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.

Contents 1 Examples 2 Degree and the classification theorem 3 Remarks 4 References

[edit] Examples

• P² — the projective plane.
• , which is a quadric.
• B_k — which is P² with k < 9 points in general position blown up.
• Any smooth cubic surface is a different description of B₆.

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 anticanonical class (equivalently, canonical class): ; equivalently, the self-intersection of the (anti-)canonical divisor.

Theorem (classification of del Pezzo surfaces).

(a) 1 ≤ d ≤ 9.

(b) (Classification) Either:

• X is isomorphic to the blow-up of a projective plane at k = 9 − d points, or
• X is isomorphic to P¹xP¹ (and d = 8)

In particular, given d:

• If d = 9, then X is isomorphic to the projective plane P² (the blow-up of P² at no points).
• If d = 8, then X is isomorphic to the blow-up of P² at one point or to P¹xP¹ (which is the blow-up of P² at "2 − 1" points: choose 2 points in P² and let L be the line between them. If we blow up the 2 points then the strict transform of L has self-intersection -1, and blowing it down yields P¹xP¹.)
• If , then X is isomorphic to the blow-up of a projective plane at k = 9 − d points.

(c) (Converse) If X is the blow-up of a projective plane at k = 9 − d points in generic position (no three points collinear, no six on a conic, no eight of them on a cubic having a node at one of them), 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 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.

[edit] References

• Yu. I. Manin Cubic Forms, Ch. 4