Del Pezzo Surface

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.

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.

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).

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.

Del Pezzo proved that a del Pezzo surface has degree d at most 9. Over an algebraically closed field, every del Pezzo surface is either a product of two projective lines (with d=8), or the blow-up of a projective plane in 9 − d points with no three collinear, no six on a conic, and no eight of them on a cubic having a node at one of them. Conversely any blowup of the plane in points satisfying these conditions is a del Pezzo surface.

The Picard group of a del Pezzo surface of degree d is the odd unimodular lattice I_1,9−d, except when the surface is a product of 2 lines when the Picard group is the even unimodular lattice II_1,1.When it is an odd lattice, the canonical element is (3, 1, 1, 1, ....), and the exceptional curves are represented by permutations of all but the first coordinate of the following vectors:

• (0, −1, 0, 0, ....) the exceptional curves of the blown up points,
• (1, 1, 1, 0, 0, ...) lines through 2 points,
• (2, 1, 1, 1, 1, 1, 0, ...) conics through 5 points,
• (3, 2, 1, 1, 1, 1, 1, 1, 0, ...) cubics through 7 points with a double point at one of them,
• (4, 2, 2, 2, 1, 1, 1, 1, 1) quartics through 8 points with double points at three of them,
• (5, 2, 2, 2, 2, 2, 2, 1, 1) quintics through 8 points with double points at all but two of them,
• (6, 3, 2, 2, 2, 2, 2, 2, 2) sextics through 8 points with double points at all except a single point with multiplicity three.