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 ${n \mapsto \binom{n+k}{k}}$.

In particular, ${F_k X}$ is a union of components of the Hilbert scheme ${\mathrm{Hilb}_X}$. The advantage of this picture is that one can apply deformation theory to understand the local structure of ${F_k X}$. In general, the tangent space to ${\mathrm{Hilb}_X}$ at a point parametrizing a subscheme ${Y \subset X}$ is given by

$\displaystyle H^0(Y, N_{X/Y}) = H^0( Y, \hom(\mathcal{I}_Y/\mathcal{I}_Y^2, \mathcal{O}_Y)),$

corresponding to the intuition that a small deformation of a subscheme ${Y \subset X}$ should be given by a family of normal vector fields on ${Y}$.

This means that we can understand the tangent space to the Fano scheme at a given subspace ${L \subset X}$; it’s

$\displaystyle T_L F_k(X) = H^0(L , \hom(\mathcal{I}_L/\mathcal{I}_L^2, \mathcal{O}_L)),$

where ${\mathcal{I}_L \subset \mathcal{O}_X}$ is the ideal cutting out ${L}$.

We can also present the Fano scheme explicitly as a subscheme of the Grassmannian. Suppose ${X}$ is cut out by sections

$\displaystyle \sigma_i \in H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(d_i));$

that is, the ${\sigma_i}$ are homogeneous polynomials whose vanishing cuts out ${X}$. Then ${F_k X}$ consists of ${k}$-planes on which these polynomials restrict to zero. More precisely, on the line bundle ${\mathbb{G}(k, r)}$, there is a tautological${k+1}$-dimensional vector bundle ${\mathcal{V}}$, which assigns to a ${k}$-plane ${L \subset \mathbb{P}^r}$ the global sections ${H^0(L, \mathcal{O}_L(1))}$; equivalently, if

$\displaystyle U \subset \mathbb{G}(k, r) \times \mathbb{P}^r,$

is the universal ${k}$-plane (the “incidence correspondence”), then the tautological bundle ${\mathcal{V}}$ can be described as

$\displaystyle \mathcal{V} = \pi_{1*} \mathcal{O}_U(1),$

which defines the vector bundle on ${\mathbb{G}(k, r)}$ described informally above. Now each ${\sigma_i}$ defines a section of ${\mathrm{Sym}^{d_i} \mathcal{V} = \pi_{1*} \mathcal{O}_U(d_i)}$ on ${\mathbb{G}(k, r)}$, and the Fano scheme is the subscheme of ${\mathbb{G}(k, r)}$ cut out by the vanishing of the ${\sigma_i}$. In favorable situations, this means that we can use the theory of Chern classes to understand the cycle in ${\mathbb{G}(k, r)}$ represented by ${F_k X}$.

2. Some dimension counting

In the case ${X \subset \mathbb{P}^r}$ is a hypersurface of degree ${d}$, the Fano scheme ${F_k X}$ is the zero locus of a single section of the vector bundle ${\mathrm{Sym}^d V }$ on ${\mathbb{G}(k, r)}$ (of dimension ${\binom{k + d_i}{k}}$), which means that we should expect the following:

• ${F_k X}$ is a subscheme of ${\mathbb{G}(k, r)}$ of codimension ${\binom{k+ d}{k}}$.
• The class of ${F_k X}$ in the Chow ring (or cohomology ring) of ${\mathbb{G}(k, r)}$ is given by the top Chern class of the vector bundle ${\mathrm{Sym}^d V}$.

While this need not be true (the section of the vector bundle ${\mathrm{Sym}^d V}$ need not be in “general position”), we can conclude the second point, with appropriate multiplicities, if the first statement holds. Using the (known) structure of the cohomology of the Grassmannian, this gives a very efficient way of solving enumerative questions related to ${F_k X}$.

