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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 and . Topologically, del Pezzo surfaces are determined by their degree except for .
If is different from 8, then a del Pezzo surface of degree , , is a generic blow up of the complex projective plane in points. But if , there are two choices: blown up in one point, , and , .
For surfaces formed by blowing up 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: ; a natural basis to choose is , where the are the “exceptional curves” obtained by blow-ups and represents the pullback of the generator of under the projection which simply collapses each exceptional curve to the corresponding point .
This has an integral statement as well. The orthogonal complement to the canonical class in the integral is isomorphic to a root lattice . These are given as , , , , or depending on the value of .
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.