View source for Fano scheme

''of a projective algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382001.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382002.png" />''

The algebraic [[Scheme|scheme]] parametrized by the family of lines lying on the subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382003.png" /> of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382004.png" />. The Fano scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382005.png" /> of a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382006.png" /> can be given as a closed subscheme of the Grassmannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382007.png" /> of lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382008.png" />. In contrast to the Fano scheme of a three-dimensional cubic (see [[Fano surface|Fano surface]]), the Fano scheme of an arbitrary projective variety does not have to be non-singular, reduced or irreducible. Thus, the [[Ruled surface|ruled surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f0382009.png" /> of lines lying on the Fermat cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f03820010.png" /> consists of 40 cones cut out by the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f03820011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f03820012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f03820013.png" /> ranges over the primitive 8th roots of unity. Each of the cones enters in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038200/f03820014.png" /> with multiplicity 2 (see [[#References|[1]]]). Thus, this Fano variety is reducible and each component of it is not reduced at a generic point.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Tennison,   "On the quartic threefold"  ''Proc. London Math. Soc.'' , '''29'''  (1974)  pp. 714–734</TD></TR></table>