For instance, if ${k = 1}$, so ${\mathbb{G}(k, r)}$ has dimension ${\dim G(2, r+1) = 2(r-1)}$, we find that the expected dimension of the Fano scheme ${F_1 X}$ of lines on ${X}$ is given by

$\displaystyle \dim X \stackrel{?}{=} 2r - d - 3.$

If ${X}$ is smooth and ${d \leq r}$, a conjecture of Debarre and de Jong states that the real dimension is always the above “expected dimension.”

If ${X}$ is general, however, the question simplifies and we can directly say something by considering the universal example again. Instead of fixing one ${X}$, the strategy is to consider all of them at once. Consider the Hilbert flag scheme ${H(k, d)}$ of pairs

$\displaystyle L \subset X \subset \mathbb{P}^r,$

where ${L}$ is a ${k}$-plane and ${X}$ is a hypersurface of degree ${d}$. By definition, the scheme ${H(k, d)}$ fibers both over the Grassmannian ${\mathbb{G}(k, r)}$ and the Hilbert scheme of degree ${d}$ hypersurfaces in ${\mathbb{P}^r}$ (which is simply a ${\mathbb{P}^{\binom{r + d}{r}-1}}$).

By definition, the fibers of ${H(k, d)}$ over the point corresponding to a hypersurface ${X \subset \mathbb{P}^r}$ is the scheme ${F_k X}$ that we are interested in. The clever trick here is to consider the fibers in the other direction, which are much simpler. The fiber of ${H(k, d)}$ over the point in ${\mathbb{G}(k, r)}$ parametrizing a ${k}$-plane ${L \subset \mathbb{P}^r}$ is the subscheme of the Hilbert scheme ${\mathbb{P}^{\binom{r+d}{r}-1}}$ consisting of hypersurfaces containing ${L}$. In other words, it is the projectivization of the kernel of the surjective map of vector bundles

$\displaystyle H^0( \mathbb{P}^r, \mathcal{O}(d)) \rightarrow \mathrm{Sym}^d \mathcal{V} ,$

where the first vector bundle is the trivial one corresponding to the vector space of degree ${d}$ polynomials.

This means that ${H(k, d)}$ is actually a projective bundle over the Grassmannian ${\mathbb{G}(k, r)}$; in particular, it is actually a smooth variety of dimension given by

$\displaystyle \dim H(k, d) = \dim \mathbb{G}(k, r) + \dim H^0(\mathbb{P}^r, \mathcal{O}(d)) - \dim H^0( \mathbb{P}^k, \mathcal{O}(d)) - 1.$

For instance, when ${k = 1}$, this works out to be

$\displaystyle \dim H(1, d) = 2(r-1) + \binom{d + r}{r} - (d+1) - 1,$

and this is mapping to a ${\mathbb{P}^{\binom{r + d}{r}-1}}$. It follows that:

Proposition 2 If the expected dimension ${2r - d - 3 < 0}$, then the general degree ${d}$ hypersurface in ${\mathbb{P}^r}$ contains no lines.

My impression is that the presence of lines (and more generally, of rational curves of higher degree) on a smooth variety ${X}$ is considered a type of “positivity” constraint on ${X}$: for instance, a spectacular theorem of Mori states that the failure of nefness of the canonical bundle (a weak form of positivity) ${X}$ implies that ${X}$ contains rational curves. Conversely, a theorem of Clemens states that general hypersurfaces of high degree ${d \geq 2r-1}$ (which are “negative” in that the canonical bundle is ample) contain no rational curves at all. In higher degree, the variety gets more and more negative, more and more complicated, and should contain fewer comparatively simple objects such as lines.

Nonetheless, it is not true that negativity in this sense corresponds precisely to the differential-geometric notion of negative curvature. For instance, a smooth hypersurface in ${\mathbb{P}^r, r \geq 3}$ has trivial fundamental group by the Lefschetz hyperplane theorem, implying (by the Cartan-Hadamard theorem) that it does not have a metric of negative curvature.