Euler sequence on complex projective space

The Euler sequence is the short exact sequence of holomorphic vector bundlesTaking top exterior powers gives $\det T{\mathbb{CP}}^n\cong \mathcal{O}(n+1)$.

Dualizing the determinant identity in the Euler sequence on complex projective space giveswhere $H$ is a hyperplane divisor.

For a smooth degree-$d$ hypersurface $X_d\subset {\mathbb{CP}}^n$, the adjunction formula and $[X_d]\cong \mathcal{O}(d)$ give

For a smooth degree-$d$ projective plane curve $X$, the adjunction formula gives $K_X\cong {\mathcal{O}}_X(d-3)$. A hyperplane section has degree $d$, so