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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 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 w1w2 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 Ls be the corresponding line on the cubic F. Let Cs be the divisor on S parametrizing lines that cut the line Ls. The self-intersection of Cs is equal to the intersection number of Cs and Ct for t a generic point. The intersection of Cs and Ct is the set of lines on F that cuts the disjoint lines Ls and Lt. Consider the linear span of Ls and Lt : 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 (Cs) 2 =Cs Ct=5. As K is numerically equivalent to 3Cs, 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.

Hodge theory

References

Bombieri, Enrico; Swinnerton-Dyer, H. P. F. (1967), "On the local zeta function of a cubic threefold", Ann. Scuola Norm. Sup. Pisa (3) 21: 1–29, MR 0212019
Clemens, C. Herbert; Griffiths, Phillip A. (1972), "The intermediate Jacobian of the cubic threefold", Annals of Mathematics. Second Series (Annals of Mathematics) 95 (2): 281–356, doi:10.2307/1970801, ISSN 0003-486X, JSTOR 1970801, MR 0302652
Fano, G. (1904), "Sul sisteme ∞2 di rette contenuto in une varietà cubica generale dello spacio a quattro dimensioni", Atti R. Accad. Sci. Torino 39: 778–792
Kulikov, Vik.S. (2001), "f/f038210", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Murre, J. P. (1972), "Algebraic equivalence modulo rational equivalence on a cubic threefold", Compositio Mathematica 25: 161–206, ISSN 0010-437X, MR 0352088