Del Pezzo surface

Summary

A del Pezzo surface is a smooth projective algebraic surface whose anticanonical divisor is ample. These are exactly the Fano varieties of dimension two. Over an algebraically closed field they are $\mathbb{P}^1 \times \mathbb{P}^1$ and the blow-ups of the plane in at most eight points in general position. So they are rational surfaces. Their degree $K^2$ runs from $1$ to $9$, and the curves of self-intersection $-1$ on them are governed by root systems of type $E$.