# 6 Hilbert schemes

[FGA 3.IV]
Grothendieck, A.
“Technique de descente et théorèmes d’existence en géométrie algébrique, IV: Les schémas de Hilbert”.
*Séminaire Bourbaki* **13** (1960–61), Talk no. 221.

## Introduction

The techniques described in FGA 3.I and FGA 3.II were, for the most part, independent of any projective hypotheses on the schemes in question. Unfortunately, they have not as of yet allowed us to solve the existence problems posed in FGA 3.II. In the current article, and the following, we will solve these problems by imposing projective hypotheses. The techniques used are typically projective, and practically make no use of any results from FGA 3.I and FGA 3.II. Here we will construct “Hilbert schemes”, which are meant to replace the use of Chow coordinates, as was mentioned in FGA 3.II, C.2. In the next article, the theory of passing to the quotient in schemes, developed in FGA 3.III, combined with the theory of Hilbert schemes, will allow us, for example, to construct Picard schemes (defined in FGA 3.II, C.3) under rather general conditions.

In summary, we can say that we now have a more or less satisfying technique of projective constructions, apart from the fact that we are still missing^{11} a theory of passing to the quotient by groups such as the projective group, acting “without fixed points” (cf. FGA 3.III, §8).
The situation even seems slightly better in analytic geometry (if we restrict to the study of “projective” analytic spaces over a given analytic space), since, for analytic spaces, the difficulty of passing to the quotient by a group that acts nicely disappears.
Either way, in algebraic geometry, as well as in analytic geometry, it remains to develop a construction technique that works without any projective hypotheses.

## 6.1 Bounded sets of sheaves: invariance properties

Let *equivalent* if there exists an extension *of finite type*.
We can thus, in the definition of classes of coherent sheaves, restrict ourselves to *algebraically closed* extensions of

Let *same*

Our aim in this section is to give a definition of certain sets of classes of sheaves, said to be *bounded*, and to show that the most standard operations

Let *Noetherian*.
For all

Let *bounded* if there exists a prescheme

This construction, by definition, sends *algebraic family* of coherent sheaves, parametrised by some

A finite union of bounded sets is bounded (take the prescheme given by the sum of the parametrising preschemes

If *flat* with respect to

The two claims that follow can be proven by *essentially* the same flatness technique;
for the primary decomposition on the fibres of a morphisms of finite type, see, in particular, [10, IV].

Let

- the family of kernels, cokernels, and images of homomorphisms
{\mathscr{F}}\to{\mathscr{F}}' (where the class of{\mathscr{F}} is inE and the class of{\mathscr{F}}' is inE' ) is bounded; - the family of extensions
{\mathscr{F}}'' of{\mathscr{F}} by{\mathscr{F}}' (where the class of{\mathscr{F}} is inE and the class of{\mathscr{F}}' is inE' ) is bounded.

*Proof*. After potentially applying a suitable base change, we can suppose that

Let

Here,

## 6.2 Bounded families and the Hilbert polynomial

In the following, we assume that

To each coherent sheaf *Hilbert polynomial* of

Now, if *flat* with respect to

Recall also that, if

From these two remarks, we reduce the “necessary” part of the following theorem:

Let

- There exists a coherent sheaf
{\mathscr{L}} onX (which we can suppose to be of the form{\mathscr{O}}_X(-n)^N ) such thatE is contained in the family of classes of coherent sheaves given by quotients of sheaves of the form{\mathscr{L}}_K ; - The Hilbert polynomials
P_{\mathscr{F}} of the sheaves{\mathscr{F}} whose class is inE are elements of a single*finite*set of polynomials.

It remains to prove the “sufficient” part, which will be a particular case of a more precise result.
For every coherent module

Under the conditions of Theorem 2.1, let

- (b
_{s}) The Poincaré polynomialsP_{\mathscr{F}} of the sheaves{\mathscr{F}} whose class is inE have coefficients*in degrees*that are bounded.\leqslant(s-1)

Under these conditions, the sheaves

Thus:

Suppose that the sheaves

The end of this section is dedicated to a sketch proof of Theorem 2.2. The key lemmas are the two following lemmas, of which the first is well known (and summarises the useful mathematical content of Chow coordinates):

Consider the structure sheaves of the subschemes

Here, the degree

Let

*Proof*. By replacing *finite* morphism

Consider the exact sequence

Since

The combination of the two lemmas above allows us to prove the following:

Suppose, under the preliminary conditions of (2.1), that, for all

*Proof*. We can suppose that the base field

*Proof*. We can now prove (2.2) by induction on the upper bound _{s}) as the _{s}), and so, by the induction hypothesis, the

## 6.3 Hilbert schemes: definition, existence theorem

Let

Now, if *contravariant functor in S'* (where

*projective*over a

*Noetherian*

*coherent*; for simplicity, we will limit ourselves to considering those

*locally Noetherian*over

Under these conditions, the contravariant functor *representable* by an

We will obtain such a decomposition in the following way.
Let *For \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S} to be representable, it is necessary and sufficient that the \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}^P be representable, and then the S-prescheme \underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}} which represents it is isomorphic to the prescheme given by the sum of the \underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}}^P that represent the functors \mathscr{Q}\kern -.5pt out_{{\mathscr{F}}/X/S}^P*.
With this, Theorem 3.1 will be a consequence of the following theorem:

With the above notation, the functor

The rest of this section is dedicated to the proof of Theorem 3.2.

