Guide to constructing the Hilbert Schemes

22/02/2026

deformation theory

moduli theory

algebraic geometry

working mathematician

Some time ago I finally decided to sit down and work through the details of constructing the Hilbert schemes of projective schemes. I figured it would be nice to try to write down some kind of a "road map" for the construction and that is what I will attempt here.

To be clear, the point of this post is not to go into the details of the construction, as there are enough excellent expositions of it written by actual experts. I myself followed the brilliant article titled Elementary introduction to representable functors and Hilbert schemes by Stein Strømme. János Kollár's book Rational Curves on Algebraic Varieties also has a very well written account of the construction. Jarod Alper's notes on moduli theory are highly recommended, too.

The point in this post is instead to try gaining the big picture view by naming the main ingredients that go into the construction and giving a quick overview of how the different pieces fit together. I will also describe the Hilbert schemes of hypersurfaces, which I hope will help with gaining intuition for the steps in the general construction.

What is the Hilbert scheme?

Informally, the Hilbert scheme $\mathrm{Hilb}_X$ of a scheme $X$ is a parameter space, whose (geometric) points correspond to closed subschemes $Z\subseteq X$ . This is expressed formally by describing the functor of points of $\mathrm{Hilb}_X$ . But in order to avoid a parameter space of infinite type, we first break the Hilbert scheme into pieces $\mathrm{Hilb}_X^P$ , where $P\in\mathbb{Q}[t]$ is an integer valued polynomial that dictates what the Hilbert polynomial of a subscheme $[Z]\in\mathrm{Hilb}_X^P$ is. We also pass to the relative setting, where we consider $X$ as a scheme over some base scheme $S$ . Thus, we define the functor $\begin{gather*} \mathrm{Hilb}_{X/S}^P\colon S\textbf{-Sch}\to\mathbf{Set}\\ \mathrm{Hilb}_{X/S}^P(T) =\left\{ \substack{\text{flat family }f\colon Z\to T\\ \text{embedded in $X$}} \,\Bigg|\, \substack{\text{the geometric fibres of $f$}\\ \text{have Hilbert poly $p$}} \right\}. \end{gather*}$ We say a flat morphism $f\colon Z\to T$ is a flat family embedded in $X$ , when there is a closed embedding $Z\hookrightarrow T\times X$ such that the below triangle commutes. Finally, we are ready to state the existence claim.

Theorem. For any projective $S$ -scheme $X$ and integer-valued polynomial $P$ , the Hilbert functor $\mathrm{Hilb}_{X/S}^P$ is representable by a projective scheme of finite type over S.

The representability of functors is rather abstract, but the general theory tells us how to make it more concrete: representability is equivalent to the existence of universal elements in the sense of category theory. In the present context it means the following: Say $\mathrm{Hilb}_{X/ S}^P$ is represented by an $S$ -scheme $H$ . Then, there is a universal family $\mathcal{Z}\to H$ , which is an element of $\mathrm{Hilb}_{X/S}^P(H)$ , such that for any element $Z\to T$ of $\mathrm{Hilb}_{X/ S}^P(T)$ , there exists a unique pullback square

Hypersurfaces

Before diving into the general construction, we can look at the case of hypersurfaces as a warm-up. The busy reader can skim this section and move quickly to the following sections. Now, fix $X=\mathbb{P}^n$ and $S=\mathrm{Spec}\,\mathbb{C}$ , say. It follows from basic calculations that the Hilbert polynomial of a hypersurface in $\mathbb{P}^n$ is $P(t)=\binom{t+n}n-\binom{t+n-d}n,$ where $d$ is the degree of the hypersurface. Conversely, any closed subscheme of $\mathbb{P}^n$ with this polynomial is a hypersurface of degree $d$ . Therefore, the closed points of $\mathrm{Hilb}_{X/S}^{P}$ are precisely the degree $d$ hypersurfaces.

Now, there is a very explicit way to define a parameter space for degree $d$ hypersurfaces in $\mathbb{P}^n$ , which we can prove represents the Hilbert functor: We take the points to be the coefficients of the homogeneous degree $d$ polynomials in $n+1$ variables. Since two polynomials define the same hypersurface, if and only if they differ by a constant multiple, we take the parameter space to be the projective space of dimension $N-1$ , where $N=\binom{n+d}{n}$ is the number of monomials of degree $d$ in $n+1$ variables. It is easy to also define a candidate for the universal family, namely $\mathcal{Z}:=\left\{(F,P)\in\mathbb{P}^{N-1}\times\mathbb{P}^n \,\big|\, F(P)=0\right\}.$

