nLab del Pezzo surface




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=1d=1 and d=9d=9. Topologically, del Pezzo surfaces are determined by their degree except for d=8d=8.

If dd is different from 8, then a del Pezzo surface of degree dd, 𝔹 k\mathbb{B}_k, is a generic blow up of the complex projective plane 2\mathbb{P}^2 in 9d9-d points. But if d=8d=8, there are two choices: 2\mathbb{P}^2 blown up in one point, 𝔹 1\mathbb{B}_1, and 1× 1\mathbb{P}^1 \times \mathbb{P}^1, .

For surfaces formed by blowing up 9d9-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.


These surfaces admit metrics of positive scalar curvature.

Homology in degree 2: dimH 2(𝔹 k,)=k+1dim H_2(\mathbb{B}_k, \mathbb{R}) = k+ 1; a natural basis to choose is {H,E 1,,E k}\{H, E_1, \ldots, E_k\}, where the E aE_a are the “exceptional curves” obtained by blow-ups and HH represents the pullback of the generator of dimH 2( 2,)dim H_2(\mathbb{P}^2, \mathbb{R}) under the projection π:𝔹 k𝔹 0= 2\pi: \mathbb{B}_k \to\mathbb{B}_0 = \mathbb{P}^2 which simply collapses each exceptional curve E aE_a to the corresponding point p ap_a \in \mathbb{P}.

This has an integral statement as well. The orthogonal complement to the canonical class in the integral H 2H_2 is isomorphic to a root lattice R rR_r. These are given as E 8E_8, E 7E_7, E 6E_6, D 5D_5, A 4A_4 or A 2×A 1A_2 \times A_1 depending on the value of kk.


See also:

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

Last revised on January 16, 2023 at 13:33:07. See the history of this page for a list of all contributions to it.