omega^1_P.
\end{align*}
Taking determinants in the Euler sequence gives
\begin{align*}
\det\bigl(\mathcal{O}_P(-1)^{\oplus(n+1)}\bigr)
\simeq
\det(\Omega^1_P) \otimes \det(\mathcal{O}_P).
\end{align*}
Since $\det(\mathcal{O}_P) \simeq \mathcal{O}_P$ and
\begin{align*}
\det\bigl(\mathcal{O}_P(-1)^{\oplus(n+1)}\bigr)
\simeq
\mathcal{O}_P(-1)^{\otimes(n+1)}
\simeq
\mathcal{O}_P(-n-1),
\end{align*}
we obtain
\begin{align*}
K_P \simeq \mathcal{O}_P(-n-1).
\end{align*}
[guided]
Let $P := \mathbb{P}^n_k$. We begin by computing the canonical bundle of the ambient projective space, because adjunction will later restrict it to the hypersurface.
The Euler exact sequence on $P$ is
\begin{align*}
0 \longrightarrow \Omega^1_P \longrightarrow \mathcal{O}_P(-1)^{\oplus(n+1)} \longrightarrow \mathcal{O}_P \longrightarrow 0.
\end{align*}
This is the standard exact sequence relating the cotangent bundle of projective space to the tautological line bundle (citing a result not yet in the wiki: Euler sequence for projective space).
Since $P$ is smooth of dimension $n$, its canonical bundle is defined by
\begin{align*}
K_P := \det \Omega^1_P.
\end{align*}
The determinant operation on a short exact sequence of locally free sheaves gives
\begin{align*}
\det\bigl(\mathcal{O}_P(-1)^{\oplus(n+1)}\bigr)
\simeq
\det(\Omega^1_P) \otimes \det(\mathcal{O}_P).
\end{align*}
The determinant of the rank-one trivial bundle is trivial:
\begin{align*}
\det(\mathcal{O}_P) \simeq \mathcal{O}_P.
\end{align*}
The determinant of the direct sum of $n+1$ copies of $\mathcal{O}_P(-1)$ is the [tensor product](/page/Tensor%20Product) of those $n+1$ line bundles:
\begin{align*}
\det\bigl(\mathcal{O}_P(-1)^{\oplus(n+1)}\bigr)
\simeq
\mathcal{O}_P(-1)^{\otimes(n+1)}
\simeq
\mathcal{O}_P(-n-1).
\end{align*}
Therefore
\begin{align*}
K_P = \det \Omega^1_P \simeq \mathcal{O}_P(-n-1).
\end{align*}
This is the ambient canonical bundle that will enter the [adjunction formula](/theorems/3878).
[/guided]
[/step]
[step:Identify the conormal bundle of the degree-$d$ hypersurface]
Let $i: X_d \hookrightarrow P$ be the closed immersion. Since $X_d$ is a hypersurface cut out by one homogeneous equation of degree $d$, its ideal sheaf in $P$ is
\begin{align*}
\mathcal{I}_{X_d} \simeq \mathcal{O}_P(-d).
\end{align*}
Restricting to $X_d$ gives the conormal bundle
\begin{align*}
\mathcal{I}_{X_d}/\mathcal{I}_{X_d}^2 \simeq i^*\mathcal{O}_P(-d) = \mathcal{O}_{X_d}(-d).
\end{align*}
Because $X_d$ is smooth over $k$, the conormal sequence is a short exact sequence of locally free sheaves on $X_d$:
\begin{align*}
0 \longrightarrow \mathcal{I}_{X_d}/\mathcal{I}_{X_d}^2
\longrightarrow i^*\Omega^1_P
\longrightarrow \Omega^1_{X_d}
\longrightarrow 0.
\end{align*}
The conormal exact sequence for a smooth closed immersion is a standard result (citing a result not yet in the wiki: conormal exact sequence).
[guided]
Let $i: X_d \hookrightarrow P$ be the closed immersion. Since $X_d$ is a hypersurface of degree $d$, it is cut out by one homogeneous polynomial of degree $d$. A single degree-$d$ equation defines a section of $\mathcal{O}_P(d)$, so the corresponding ideal sheaf is
\begin{align*}
\mathcal{I}_{X_d} \simeq \mathcal{O}_P(-d).
\end{align*}
The conormal bundle is the quotient of the ideal by its square, viewed as a sheaf on $X_d$. Therefore
\begin{align*}
\mathcal{I}_{X_d}/\mathcal{I}_{X_d}^2 \simeq i^*\mathcal{O}_P(-d) = \mathcal{O}_{X_d}(-d).
