Fano surface
# Fano surface

In algebraic geometry, a Fano surface is a surface of general type (in particular, not a Fano variety) whose points index the lines on a non-singular cubic threefold. They were first studied by Fano (1904).

Hodge diamond:

| | | 1 | | |
| - | - | - | - | - |
| | 5 | | 5 | |
| 10 | | 25 | | 10 |
| | 5 | | 5 | |
| | | 1 | | |

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 in 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 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 $K^2=45.$ 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 $3C_s,$ 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.

