Дель Пеццо беттері және олардың геометриясы

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, whose canonical class is big. 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 , which are the del Pezzo surfaces of degree at least 3.

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 P2. Its anticanonical embedding is the degree 3 Veronese embedding into P9 using the linear system of cubics.

A weak del Pezzo surface is a complete non singular surface with anticanonical bundle that is nef and big. The blowdown of any (−1) curve on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 more. The blowup of any point on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 less, provided that the point does not lie on a −2 curve and the degree is greater than 1. Any curve on a weak del Pezzo surface has self intersection number at least −2. The number of curves with self intersection number −2 is at most 9−d, and the number of curves with self intersection number −1 is finite.