Del Pezzo surface

From Free net encyclopedia

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

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

Examples

• <math>B_0=CP^2</math> — the complex projective plane.
• <math>CP^1 \times CP^1</math>, which is a quadric.
• <math>B_k</math> — which is <math>CP^2</math> with k < 9 points in general position blown up.
• Any smooth cubic surface is a different description of <math>B_6</math>.

The surface <math>B_9</math> is not a del Pezzo surface anymore, as its anticanonical divisor has intersection number 0 with itself.

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. 4Template:Geometry-stub