Fano surface

The surface parametrized by the family of lines lying on a non-singular cubic surface $V_3\subset P^4$. G. Fano studied the family of lines $F(V_3)$ on a three-dimensional cubic [1].

Through a generic point of a non-singular cubic $V_3\subset P^4$ there pass exactly 6 lines lying on it, and the Fano surface $F(V_3)$ is a non-singular irreducible reduced algebraic surface of geometric genus $p_g=10$ and irregularity $q=5$, with topological Euler characteristic (in case $k=\mathbf C$) equal to 27. From the Fano surface $F(V_3)$ one can reconstruct the cubic $V_3$ (see [2]).

References

[1]	G. Fano, "Sul sisteme $\infty^2$ 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 $P^4$" 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