Let

\operatorname{R}^i f'_*({\mathscr{G}}(n))=0 fori>0 andn\geqslant\nu ;\operatorname{R}^i f'_*({\mathscr{H}}(n))=0 fori>0 andn\geqslant\nu ;f'_*({\mathscr{H}}(\nu+k))={\mathcal{S}}'_k f'_*({\mathscr{H}}(\nu)) fork\geqslant 0 .

For this last condition, we suppose that *loc. cit.*).
In other words, * A_\nu(S') is a contravariant functor in S', and in a precise sense a subfunctor of A(S')=\mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}^P(S')*.
For varying

*For the functor*A to be representable, it is necessary and sufficient that the functors A_\nu be representable, and then the S -prescheme Q that represents A is the increasing union of the opens Q_\nu that represent the A_\nu .

Let

It follows from (a) that we have

a’.

and it follows from (b),for

a’’.

Also, the knowledge of this quotient module, for *injective map*
*functorial* homomorphism

This claim is equivalent to the following (compare with [8, IV]):
If we have a quotient module

M_{\nu+k}/{\mathcal{S}}_k R_\nu is locally free of rankP(\nu+k) fork\geqslant 0 .- Both the subsheaf
{\mathscr{H}} of{\mathscr{F}} defined by the graded submoduleR_\bullet=\sum_{k\geqslant 0}{\mathscr{S}}_kR_\nu ofM_\bullet (cf. [10, II, §3]) and the quotient{\mathscr{G}}={\mathscr{F}}/{\mathscr{H}} satisfy conditions (a) and (b) above.

These conditions are clearly necessary, and if they are satisfied then the sheaf *flat* over *surjective* homomorphism of locally free modules of equal rank, and thus an isomorphism, and thus

Criteria (i) and (ii) above apply equally to the situation obtained after a change of base

Let

We can evidently suppose that

For every integer

*Proof*. We know [9, IV] that *flat* over

In particular, the set

is open, since it is the intersection of the

Let

*Proof*. Indeed, the above reasoning leads us to the case where we have the inequality (

*Proof*. Returning to the proof of (3.4) where we left off, we denote by *closed* subprescheme of

We have thus proven that * \mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}^P is representable by an S-prescheme Q that is an increasing union of open quasi-projective subpreschemes Q_\nu over S.*
To go further, we need to invoke Theorem 2.1, whence we easily conclude that

*quasi-compact*(since it is the image of a prescheme

*proper*over

Let

*Proof*. Indeed, if

Theorem 3.2, and thus Theorem 3.1, is now completely proven.

The proof also shows, at the same time, the following:

Under the conditions of (3.2), let

*Proof*. Indeed, we can reduce, as in (3.2), to the case where

The most important application of (3.2) is in the case where *Hilbert prescheme* of *Hilbert prescheme of index P*, respectively.
The terminology is justified by the role played in the theory by the Hilbert polynomials.
Their difference in nature with the classical Chow varieties (meant to parametrise cycles, not varieties) is analogous to that between

*the Chow ring*of classes of cycles of a variety and

*the ring of classes of sheaves*of the variety (as is introduced in the Riemann–Roch theorem [1]); we note that, when

We also note that the construction of

*[Comp.]*
The study of connected components of Hilbert schemes over an algebraically closed field was done by Hartshorne, who proves that the

## 6.4 Variants

Under the conditions of (3.1), let

U be open inX , and denote byA' the subfunctor ofA=\mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}} such thatA'(S') is the set of quotient modules{\mathscr{G}} of{\mathscr{F}}' that are flat overS' and whose support is contained insideU' . We immediately see thatA' is representable by an open subset of the prescheme\underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}} that representsA . It follows that Theorems (3.1) and (3.2) remain true if we suppose thatX is*quasi-projective*overS instead of projective overS , as long as we also replace in the conclusions the word “projective” by “quasi-projective”, and use\mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}}(S') to mean the set of coherent quotients{\mathscr{G}} of{\mathscr{F}}' that are flat overS' and*whose support is proper over*S' .Generally we can impose all sorts of supplementary natural conditions on the quotients

{\mathscr{G}} of{\mathscr{F}}'={\mathscr{F}}\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_{S'} that are flat overS' and stable under base change, thus obtaining as many subfunctors of\mathscr{Q}\kern -.5pt out_{{{\mathscr{F}}/X/S}} as we want to represent. The usual criteria allow us, in many cases, to prove that we again obtain functors that are representable by open subsets of\underline{\operatorname{Quot}}_{{{\mathscr{F}}/X/S}} . This is, in particular, the case if we impose one of the following additional properties:

