Difference between revisions of "Fano surface"
From Encyclopedia of Mathematics
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| − | The surface parametrized by the family of lines lying on a non-singular cubic surface | + | {{TEX|done}} |
| + | 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 | + | 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Fano, "Sul sisteme | + | <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> |
Latest revision as of 15:54, 17 July 2014
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 |
How to Cite This Entry:
Fano surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fano_surface&oldid=32484
Fano surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fano_surface&oldid=32484
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article