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 .
Hodge diamond:
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<sup>4</sup> carries many remarkable geometric properties. The surface S is naturally embedded into the grassmannian of lines G(2,5) of P<sup>4</sup>. 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<sup>4</sup> 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<sub>1</sub>, w<sub>2</sub> be two 1-forms on S. The canonical divisor K on S associated to the canonical form w<sub>1</sub> ⧠w<sub>2</sub> parametrizes the lines on F that cut the plane P={w<sub>1</sub>=w<sub>2</sub>=0} into P<sup>4</sup>. Using w<sub>1</sub> and w<sub>2</sub> such that the intersection of P and F is the union of 3 lines, one can recover the fact that K<sup>2</sup>=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<sub>s</sub> be the corresponding line on the cubic F. Let C<sub>s</sub> be the divisor on S parametrizing lines that cut the line L<sub>s</sub>. The self-intersection of C<sub>s</sub> is equal to the intersection number of C<sub>s</sub> and C<sub>t</sub> for t a generic point. The intersection of C<sub>s</sub> and C<sub>t</sub> is the set of lines on F that cuts the disjoint lines L<sub>s</sub> and L<sub>t</sub>. Consider the linear span of L<sub>s</sub> and L<sub>t</sub> : it is an hyperplane into P<sup>4</sup> 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<sub>s</sub>) <sup>2</sup> =C<sub>s</sub> C<sub>t</sub>=5. As K is numerically equivalent to 3C<sub>s</sub>, we obtain K <sup>2</sup> =45.
c) The natural composite map: S -> G(2,5) -> P<sup>9</sup> is the canonical map of S. It is an embedding.