Let be a subvariety (or scheme). A natural question one might ask is whether contains lines, or more generally, planes and, if so, what the family of such look like. For example, if is a nonsingular quadric surface, then has two families of lines (or “rulings”) that sweep out ; this corresponds to the expression
imbedded in via the Segre embedding. For a nonsingular cubic surface in , it is a famous and classical result of Cayley and Salmon that there are twenty-seven lines. In this post and the next, I’d like to discuss this result and more generally the question of planes in hypersurfaces.
Most of this material is classical; I recently learned it from Eisenbud-Harris’s (very enjoyable) draft textbook 3264 and All That.
1. Varieties of planes
Let be a variety. There is a natural subset of the Grassmannian of -planes in (i.e., -dimensional subspaces of ) that parametrizes those -planes which happen to be contained in . This is called the Fano variety.
However, the Fano variety has a natural (and possibly nonreduced) subscheme structure that arises from its interpretation as the solution to a moduli problem, so perhaps it should be called a Fano scheme. The first observation is that the itself has a moduli interpretation: it is the Hilbert scheme of -dimensional subschemes of consisting of subschemes whose Hilbert polynomial is given by ; such a subscheme is necessarily a linear subspace.
This suggests that we should think of the Fano scheme as a Hilbert scheme.
Definition 1 The Fano scheme of is the subscheme of parametrizing subschemes whose Hilbert polynomial is . (more…)