Del Pezzo surface

Remarks

The degree satisfies $1 \le d \le 9$. Classification: $X$ is either $\mathbb{P}^1 \times \mathbb{P}^1$, of degree $8$, or the blow-up of $\mathbb{P}^2$ at $9 - d$ points in general position. Here general position means three conditions. No three of the points lie on a line, and no six lie on a conic. No eight lie on a cubic that is singular at one of them. Example: $\mathbb{P}^2$ has $K = -3L$ for a line $L$, so $-K = 3L$ is ample and $\mathbb{P}^2$ is a del Pezzo surface of degree $9$. Example: a smooth cubic surface $X \subset \mathbb{P}^3$ has $K_X = -H$ by adjunction, where $H$ is a plane section, so $-K_X$ is ample and $d = H^2 = 3$. Non-example: blow up $\mathbb{P}^2$ at three points on a line $L$, and let $\tilde L$ be the strict transform of $L$. Then $\tilde L^2 = 1 - 3 = -2$, and adjunction gives $K \cdot \tilde L = -2 - \tilde L^2 = 0$. So $-K \cdot \tilde L = 0$, and $-K$ is not ample. Some authors allow mild singularities, such as ADE singularities, and call the smooth ones nonsingular del Pezzo surfaces.