Adjunction formula

For a smooth hypersurface $X$ in a complex manifold $Y$,

For a degree-$d$ smooth surface $X\subset {\mathbb{P}}^3$ and a smooth rational curve $C$ of degree $a$, the canonical bundle is $K_X={\mathcal{O}}_X(d-4)$. The adjunction formula gives $(K_X+C)\cdot C=-2$, hence $C^2=-2-(d-4)a$. In particular a line has self-intersection $2-d$.

For regular homogeneous equations of degrees $d_1,\dots ,d_r$ in ${\mathbb{P}}^n$, the normal bundle of a regular zero locus is ${⨁}_i\mathcal{O}(d_i){|}_V$. The Euler sequence gives $K_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$, hence $K_V=\mathcal{O}({\sum }_i{d}_i-n-1){|}_V$. Equations with unreduced multiplicities do not have this conclusion for the underlying reduced submanifold.

For a smooth complex submanifold $V\subset M$, the tangent exact sequence gives $\det TM{|}_V=\det TV\otimes \det N_V$. Dualizing yields $K_V=K_M{|}_V\otimes \det N_V$. This form of adjunction formula applies in any codimension.