Fano variety

Remarks

In divisor notation, $X$ is Fano when $-K_X$ is ample. Example: on $\mathbb{P}^n$ one has $\omega = \mathcal{O}(-n-1)$, so $\omega^{-1} = \mathcal{O}(n+1)$ is ample and $\mathbb{P}^n$ is Fano. More generally, a smooth hypersurface $X \subset \mathbb{P}^n$ of degree $d \ge 1$, with $n \ge 2$, has $\omega_X = \mathcal{O}_X(d - n - 1)$ by the adjunction formula, so it is Fano exactly when $d \le n$. Non-example: a smooth plane cubic curve has $\omega = \mathcal{O}(3 - 3) = \mathcal{O}$, and the trivial bundle on a curve is not ample. Many authors also allow mild singularities, such as terminal or log terminal ones, and then ask that $-K_X$ be an ample $\Q$-Cartier divisor. Over $\C$, smooth Fano varieties are simply connected and rationally connected, and in each dimension they form finitely many deformation families.