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.

1 Examples
2 Degree and the Classification Theorem
3 Remarks
4 Reference

[edit] Examples

$P 2$ — the projective plane.
$P^1 \times P^1$ , which is a quadric.
$B k$ — which is $P 2$ with k < 9 points in general position blown up.
Any smooth cubic surface is a different description of $B 6$ .

The surface $B 9$ 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 = 9, then X is isomorphic to the projective plane $P 2$ .

If d = 8, then X is isomorphic to $P 1 x P 1$ or the blow-up of $P 2$ 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 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.