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 **P**4 carries many remarkable geometric properties. The surface S is naturally embedded into the grassmannian of lines G(2,5) of **P**4. 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 **P**4 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 *w*_{1}, *w*_{2} be two 1-forms on S. The canonical divisor K on S associated to the canonical form *w*_{1} ∧ *w*_{2} parametrizes the lines on F that cut the plane P={*w*_{1}=*w*_{2}=0} into **P**4. Using *w*_{1} and *w*_{2} 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 **P**4 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 3*C*_{s}, we obtain K 2 =45.

c) The natural composite map: S -> G(2,5) -> **P**9 is the canonical map of S. It is an embedding.

### Famous quotes containing the word surface:

“The *surface* of the ground in the Maine woods is everywhere spongy and saturated with moisture.”

—Henry David Thoreau (1817–1862)