\end{align*}
The smoothness hypothesis is used here to ensure that the cotangent sheaf $\Omega^1_{X_d}$ is locally free and that the conormal sequence is an exact sequence of locally free sheaves:
\begin{align*}
0 \longrightarrow \mathcal{I}_{X_d}/\mathcal{I}_{X_d}^2
\longrightarrow i^*\Omega^1_P
\longrightarrow \Omega^1_{X_d}
\longrightarrow 0.
\end{align*}
This is the standard conormal exact sequence for a smooth closed immersion (citing a result not yet in the wiki: conormal exact sequence). The point of this sequence is that it compares the cotangent bundle of $X_d$ with the restricted cotangent bundle of $P$, with the conormal line bundle measuring the one normal direction cut out by the hypersurface equation.
[/guided]
[/step]
[step:Take determinants in the conormal sequence to obtain adjunction]
Taking determinants in
\begin{align*}
0 \longrightarrow \mathcal{O}_{X_d}(-d)
\longrightarrow i^*\Omega^1_P
\longrightarrow \Omega^1_{X_d}
\longrightarrow 0
\end{align*}
gives
\begin{align*}
\det(i^*\Omega^1_P)
\simeq
\det(\mathcal{O}_{X_d}(-d)) \otimes \det(\Omega^1_{X_d}).
\end{align*}
Since $\mathcal{O}_{X_d}(-d)$ is a line bundle,
\begin{align*}
\det(\mathcal{O}_{X_d}(-d)) \simeq \mathcal{O}_{X_d}(-d).
\end{align*}
Also
\begin{align*}
\det(i^*\Omega^1_P) \simeq i^*\det(\Omega^1_P) = i^*K_P,
\end{align*}
and
\begin{align*}
\det(\Omega^1_{X_d}) = K_{X_d}.
\end{align*}
Thus
\begin{align*}
i^*K_P \simeq \mathcal{O}_{X_d}(-d) \otimes K_{X_d}.
\end{align*}
Tensoring both sides with $\mathcal{O}_{X_d}(d)$ yields
\begin{align*}
K_{X_d} \simeq i^*K_P \otimes \mathcal{O}_{X_d}(d).
\end{align*}
[guided]
We now take determinants in the conormal sequence. The sequence is
\begin{align*}
0 \longrightarrow \mathcal{O}_{X_d}(-d)
\longrightarrow i^*\Omega^1_P
\longrightarrow \Omega^1_{X_d}
\longrightarrow 0.
\end{align*}
For a short exact sequence of locally free sheaves
\begin{align*}
0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0,
\end{align*}
the determinant line bundles satisfy
\begin{align*}
\det B \simeq \det A \otimes \det C.
\end{align*}
Applying this determinant formula with
\begin{align*}
A = \mathcal{O}_{X_d}(-d), \qquad
B = i^*\Omega^1_P, \qquad
C = \Omega^1_{X_d},
\end{align*}
we obtain
\begin{align*}
\det(i^*\Omega^1_P)
\simeq
\det(\mathcal{O}_{X_d}(-d)) \otimes \det(\Omega^1_{X_d}).
\end{align*}
Now each determinant has a concrete interpretation. Since $\mathcal{O}_{X_d}(-d)$ is already a line bundle,
\begin{align*}
\det(\mathcal{O}_{X_d}(-d)) \simeq \mathcal{O}_{X_d}(-d).
\end{align*}
Since determinants commute with pullback for locally free sheaves,
\begin{align*}
\det(i^*\Omega^1_P) \simeq i^*\det(\Omega^1_P) = i^*K_P.
\end{align*}
Finally, because $X_d$ is smooth, its canonical bundle is
\begin{align*}
K_{X_d} := \det(\Omega^1_{X_d}).
\end{align*}
Substituting these identifications into the determinant formula gives
\begin{align*}
i^*K_P \simeq \mathcal{O}_{X_d}(-d) \otimes K_{X_d}.
\end{align*}
Tensoring both sides with the inverse line bundle $\mathcal{O}_{X_d}(d)$ gives the adjunction form
\begin{align*}
K_{X_d} \simeq i^*K_P \otimes \mathcal{O}_{X_d}(d).
\end{align*}
[/guided]
[/step]
[step:Substitute the ambient canonical bundle and simplify the twist]
From the first step,
\begin{align*}
K_P \simeq \mathcal{O}_P(-n-1).
\end{align*}
Pulling back along $i$ gives
\begin{align*}
i^*K_P \simeq i^*\mathcal{O}_P(-n-1) = \mathcal{O}_{X_d}(-n-1).
\end{align*}
Substituting this into the adjunction isomorphism gives
\begin{align*}
K_{X_d}
&\simeq \mathcal{O}_{X_d}(-n-1) \otimes \mathcal{O}_{X_d}(d) \\
&\simeq \mathcal{O}_{X_d}(d-n-1).
\end{align*}
This is the asserted canonical bundle formula.
[/step]
