Examples
Degree 1: they have 240 (−1)-curves corresponding to the roots of an E8 root system. They form an 8-dimensional family. The anticanonical divisor is not very ample. The linear system |−2K| defines a degree 2 map from the del Pezzo surface to a quadratic cone in P3, branched over a nonsingular genus 4 curve cut out by a cubic surface.
Degree 2: they have 56 (−1)-curves corresponding to the minuscule vectors of the dual of the E7 lattice. They form a 6-dimensional family. The anticanonical divisor is not very ample, and its linear system defines a map from the del Pezzo surface to the projective plane, branched over a quartic plane curve. This map is generically 2 to 1, so this surface is sometimes called a del Pezzo double plane. The 56 lines of the del Pezzo surface map in pairs to the 28 bitangents of a quartic.
Degree 3: these are essentially cubic surfaces in P3; the cubic surface is the image of the anticanonical embedding. They have 27 (−1)-curves corresponding to the minuscule vectors of one coset in the dual of the E6 lattice, which map to the 27 lines of the cubic surface. They form a 4-dimensional family.
Degree 4: these are essentially Segre surfaces in P4, given by the intersection of two quadrics. They have 16 (−1)-curves. They form a 2-dimensional family.
Degree 5: they have 10 (−1)-curves corresponding to the minuscule vectors of one coset in the dual of the A4 lattice. There is up to isomorphism only one such surface, given by blowing up the projective plane in 4 points with no 3 on a line.
Degree 6: they have 6 (−1)-curves. There is up to isomorphism only one such surface, given by blowing up the projective plane in 3 points not on a line. The root system is A2 × A1
Degree 7: they have 3 (−1)-curves. There is up to isomorphism only one such surface, given by blowing up the projective plane in 2 distinct points.
Degree 8: they have 2 isomorphism types. One is a Hirzebruch surface given by the blow up of the projective plane at one point, which has 1 (−1)-curves. The other is the product of two projective lines, which is the only del Pezzo surface that cannot be obtained by starting with the projective plane and blowing up points. Its Picard group is the even 2-dimensional unimodular indefinite lattice II1,1, and it contains no (−1)-curves.
Degree 9: The only degree 9 del Pezzo surface is the P2. Its anticanonical embedding is the degree 3 Veronese embedding into P9 using the linear system of cubics.
Read more about this topic: Del Pezzo Surface
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