# 6 Hilbert schemes

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.

**TO-DO:**

**additif****errata**

## Introduction

The techniques described in [15] 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 [15]. In the current exposé, 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 [15]. Here we will construct “Hilbert schemes”, which are meant to replace the use of Chow coordinates, as was mentioned in [15]. In the next exposé, the theory of passing to the quotient in schemes, developed in [15], combined with the theory of Hilbert schemes, will allow us, for example, to construct Picard schemes (defined in [15]) 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. [15]).
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 **(TO-DO: ? is this right?)**.
This defines an equivalence relation, and we are interested in the equivalence classes of sheaves under this relation, and of sets of such equivalence classes.
Note that, if *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, [18].

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

### References

[15] A. Grothendieck. “Technique de descente et théorèmes d’existence en géométrie algébrique, I, II, III.” *Séminaire Bourbaki*. **12, 13** (n.d.), Talks no. 190, 195, and 212.

[16] A. Grothendieck. *Séminaire de géométrie algébrique, I, II, III, IV*. Paris, Institut des Hautes Études Scientifiques, n.d. (n.d.).

[18] A. Grothendieck, J. Dieudonné. “Eléments de géométrie algébrique, I, II, III, IV.” *Publications Mathématiques de L’Institut Des Hautes Études Scientifiques*. (n.d.).

See the addendum at the end of this exposé.↩︎