We can do the construction a bit more concisely, which will help with generalisation. Consider any degree $d$ hypersurface $Y\subset\mathbb{P}^n$ . Now, take its closed subscheme exact sequence, twist by $d$ , and take global sections to obtain a short exact sequence $0\to \mathrm{H}^0(\mathbb{P}^n,\mathscr{I}_{Y}(d)) \to \mathrm{H}^0(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(d)) \to \mathrm{H}^0(\mathbb{P}^n,\mathscr{O}_{Y}(d))$ It is easy to see that the global sections of $\mathscr{I}_Y(d)$ define a 1-dimensional vector subspace of $\mathrm{H}^0(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(d))$ . Conversely, any 1-dimensional vector subspace generates an ideal sheaf of a degree $d$ hypersurface. Therefore, we set $\mathrm{H}:=\mathbb{P}(\mathrm{H}^0(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(d))).$ Note that this indeed coincides with the projective space $\mathbb{P}^{N-1}$ .

To show that $\mathrm{H}$ represents $\mathrm{Hilb}_{X/ S}^P$ , we show that $\mathcal{Z}\to \mathrm{H}$ is a universal family. Thus, fix a flat family $f\colon Z\to T$ of hypersurfaces embedded in $\mathbb{P}^n$ . The aim is to find a morphism $\varphi\colon T\to \mathrm{H}$ such that $Z$ is the pullback of $\mathcal{Z}$ . It is easy to see what this morphism should do to closed points. Given a closed point $t\in T$ , we know that $\varphi(t)$ should be the projective point with coordinates given by the coefficient of some homogeneous polynomial cutting out the fibre $Z_t$ in the family. Now, we need to show that one can choose these coefficients so that they "vary regularly" as we change the point $t$ .

Since defining a morphism can be done affine-locally, we can assume $T=\mathrm{Spec}\,A$ for some $\mathbb{C}$ -algebra $A$ . Now, since the family $f\colon Z\to T$ is flat, the scheme $Z$ is a hypersurface in $\mathbb{P}^n\times T$ cut out by some polynomial $\sum_{\substack{I\subset \mathbb{N}^{n+1}\\ \lvert I\rvert=d}}a_{I}X^I\in A[X_0,\ldots,X_{n}],$ i.e. a polynomial that is homogeneous of degree $d$ in the coordinates on $\mathbb{P}^n$ with coefficients in $A$ . The coefficients $a_I$ can be thought of as regular functions on $T$ , and if we "evaluate them" at closed points, they give the coefficients of the polynomials cutting out the corresponding fibres in the family. Thus, they define a morphism $T\to \mathrm{H}$ , whose values at the closed points are exactly what we wanted.

To use fancier language again, we can consider taking the pushforward of the twisting of the closed subscheme exact sequence for $Z\subset\mathbb{P}^n\times T$ to obtain $0\to \mathrm{pr}_{\ast}\mathscr{I}_{Z}(d)\to \mathrm{pr}_{\ast}\mathscr{O}_{\mathbb{P}^n\times T}(d)\to \mathrm{pr}_{\ast}\mathscr{O}_{Z}(d)\to0,$ where $\mathrm{pr}$ is the projection $\mathbb{P}^n\times T\to T$ . While taking pushforwards is not exact in general, it is in this case by cohomology and base change, which is discussed below and which you will meet many times in the course of the proof. Cohomology and base change also tells us that the first map is an inclusion of a line bundle. Then, one obtains from this a morphism $\varphi\colon T\to \mathrm{H}$ after noting that the projective space $\mathbb{P}(\mathrm{H}^0(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(d)))$ represents the functor of such line subbundles. It remains to check that $f\colon Z\to T$ is the pullback of the universal family $\mathcal{Z}\to \mathrm{H}$ along $\varphi$ . This follows quite directly from two applications of cohomology and base change and I leave it as an exercise.

Ingredients

The proof of the representability uses three ingredients that have names outside of the construction:

Cohomology and base change,
Castelnuovo-Mumford regularity, and
Flattening stratifications.

You can choose to either study them independently before tackling the proof, or taking them as black boxes at first. In any case, I will attempt to give a short description of each ingredient here.

Cohomology and base change

As in the construction in the hypersurface case, we will be considering pushforwards of coherent sheaves on families $\pi\colon Z\to T$ . Then, there are two basic questions we need to answer for a fixed coherent sheaf $\mathscr{F}$ on $Z$ :

What does the direct image $\pi_\ast\mathscr{F}$ look like?
How does the direct image behave under pullback along some $U\to T$ ?

