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…)

Let ${C \subset \mathbb{P}^r}$ be a (smooth) curve in projective space of some degree ${d}$. We will assume that ${C}$ is nondegenerate: that is, that ${C}$ is not contained in a hyperplane. In other words, one has an abstract algebraic curve ${C}$, and the data of a line bundle ${\mathcal{L} = \mathcal{O}_C(1)}$ of degree ${d}$ on ${C}$, and a subspace ${V \subset H^0( \mathcal{L})}$ of dimension ${r+1}$ such that the sections in ${V}$ have no common zeros in ${C}$.

In this post, I’d like to discuss a useful condition on such an imbedding, and some of the geometry that it leads to. Most of this material is, once again, from ACGH’s book Geometry of algebraic curves.

1. Projective normality

In general, there are two natural commutative graded rings one can associate to this data. First, one has the homogeneous coordinate ring of ${C}$ inside ${\mathbb{P}^r}$. The curve ${C \subset \mathbb{P}^r}$ is defined by a homogeneous ideal ${I \subset k[x_0, \dots, x_r]}$ (consisting of all homogeneous polynomials whose vanishing locus contains ${C}$). The homogeneous coordinate ring of ${C}$ is defined via $\displaystyle S = k[x_0, \dots, x_r]/I;$

it is an integral domain. Equivalently, it can be defined as the image of ${k[x_0, \dots, x_r] = \bigoplus_{n = 0}^\infty H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n))}$ in ${\bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n))}$. But that in turn suggests another natural ring associated to ${C}$, which only depends on the line bundle ${\mathcal{L}}$ and not the projective imbedding: that is the ring $\displaystyle \widetilde{S} = \bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n)),$

where the multiplication comes from the natural maps ${H^0(\mathcal{M}) \otimes H^0(\mathcal{N}) \rightarrow H^0( \mathcal{M} \otimes \mathcal{N})}$ for line bundles ${\mathcal{M}, \mathcal{N}}$ on ${C}$. One has a natural map $\displaystyle S \hookrightarrow \widetilde{S},$

which is injective by construction. Moreover, since higher cohomology always vanishes after enough twisting, the map ${S \rightarrow \widetilde{S}}$ is surjective in all large dimensions.

Definition 1 The curve ${C \subset \mathbb{P}^r}$ is said to be projectively normal if the map ${S \hookrightarrow \widetilde{S}}$ is an isomorphism. (more…)