nLab del Pezzo surface

Idea

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such a surface has a degree defined as the self-intersection of the canonical divisor. The possible degrees run between $d=1$ and $d=9$. Topologically, del Pezzo surfaces are determined by their degree except for $d=8$.

If $d$ is different from 8, then a del Pezzo surface of degree $d$, $\mathbb{B}_k$, is a generic blow up of the complex projective plane $\mathbb{P}^2$ in $9-d$ points. But if $d=8$, there are two choices: $\mathbb{P}^2$ blown up in one point, $\mathbb{B}_1$, and $\mathbb{P}^1 \times \mathbb{P}^1$, .

For surfaces formed by blowing up $9-d$ points, no three may be collinear, no six lie on a conic, and no eight of them lie on a cubic having a node at one of them. Conversely any blowup of the plane in points satisfying these conditions is a del Pezzo surface.

Properties

These surfaces admit metrics of positive scalar curvature.

Homology in degree 2: $dim H_2(\mathbb{B}_k, \mathbb{R}) = k+ 1$; a natural basis to choose is $\{H, E_1, \ldots, E_k\}$, where the $E_a$ are the “exceptional curves” obtained by blow-ups and $H$ represents the pullback of the generator of $dim H_2(\mathbb{P}^2, \mathbb{R})$ under the projection $\pi: \mathbb{B}_k \to\mathbb{B}_0 = \mathbb{P}^2$ which simply collapses each exceptional curve $E_a$ to the corresponding point $p_a \in \mathbb{P}$.

This has an integral statement as well. The orthogonal complement to the canonical class in the integral $H_2$ is isomorphic to a root lattice $R_r$. These are given as $E_8$, $E_7$, $E_6$, $D_5$, $A_4$ or $A_2 \times A_1$ depending on the value of $k$.

Related concepts

• mysterious duality

References

See also:

• Wikipedia, Del Pezzo surface

On exceptional structures (exceptional Lie algebras and sporadic finite simple groups) via del Pezzo surfaces:

• Yang-Hui He, John McKay, Sporadic and Exceptional [arXiv:1505.06742]