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Difference between revisions of "Fano surface"

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The surface parametrized by the family of lines lying on a non-singular cubic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382101.png" />. G. Fano studied the family of lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382102.png" /> on a three-dimensional cubic [[#References|[1]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382103.png" /> there pass exactly 6 lines lying on it, and the Fano surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382104.png" /> is a non-singular irreducible reduced algebraic surface of [[Geometric genus|geometric genus]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382105.png" /> and [[Irregularity|irregularity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382106.png" />, with topological [[Euler characteristic|Euler characteristic]] (in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382107.png" />) equal to 27. From the Fano surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382108.png" /> one can reconstruct the cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f0382109.png" /> (see [[#References|[2]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f03821010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038210/f03821011.png" />"  ''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>
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<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
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article