Del Pezzo surface

In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class.

They are named for Pasquale del Pezzo who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree n embedding in n-dimensional projective space (del Pezzo 1887), which are the del Pezzo surfaces of degree at least 3.

A del Pezzo surface is a complete non-singular surface with ample anticanonical bundle. There are some variations of this definition that are sometimes used. Sometimes del Pezzo surfaces are allowed to have singularities. They were originally assumed to be embedded in projective space by the anticanonical embedding, which restricts the degree to be at least 3.

The degree d of a del Pezzo surface X is by definition the self intersection number (K, K) of its canonical class K.

Any curve on a del Pezzo surface has self intersection number at least −1. The number of curves with self intersection number −1 is finite and depends only on the degree (unless the degree is 8).

A (−1)-curve is a rational curve with self intersection number −1. For d > 2, the image of such a curve in projective space under the anti-canonical embedding is a line.

The blowdown of any (−1)-curve on a del Pezzo surface is a del Pezzo surface of degree 1 more. The blowup of any point on a del Pezzo surface is a del Pezzo surface of degree 1 less, provided that the point does not lie on a (−1)-curve and the degree is greater than 2. When the degree is 2, we have to add the condition that the point is not fixed by the Geiser involution, associated to the anti-canonical morphism.

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