Adjunction formula

Remarks

The idea of the proof: for $Y$ smooth with ideal sheaf $\mathcal{I}$, the conormal sequence $0 \to \mathcal{I}/\mathcal{I}^2 \to \Omega_X|_Y \to \Omega_Y \to 0$ is an exact sequence of vector bundles. Taking top exterior powers gives $\omega_X|_Y \cong \omega_Y \otimes \mathcal{I}/\mathcal{I}^2$. Since $\mathcal{I} = \mathcal{O}_X(-Y)$, the conormal bundle is $\mathcal{I}/\mathcal{I}^2 \cong \mathcal{O}_X(-Y)|_Y$, and the formula follows.

Example: let $C \subseteq \mathbb{P}^2_k$ be a smooth plane curve of degree $d$. Then $K_{\mathbb{P}^2} = -3H$ for a line $H$ and $C \sim dH$, so $K_C = (d-3)H|_C$ has degree $d(d-3)$. Since $\deg K_C = 2g - 2$, this gives the genus $g = (d-1)(d-2)/2$.

For a smooth closed subvariety $Y$ of any codimension $r$, the same argument gives $\omega_Y \cong \omega_X|_Y \otimes \det N_{Y/X}$, where $N_{Y/X}$ is the normal bundle.

Going deeper

For a student

The formula is the standard way to compute the genus of curves on surfaces and the canonical class of complete intersections. For a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ it gives $\omega_Y \cong \mathcal{O}_Y(d - n - 1)$, which decides whether $Y$ is Fano, Calabi–Yau or of general type. On a surface $X$ it gives the genus formula $2g - 2 = C \cdot (C + K_X)$ for a smooth curve $C$.