The space $\mathbb{P}^n$ serves as a parameter space for all lines through the origin in $\mathbb{A}^{n+1}$. More generally, the Grassmannian $Gr(k+1, n+1)$ is a parameter space for closed $k$-dimensional degree 1 subschemes of $\mathbb{P}^n$. A natural question to ask is: are there schemes which serve as parameter spaces for higher degree subschemes of a given projective scheme?

We provide some background and then introduce the notion of a Hilbert scheme, with some examples. Throughout we work over $\mathbb{C}$.

Hilbert polynomials

Let $X \subset \mathbb{P}^n$ be a projective scheme. For $d \geq 0$, we denote by $\mathcal O_X(d)$ the pullback of the bundle $\mathcal O_{\mathbb{P}^n}(d)$. The Hilbert function of $X$ is defined to be

\[\begin{aligned} h_X(d) := h^0(X, \mathcal O_X(d)), \end{aligned}\]

the dimension of the space of global sections.

Example 1. The global sections of $\mathcal O_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$ are precisely the homogeneous degree $d$ polynomials in $n+1$ variables. Hence we see that

\[\begin{aligned} h_{\mathbb{P}^n}(d) = \binom{n+d}{d} = \frac{(d+1)\ldots(d+n)}{n!}, \end{aligned}\]

which is a degree $n$ polynomial (in $d$) with leading coefficient $1/n!$.

Example 2. If $X$ is a point in $\mathbb{P}^n$, then the restriction of any line bundle to $X$ is just a one-dimensional vector space, so

\[\begin{aligned} h_{point}(d) = 1 \qquad \text{for all } d. \end{aligned}\]

Clearly this does not depend on the point chosen. Thus all points on $\mathbb{P}^n$ have the same Hilbert function. If $X = {p_1, \ldots, p_r}$ is a set of $r$ points in $\mathbb{P}^n$, then $h_X(d) \leq r$ for all $d$ (because sections of $\mathcal O_X(d)$ can only take values in the $r$-dimensional space $\bigoplus_i \mathcal O_{p_i}(d)$). Moreover, for large enough $d$, we can find a homogeneous degree $d$ polynomial which vanishes at all points of $X$ except one, so it follows that, for large enough $d$, $h_X(d) = r$.

Hilbert showed (in the more general setting of finitely generated graded commutative algebras) the following:

Theorem 1. (18.6.1, The Rising Sea)
For large enough $d$, $h_X(d)$ is a polynomial with degree equal to dim $X$.

The polynomial mentioned in the theorem is called the Hilbert polynomial of $X$, denoted by $p_X(d)$.

Example 3. It follows from Example 2 that for $X = {p_1, \ldots, p_r}$, $p_X(d) = r$. Thus the Hilbert polynomial of a 0-dimensional subscheme is the cardinality of its underlying set of points. Furthermore, the theorem implies that the only schemes with Hilbert polynomial 1 are points. We conclude that the collection of subschemes of $\mathbb{P}^n$ with Hilbert polynomial 1 is a scheme: $\mathbb{P}^n$ itself!

Example 4. Suppose $X$ is a hypersurface in $\mathbb{P}^n$ given by the vanishing of a homogeneous polynomial $f$ of degree $r$. Then we have the short exact sequence of sheaves on $\mathbb{P}^n$

\[\begin{aligned} 0 \to \mathcal O_{\mathbb{P}^n}(d-r) \to \mathcal O_{\mathbb{P}^n}(d)\to f_*\mathcal O_X(d)\to 0 \end{aligned}\]

where the inclusion is given by multiplication by $f$ and the surjection follows from the fact that $X$ is a closed subscheme. When $d > r$, we know that the higher cohomology groups of $\mathcal O_{\mathbb{P}^n}(d-r)$ vanish, so we get an exact sequence

\[\begin{aligned} 0 \to H^0(\mathcal O_{\mathbb{P}^n}(d-r)) \to H^0(\mathcal O_{\mathbb{P}^n}(d)) \to H^0(f_*\mathcal O_X(d)) \to 0 \end{aligned}\]

from which we see that $p_X(d) = h_{\mathbb{P}^n}(d) - h_{\mathbb{P}^n}(d-r)$.

Example 5. Suppose $C \subset \mathbb{P}^2$ is a degree $r$ plane curve. Using the previous example and Example 1, we see that the Hilbert polynomial of $C$ is given by

\[h_C(d) = \dfrac{(d+1)(d+2)}{2} - \dfrac{(d-r+1)(d-r+2)}{2} = rd + \dfrac{3r-r^2}{2}.\]

In particular, the coefficient of the leading term is the degree of the curve.
For $r = 1$ (lines), $h_C(d) = d+1$.
For $r = 2$ (conics), $h_C(d) = 2d+1$.

As seen in this last example, the degree of a curve can be read off from its Hilbert polynomial. This turns out to be true even for arbitrary projective schemes, and consequently, instead of asking for a parameter space for subschemes of a given degree, we can ask for a parameter space for subschemes having a given Hilbert polynomial.

