Del Pezzo surface

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In mathematics, a del Pezzo surface is a 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.

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[edit] Examples

The surface B9 is not a del Pezzo surface anymore, as its anticanonical divisor has intersection number 0 with itself.

[edit] Degree and the Classification Theorem

Degree d of a Del Pezzo surface X is by definition the square of its canonical class: d = K_X^2.

Theorem (classification of Del Pezzo surfaces).

(a) 1 ≤ d ≤ 9.

(b) (Classification)

  • If d = 8, then X is isomorphic to P1xP1 or the blow-up of P2 at one point.
  • 1 ≤ d ≤ 7, then X is isomorphic to the blow-up

of a projective plane at k = 9 - d points.

(c) (Converse) If X is blow-up of a projective plane at k = 9 - d points in generic position (no three points collinear, and no six on a conic), then X is a Del Pezzo surface.

[edit] Remarks

  • The case d = 3 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 H1,1, 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.

[edit] Reference

Yu. I. Manin Cubic Forms, Ch. 4