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Canonical bundle of complex projective space

{title2=$K_{X_d}\cong\mathcal O_{X_d}(d-n-1)$}

For a smooth degree-$d$ hypersurface $X_d\subset\mathbb{CP}^n$, the <Adjunction formula> and $[X_d]\cong\mathcal O(d)$ give
$$
K_{X_d}\cong\bigl(K_{\mathbb{CP}^n}\otimes[X_d]\bigr)|_{X_d}
\cong\mathcal O_{X_d}(d-n-1).
$$

Canonical bundle of a smooth projective hypersurface

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 159 / 2 / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 118 / 2 / Solution

Canonical bundle of a smooth projective hypersurface ( $K_{X_{d}} ≅ O_{X_{d}} (d - n - 1)$ )

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Canonical bundle of a smooth projective hypersurface (KXd​​≅OXd​​(d−n−1))

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Canonical bundle of a smooth projective hypersurface ( $K_{X_{d}} ≅ O_{X_{d}} (d - n - 1)$ )

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