Flat families

Our aim is to construct a “universal family” $U \to \mathscr{H}_P$ with the following properties:

  1. $\mathscr{H}_P$ parameterizes all closed subschemes of $\mathbb{P}^n$ with Hilbert polynomial $P$.

  2. Given any projective scheme $X$ and a map $E \to X$ whose fibers are closed subschemes of $\mathbb{P}^n\times X$ with Hilbert polynomial $P$, there is a unique morphism $X \to \mathscr{H}_P$ such that $E$ is isomorphic to the pullback of $U$ under this morphism.

For arbitrary morphisms $E \to X$, the degrees, dimensions and Hilbert polynomials can jump from fiber to fiber. In order to achieve the aims above, we need to restrict the set of morphisms (and therefore also subschemes of $X$) we consider.

A family of schemes is defined to be a morphism $\pi:E\to X$ of schemes. Such a family is said to be a flat family if the morphism is flat. The crucial results are then the following

Theorem 2. (Proposition III-56, The Geometry of Schemes)
A family of closed subschemes of a projective space over a reduced base is flat if and only if all fibers have the same Hilbert polynomial.

Theorem 3. (Corollary 24.7.2, The Rising Sea) If $\pi:X \to Y$ is a projective flat morphism of locally Noetherian schemes and $Y$ is connected, then all fibers have the same degree, dimension and Hilbert polynomial.

Another crucial property of flatness that we will use is that it is preserved under base change / pullbacks.

Hilbert functors

Given a scheme $X$, its functor of points is defined to be the functor

\[\begin{aligned} \phi_X:\texttt{Schemes}^{op} &\to \texttt{Sets}\\ Y &\mapsto \text{Mor}(Y, X)\\ (f:Y \to Z) &\mapsto (g \mapsto g\circ f). \end{aligned}\]

If we denote by $\mathcal{C}$ the category of contravariant functors from $\texttt{Schemes}$ to $\texttt{Sets}$ (whose morphisms are natural transformations), we get a functor

\[\begin{aligned} \Phi:\texttt{Schemes} &\to \mathcal{C}\\ X \mapsto \phi_X. \end{aligned}\]

It is a consequence of the Yoneda lemma ([@schemes] Lemma VI-1) that $\Phi$ is fully faithful. We call a functor $\phi \in \mathcal{C}$ representable if there exists a scheme $X$ such that $\phi = \phi_X$. Again by the Yoneda lemma, such an $X$ (if it exists) must be unique.

Now suppose $P$ is a polynomial. The Hilbert functor $h_P$ is the functor that associates to a scheme $X$ the set of subschemes $Y \subset \mathbb{P}^n \times X$ flat over $X$ whose fibers over points of $X$ have Hilbert polynomial $P$. This is a contravariant functor because, if $Y \subset \mathbb{P}^n\times X$ is a subscheme flat over $X$ and $Z \to X$ is a morphism, then $Z\times_{X}Y \subset \mathbb{P}^n\times Z$ is flat over $Z$, and the fibers have the same Hilbert polynomial $P$. Thus $h_P$ is in $\mathcal{C}$. Is it representable? Grothendieck showed that it is:

Theorem 4. (Grothendieck)
There exists a scheme $\mathscr{H}_P$ whose functor of points is the functor $h_P$.

The scheme $\mathscr{H}_P$ is called the Hilbert scheme associated to the polynomial $P$. We will not prove this theorem, but will discuss a few examples.

Example 6. This is just a restatement of Example 3. For the constant polynomial $P = 1$, the Hilbert scheme $\mathscr{H}_P$ is $\mathbb{P}^n$.

Example 7. We have already seen that the Hilbert polynomial for lines in $\mathbb{P}^2$ is $d+1$. Since we know that the degree of the Hilbert polynomial is the dimension of the subscheme, and the coefficient of the leading term (for curves) the degree, it follows that any subscheme of $\mathbb{P}^2$ with Hilbert polynomial $P(d) = d+1$ must be a line. Therefore the Hilbert scheme $\mathscr{H}_P$ consists of the collection of lines in $\mathbb{P}^2$, which is the Grassmannian $Gr(2,3)$, and therefore isomorphic to $\mathbb{P}^2$ itself.

Example 8. The Hilbert polynomial for conics in $\mathbb{P}^2$ is $P(d) = 2d+1$. Again, as above, we know that any closed subscheme of $\mathbb{P}^2$ with this Hilbert polynomial must be a conic. A homogeneous degree 2 polynomial in $\mathbb{P}^2$ looks like $a_1x^2 + a_2y^2 + a_3z^2 + a_4xy + a_5yz + a_6zx$, where not all the $a_i$’s are simultaneously zero. So such polynomials are parameterized by $\mathbb{C}^6\setminus { 0}$. Moreover, any two such polynomials give the same curve if one is obtained by a rescaling of the other. We conclude that the Hilbert scheme $\mathscr{H}_p$ is isomorphic to $\mathbb{P}^5$.