Del Pezzo surface

In mathematics, a del Pezzo surface is a complex two-dimensional Fano variety, i.e. an algebraic surface with ample vector bundle anticanonical divisor class.

The name is for Pasquale del Pezzo (1859-1936), an Italy mathematician from Naples. He initiated the study of these surfaces around 1887.

= Examples =

B_0=CP^2 — the complex projective plane.

CP^1 imes CP^1, which is a Quadric.

B_k — which is CP^2 with k < 9 points in general position blown up.

Any cubic surface is a different description of B_6.

The surface B_9 is not a del Pezzo surface anymore. B_9 must be non-compact, and can be visualized as one half of the K3 surface.

A Del Pezzo surface has a degree d : the projective plane case is d = 9 and the quadric case d = 8. The other possible cases are those for d = 9 − k with 3 ≤ k ≤ 8, and general position here meaning no three points collinear, and no six on any conic. The case k = 6 is that of cubic surfaces. There is interest in the intersection theory of curves on a Del Pezzo surface, represented by the Picard group of divisor classes or the Hodge space H 1,1, because of the connection with root systems of the ADE classification, in the various cases. This has commonly been invoked in work on string theory.

=Reference=

Yu. I. Manin Cubic Forms , Ch. 4