- The dimensions of the prime cycles associated to the modules
{\mathscr{G}}_{s'} (fors'\in S' ) that are induced on the fibresX'_{s'} belong to a given set of integers. - (In the case where
{\mathscr{F}}={\mathscr{O}}_X , and thus{\mathscr{G}} corresponds to a closed subpreschemeY ofX' );Y is a*simple*prescheme [9, IV] overS , resp.*normal*overS (i.e. the fibresY_{s'} are normal “overk(s) ”, i.e. are normal under any extension of base field), resp. (ifX is flat overS ) are local completek -intersections inX with respect toS (i.e. the fibresY_{s'} are local complete intersections in theX_{s'} )

Other conditions would involve properties of a cohomological nature on the modules

- The preschemes
\underline{\operatorname{Hom}}_S(X,Y) ,\prod_{X/S}Z/S , and\underline{\operatorname{Isom}}_S(X,Y) , defined in FGA 3.II, C.2 exist thanks to suitable projective hypotheses, and can be realised as opens in suitable Hilbert preschemes. Since we have\underline{\operatorname{Hom}}_S(X,Y)=\prod_{X/S}((X\times Y)/X) , the case of\operatorname{Hom}_S(X,Y) reduces to that of\prod_{X/S}(Z/X) . We then note that, for allS' overS , the set of sections ofZ'=Z\times_S S' overX'=X\times_S S' is in bijective correspondence with the set of subpreschemes\Gamma ofZ (necessarily closed ifZ is separated overX ) such that the morphism\Gamma\to X' induced byZ'\to X' is an isomorphism. In this way,*if*ThusX is flat and proper overS , andZ quasi-projective overS , then\prod_{X/S}(Z/X) exists and is realised as an open subprescheme of\underline{\operatorname{Hilb}}_{Z/S} .*if*IfX is projective and flat overS , andY quasi-projective overS , then\underline{\operatorname{Hom}}_S(X,Y) exists and is realised as an open subprescheme of\underline{\operatorname{Hilb}}_{(X\times_S Y)/S} .X andY are both projective overS , then it immediately follows that\underline{\operatorname{Isom}}_S(X,Y) also exists, and is represented by an open subset of\underline{\operatorname{Hom}}_S(X,Y) . Similarly, ifX is flat and projective overS , andY quasi-projective overS , then theS -prescheme\underline{\operatorname{Imm}}_S(X/Y) that corresponds to the subfunctor of the functor represented by\underline{\operatorname{Hom}}_S(X,Y) that corresponds toS' -homomorphismsX'\to Y' that are immersions is also representable by an open subset of\underline{\operatorname{Hom}}_S(X,Y) .

Let

## 6.5 Differential study of Hilbert schemes

We have the following result:

Let

We thus conclude:

Suppose that there exists locally on

The existence of a global extension is thus guaranteed, in particular, if

Suppose that

Then the Zariski tangent space of the fibre

The result giving the Zariski tangent space can be generalised, and gives a characterisation, for a given

If, in (5.1), we have *conormal sheaf* of

Under the conditions of (5.3), suppose that

This statement applies in particular when *one* single equation, i.e. is a positive “Cartier divisor”.
Then *characteristic homomorphism*” (from the former to the latter).
It was not defined when *reduced* structure.
The tangent space of *subspace* of the tangent space of *field of complex numbers*) where even at the generic point of *This thus implies that D is not integral even at the generic point of the irreducible component in question.*
This shows in a particularly striking manner how varieties with nilpotent elements are necessary for understanding the most classical phenomena of the theory of surfaces.

*[Comp.]*
Concerning the example of Zappa, we note that Mumford has even constructed an irreducible component of the Hilbert scheme for *Séminaire Cartan 13, 1960/61*) is non-reduced at all its points.

We have given, in (5.4), a criterion for simplicity, which applies in particular to schemes of divisors.
Kodaira gave a different criterion in [12], given by the vanishing of *a field k of characteristic 0* is simple over

To finish, we give the following result, which plays an important role in the differential study of fibred spaces:

Let

*Proof*. This is an immediate consequence of the definition, and of the usual criterion of simplicity [9, III, §3.1].

Note that if *bundle of germs of sections of order n of X over T* (with respect to

## 6.6 Relation to the notion of norm and symmetric products

Let *symmetric*

Now suppose that we have a coherent module

- If
X is simple overS , then the norm morphism defines an isomorphism from the open of\underline{\operatorname{Hilb}}_{X/S}^n that corresponds to the classification of étale covers of rankn contained insideX (cf. §4.b) to the open of\operatorname{Symm}_S^n(X) that corresponds to then -cycles without multiple components. - If furthermore
X is of relative dimension1 overS , then the norm morphism even defines an isomorphism from\underline{\operatorname{Hilb}}_{X/S}^n to\operatorname{Symm}_{X/S}^n .

This second fact is due to the fact that a subscheme of dimension

## 6.7 Supplements and questions

As remarked by J.-P. Serre, it follows from a well-known example of Nagata that we can find a scheme

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See the addendum at the end of this article.↩︎