Fano Surface

Fano surfaces are perhaps the simplest and most studied examples of irregular surfaces of general type that are not related to a product of two curves and are not a complete intersection of divisors into an Abelian variety.

The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties. The surface S is naturally embedded into the grassmannian of lines G(2,5) of P4. Let U be the restriction to S of the universal rank 2 bundle on G. We have the:

Tangent bundle Theorem (Fano, Clemens-Griffiths, Tyurin): The tangent bundle of S is isomorphic to U.

This is a quite interesting result because, a priori, there should be no link between these two bundles. It has many powerful applications. By example, one can recover the fact that that the cotangent space of S is generated by global sections. This space of global 1-forms can be identified with the space of global sections of the tautological line bundle O(1) restricted to the cubic F and moreover:

Torelli-type Theorem : Let g' be the natural morphism from S to the grassmannian G(2,5) defined by the cotangent sheaf of S generated by its 5 dimensional space of global sections. Let F' be the union of the lines corresponding to g'(S). The threefold F' is isomorphic to F.

Thus knowing a Fano surface S, we can recover the threefold F. By the Tangent Bundle Theorem, we can also understand geometrically the invariants of S:

a) Recall that the second Chern number of a rank 2 vector bundle on a surface is the number of zeroes of a generic section. For a Fano surface S, a 1-form w defines also a hyperplane section {w=0} into P4 of the cubic F. The zeros of the generic w on S corresponds bijectively to the numbers of lines into the smooth cubic surface intersection of {w=0} and F, therefore we recover that the second Chern class of S equals 27.

b) Let w₁, w₂ be two 1-forms on S. The canonical divisor K on S associated to the canonical form w₁ ∧ w₂ parametrizes the lines on F that cut the plane P={w₁=w₂=0} into P4. Using w₁ and w₂ such that the intersection of P and F is the union of 3 lines, one can recover the fact that K2=45. Let us give some details of that computation: By a generic point of the cubic F goes 6 lines. Let s be a point of S and let L_s be the corresponding line on the cubic F. Let C_s be the divisor on S parametrizing lines that cut the line L_s. The self-intersection of C_s is equal to the intersection number of C_s and C_t for t a generic point. The intersection of C_s and C_t is the set of lines on F that cuts the disjoint lines L_s and L_t. Consider the linear span of L_s and L_t : it is an hyperplane into P4 that cuts F into a smooth cubic surface. By well known results on a cubic surface, the number of lines that cuts two disjoints lines is 5, thus we get (C_s) 2 =C_s C_t=5. As K is numerically equivalent to 3C_s, we obtain K 2 =45.

c) The natural composite map: S -> G(2,5) -> P9 is the canonical map of S. It is an embedding.