Cohomology and base change gives answers to these questions. To set things up, suppose we have a pullback square of schemes, where $\pi$ is proper. Then, for any coherent sheaf $\mathscr{F}$ on $Z$ , we have the so-called push-pull morphism $\Phi\colon \varphi^\ast(\mathrm{R}^i\pi_{\ast}\mathscr{F}) \to \mathrm{R}\tilde{\pi}_{\ast}\tilde{\varphi}^\ast\mathscr{F}.$ Cohomology and base change gives sufficient conditions for when the higher direct image $\mathrm{R}^i\pi_{\ast}\mathscr{F}$ is locally free and the push-pull morphism $\Phi$ is an isomorphism.

Castelnuovo-Mumford regularity

When studying coherent sheaves, we are interested in when their cohomology vanishes. Regularity quantifies this; We say a coherent sheaf $\mathscr{F}$ on $\mathbb{P}^n$ is $m$ -regular, if $\mathrm{H}^i(\mathbb{P}^n,\mathscr{F}(m-i))=0$ for all positive $i$ . A result of Castelnuovo states that $m$ -regularity implies $m'$ -regularity for all $m'\geq m$ . Thus, one should imagine tabulating all of the cohomology groups of the twists of $\mathscr{F}$ on a 2-dimensional grid. Then, one can draw a staircase of width $m-1$ and height $m-1$ such that the cohomolgy groups vanish outside it.

The crucial property of $m$ -regularity for the construction of the Hilbert scheme is the following theorem of Mumford.

Theorem. For any integer-valued polynomial $P$ , there is $m_0$ such that for all closed subschemes $Z\subseteq\mathbb{P}^n$ with Hilbert polynomial $P$ , the ideal sheaves $\mathscr{I}_Z$ are $m_0$ -regular.

Flattening strata

When I heard the name "flattening stratification" for the first time, it sounded scary for some reason. Turns out, it is a very natural concept. Suppose we have a coherent sheaf $\mathscr{F}$ on projective space $\mathbb{P}^n_S$ over some Noetherian scheme $S$ . To put it crudely, a flattening stratification of $\mathscr{F}$ is a finite "partition" $\{S_i\}$ of $S$ by locally closed subschemes such that $\mathscr{F}$ is flat over $S_i$ for all $i$ .

The stratification is constructed using the Fitting ideals of twists of $\mathrm{pr}_{\ast}\mathscr{F}$ . Here is what the Fitting ideals do: They form a sequence $0\subseteq \mathrm{Fitt}_{0}(\mathrm{pr}_{\ast}\mathscr{F}(m)) \subseteq \mathrm{Fitt}_{1}(\mathrm{pr}_{\ast}\mathscr{F}(m)) \subseteq \mathrm{Fitt}_{2}(\mathrm{pr}_{\ast}\mathscr{F}(m))\subseteq\cdots$ of ideals defined by taking minors of a presentation of $\mathrm{pr}_{\ast}\mathscr{F}(m)$ . These define a corresponding chain of subschemes of $S$ . If we define subschemes $S_{m,i}$ by taking differences of consecutive subschemes in the chain $S_{m,i}=V(\mathrm{Fitt}_{i}(\mathrm{pr}_{\ast}\mathscr{F}(m))) \setminus V(\mathrm{Fitt}_{i+1}(\mathrm{pr}_{\ast}\mathscr{F}(m))),$ then $\mathrm{pr}_{\ast}\mathscr{F}(m)$ restricts to a locally free sheaf of rank $i$ on $S_{m,i}$ . In conclusion, we can stratify $S$ by subschemes, where $\mathscr{F}$ becomes flat by startifying it into subschemes, where these twists become locally free.

The general construction

Having named the ingredients, we are ready to move onto the road map for the general construction. We will specialise right away to the case $X=\mathbb{P}^n_{\mathbb{Z}}$ and $S=\mathbb{Z}$ , since it is not hard to reduce the general statement to this special case. The general construction has similarities with the construction in the case of hypersurfaces. This time, instead of identifying subschemes of $X$ by points in some projective space, we identify them with the points in some Grassmannian. Here is the broad overview.

