View source for Fano surface

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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 [[#References|[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|geometric genus]] $p_g=10$ and [[Irregularity|irregularity]] $q=5$, with topological [[Euler characteristic|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 [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Clemens,   P. Griffiths,   "The intermediate Jacobian of the cubic threefold"  ''Ann. of Math.'' , '''95'''  (1972)  pp. 281–356</TD></TR></table>