Fano scheme

of a projective algebraic variety $X$ over a field $k$

The algebraic scheme parametrized by the family of lines lying on the subvariety $X$ of the projective space $P^n$. The Fano scheme $F(X)$ of a projective variety $X$ can be given as a closed subscheme of the Grassmannian $G(2,n+1)$ of lines in $P^n$. 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 $R$ of lines lying on the Fermat cubic $\sum_{i=0}^4x_i^4=0$ consists of 40 cones cut out by the hyperplanes $x_i=\zeta x_j$, $i\neq j$, where $\zeta$ ranges over the primitive 8th roots of unity. Each of the cones enters in $R$ 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