Construct a morphism from the Hilbert scheme into a Grassmannian
Show that the morphism is an embedding
Show that the Hilbert scheme is projective

Mapping into a Grassmannian

The Grassmannian of a coherent sheaf $\mathscr{F}$ on $\mathrm{Spec}\,\mathbb{Z}$ is the scheme, whose $T$ -points are locally free quotients of $\mathscr{F}_T$ of rank $r$ for some fixed $r$ , where $\mathscr{F}_T$ is the pullback along $T\to \mathrm{Spec}\,\mathbb{Z}$ . Thus, we need to find a suitable coherent sheaf $\mathscr{F}$ such that for each family $f\colon Z\to T$ in $\mathrm{Hilb}_{X/S}^P(T)$ , we can define an associated quotient of $\mathscr{F}_T$ . To do this, we start with taking the close subscheme exact sequence of $Z$ , twisting it, and pushing it forward to $T$ , like we did in the construction of the Hilbert scheme of hypersurfaces.

In that hypersurface case we twisted by the degree, but in the general situation we twist by the integer $m_0$ from Mumford's theorem applied to the Hilbert polynomial $P$ . By taking the derived functor long exact sequence, we get $0\longrightarrow\text{pr}_\ast\mathscr{I}_{Z}(m_0) \longrightarrow\text{pr}_\ast\mathscr{O}_{\mathbb{P}^n_T}(m_0) \longrightarrow\text{pr}_\ast\mathscr{O}_{Z}(m_0) \longrightarrow\mathrm{R}^1\text{pr}_\ast\mathscr{I}_{Z}(m_0) \longrightarrow\cdots$ where $\mathrm{pr}$ denotes the projection $\mathbb{P}^n_T\to T$ . The next step is to use $m_0$ -regularity of $\mathscr{I}_Z$ along with cohomology and base change to show that $\mathrm{R}^1\text{pr}_\ast\mathscr{I}_{Z}(m_0)=0$ and that $\text{pr}_\ast\mathscr{O}_{Z}(m_0)$ is locally free of rank $P(m_0)$ . Finally, noting $\text{pr}_\ast\mathscr{O}_{\mathbb{P}^n_T}(m_0)\cong\mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)_T$ , we have obtained a $T$ -point $\mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)_T \longrightarrow\mathrm{pr}_{\ast}\mathscr{O}_{Z}(m_{0})$ of the Grassmannian of $\mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)$ of rank $P(m_0)$ quotients. This process defines a morphism $\mathrm{Hilb}_{X /S}^P\to \mathrm{Grass}^{P(m_{0})}\left(\mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)\right)$ once we have shown that this assignment is natural.

Embedding into the Grassmannian

We show the above morphism is an embedding by constructing a subscheme of the Grassmannian using flattening strata and showing that the morphism is an isomorphism unto this subscheme. The first step is constructing a coherent sheaf on the Grassmannian, the flattening strata of which we look at later.

To start, denote the Grassmannian by $G:=\mathrm{Grass}^{P(m_{0})}\left(\mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)\right)$ and recall that it carries a universal short exact sequence $0\longrightarrow\mathscr{K}\longrightarrow \mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)_{G} \longrightarrow\mathscr{Q}\longrightarrow 0$ where $\mathscr{Q}$ is a locally free sheaf on $G$ of rank $P(m_0)$ . The goal here is to "realise" the kernel as the ideal sheaf of a closed subscheme of $\mathbb{P}^n_G$ . We do this by considering the composition $\mathrm{pr}^\ast\mathscr{K}\longrightarrow \mathrm{H}^0\left(\mathscr{O}_{\mathbb{P}^n_\mathbb{Z}}(m_0)\right)_{\mathbb{P}^n_{G}} \longrightarrow\mathscr{O}_{\mathbb{P}^n_{\mathbb{Z}}}(m_{0})_{\mathbb{P}^n_{G}}$ The image of this morphism is the ideal sheaf of some closed subscheme $\mathscr{V}\subseteq\mathbb{P}^n_{G}$ . Note that this is not necessarily flat over $G$ , so we consider the flattening strata of $\mathscr{O}_{\mathscr{V}}$ in $G$ . Let $G_P$ be the stratum, where the closed fibres of $\mathscr{V}$ have Hilbert polynomial equal to $P$ . Now, it remains to show that the morphism we constructed above maps into $G_P$ and that there is an inverse morphism from $G_P$ . In fact, since $\mathscr{V}\vert_{G_{P}}\to G_{P}$ is a flat family embedded in $\mathbb{P}^n_{G_P}$ , there is a unique morphism $G_{P}\to \mathrm{Hilb}_{X /S}^P$ by the definition of the Hilbert scheme, and this is precisely the inverse morphism.

Hilbert scheme is projective

Showing projectivity is the easiest step. Since the Grassmannian is projective, it suffices to show that the Hilbert scheme is complete. But this can be checked directly from the definition of the Hilbert scheme by using the valuative criterion of properness. This finishes the proof.