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del Pezzo surface

Contents

Contents

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=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, ๐”น k\mathbb{B}_k, is a generic blow up of the complex projective plane โ„™ 2\mathbb{P}^2 in 9โˆ’d9-d points. But if d=8d=8, there are two choices: โ„™ 2\mathbb{P}^2 blown up in one point, ๐”น 1\mathbb{B}_1, and โ„™ 1ร—โ„™ 1\mathbb{P}^1 \times \mathbb{P}^1, .

For surfaces formed by blowing up 9โˆ’d9-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: dimH 2(๐”น k,โ„)=k+1dim H_2(\mathbb{B}_k, \mathbb{R}) = k+ 1; a natural basis to choose is {H,E 1,โ€ฆ,E k}\{H, E_1, \ldots, E_k\}, where the E aE_a are the โ€œexceptional curvesโ€ obtained by blow-ups and HH represents the pullback of the generator of dimH 2(โ„™ 2,โ„)dim H_2(\mathbb{P}^2, \mathbb{R}) under the projection ฯ€:๐”น kโ†’๐”น 0=โ„™ 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 aโˆˆโ„™p_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 2ร—A 1A_2 \times A_1 depending on the value of kk.

References

See also:

Last revised on November 29, 2020 at 04:34:51. See the history of this page for a list of all contributions to it.