Fano surface

The surface parametrized by the family of lines lying on a non-singular cubic surface . G. Fano studied the family of lines on a three-dimensional cubic [1].

Through a generic point of a non-singular cubic there pass exactly 6 lines lying on it, and the Fano surface is a non-singular irreducible reduced algebraic surface of geometric genus and irregularity , with topological Euler characteristic (in case ) equal to 27. From the Fano surface one can reconstruct the cubic (see [2]).

References

[1]	G. Fano, "Sul sisteme di rette contenuto in une varietà cubica generale dello spacio a quattro dimensioni" Atti R. Accad. Sci. Torino , 39 (1903–1904) pp. 778–792
[2]	A.N. [A.N. Tyurin] Tjurin, "On the Fano surface of a nonsingular cubic in " Math. USSR Izv. , 4 : 6 (1960) pp. 1207–1214 Izv. Akad. Nauk SSSR Ser. Mat. , 34 : 6 (1970) pp. 1200–1208
[3]	C. Clemens, P. Griffiths, "The intermediate Jacobian of the cubic threefold" Ann. of Math. , 95 (1972) pp. 281–356