Let ${X \subset \mathbb{P}^r}$ be a subvariety (or scheme). A natural question one might ask is whether ${X}$ contains lines, or more generally, planes ${\mathbb{P}^{k} \subset X \subset \mathbb{P}^r}$ and, if so, what the family of such look like. For example, if ${Q \subset \mathbb{P}^3}$ is a nonsingular quadric surface, then ${Q}$ has two families of lines (or “rulings”) that sweep out ${Q}$; this corresponds to the expression

$\displaystyle Q \simeq \mathbb{P}^1 \times \mathbb{P}^1,$

imbedded in ${\mathbb{P}^3}$ via the Segre embedding. For a nonsingular cubic surface in ${\mathbb{P}^3}$, 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 ${X \subset \mathbb{P}^r}$ be a variety. There is a natural subset of the Grassmannian ${\mathbb{G}(k, r)}$ of ${k}$-planes in ${\mathbb{P}^r}$ (i.e., ${k+1}$-dimensional subspaces of ${\mathbb{C}^{r+1}}$) that parametrizes those ${k}$-planes which happen to be contained in ${X}$. 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 ${\mathbb{G}(k, r)}$ itself has a moduli interpretation: it is the Hilbert scheme of ${k}$-dimensional subschemes of ${\mathbb{P}^r}$ consisting of subschemes whose Hilbert polynomial is given by ${n \mapsto \binom{n+k}{k}}$; 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 ${F_k X}$ of ${X}$ is the subscheme of ${\mathrm{Hilb}_X}$ parametrizing subschemes ${L \subset X}$ whose Hilbert polynomial is ${n \mapsto \binom{n+k}{k}}$. (more…)