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…)

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