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Fano scheme

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of a projective algebraic variety over a field

The algebraic scheme parametrized by the family of lines lying on the subvariety of the projective space . The Fano scheme of a projective variety can be given as a closed subscheme of the Grassmannian of lines in . In contrast to the Fano scheme of a three-dimensional cubic (see Fano surface), the Fano scheme of an arbitrary projective variety does not have to be non-singular, reduced or irreducible. Thus, the ruled surface of lines lying on the Fermat cubic consists of 40 cones cut out by the hyperplanes , , where ranges over the primitive 8th roots of unity. Each of the cones enters in with multiplicity 2 (see [1]). Thus, this Fano variety is reducible and each component of it is not reduced at a generic point.

References

[1] B. Tennison, "On the quartic threefold" Proc. London Math. Soc. , 29 (1974) pp. 714–734
How to Cite This Entry:
Fano scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fano_scheme&oldid=32660
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article