13
$\begingroup$

Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?

$\endgroup$
  • 2
    $\begingroup$ That follows from the following MathOverflow answer: mathoverflow.net/questions/268764/… You need to combine that with a little argument using Chow's Lemma. The upshot is that there are countably many proper, smooth morphisms $(\pi_i:X_i\to B_i)_i$ of smooth, separated $\mathbb{C}$-schemes such that every proper smooth $\mathbb{C}$-scheme is a fiber of (at least) one morphism $\pi_i$. Thus, that scheme embeds in $X_i$. $\endgroup$ – Jason Starr Mar 29 '19 at 18:05
12
$\begingroup$

I am just posting my comment as one answer. So long as you are only asking about schemes (rather than complex analytic spaces), you can avoid the hard analysis from the previous MathOverflow answer. The argument below sketches this. The main additional detail beyond Hilbert scheme techniques is a strong variant of Chow's Lemma. Let $k$ be an algebraically closed field (soon we will assume it has characteristic $0$).

Chow's Lemma. Let $X$ be a separated, finitely presented scheme over a field $k$, and let $U\subset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $\nu:\widetilde{X}\to X$ such that $\widetilde{X}$ is a quasi-projective $k$-scheme and such that the restriction of $\nu$ over $U$ is an isomorphism.

There may be an earlier source, but the source that I know is the following article.

MR0308104 (46 #7219)
Raynaud, Michel; Gruson, Laurent
Critères de platitude et de projectivité. Techniques de "platification'' d'un module.
Invent. Math. 13 (1971), 1–89.

Finally, the very last step of the argument requires Nagata compactification.

Nagata compactification Every separated, finite type $k$-scheme is isomorphic to a dense open subscheme of a proper $k$-scheme.

Definition. For every separated morphism that restricts as an isomorphism over a dense open in the target, the isomorphism locus is the maximal open subscheme of the target over which the morphism is an isomorphism. For a separated, flat, finite type morphism of schemes, $$\rho:\mathcal{X}\to B,$$ a Chow covering is an ordered $n$-tuple (for some integer $n\geq 0$) of pairs of $B$-morphisms, $$(\nu_\ell:\widetilde{\mathcal{X}}_\ell \to \mathcal{X}, e_\ell:\widetilde{\mathcal{X}}_\ell \to \mathbb{P}^m_B),$$ where each $e_\ell$ is a locally closed immersion of flat $B$-schemes and where the morphisms $\nu_\ell$ are strongly projective morphisms whose isomorphism loci give an open covering of $\mathcal{X}$.

Corollary. Every separated, finite type $k$-scheme has a Chow covering.

Proof. Since $X$ is quasi-compact, there exists a finite covering of $X$ by open affine subschemes. By Chow's Lemma, for each open affine, there exists a strongly projective $k$-morphism from a quasi-projective $k$-scheme to $X$ whose isomorphism locus contains that open affine. QED

Hypothesis. The field $k$ has characteristic $0$.

Definition. A smooth Chow covering is a Chow covering such that every $B$-scheme $\widetilde{\mathcal{X}}_\ell$ is smooth over $B$.

Corollary. Every smooth, separated, quasi-compact $k$-scheme has a smooth Chow covering.

Proof. This follows from the previous corollary and Hironaka's Theorem. QED

Notation. For every smooth Chow covering over $B$, denote by $\widetilde{\mathcal{X}}$ the disjoint union of the $B$-schemes $\widetilde{\mathcal{X}}_\ell$. Denote by $\nu:\widetilde{\mathcal{X}}\to \mathcal{X}$ the unique $B$-morphism whose restriction to each component $\widetilde{\mathcal{X}}_\ell$ equals $\nu_\ell$. Denote by $\mathcal{Y}$ the disjoint union over all ordered pairs $(j,\ell)$ with $1\leq j,\ell\leq n$ of the closed subscheme of the product, $$\mathcal{Y}_{j,\ell}:= \widetilde{\mathcal{X}}_j\times_{\mathcal{X}} \widetilde{\mathcal{X}}_\ell \subseteq \widetilde{\mathcal{X}}_j\times_{B} \widetilde{\mathcal{X}}_\ell,$$ together with its two projections, $$\text{pr}_{1,(j,\ell)}:\mathcal{Y}_{j,\ell}\to \widetilde{\mathcal{X}}_j, \ \ \text{pr}_{2,(j,\ell)}:\mathcal{Y}_{j,\ell}\to \widetilde{\mathcal{X}}_\ell.$$ Denote the disjoint union of these morphisms by $$\text{pr}_1:\mathcal{Y} \to \widetilde{\mathcal{X}}, \ \ \text{pr}_2:\mathcal{Y}\to \widetilde{\mathcal{X}}.$$ For every $\ell=1,\dots,n$, denote the diagonal morphism by $$\delta_\ell:\widetilde{\mathcal{X}}_\ell \to \mathcal{Y}_{\ell,\ell}.$$ Denote the disjoint union of these morphisms by $$\delta:\widetilde{\mathcal{X}}\to \mathcal{Y}.$$ For every $(j,\ell)$, denote the involution of $B$-schemes that transposes the factors by $$\sigma_{j,\ell}:\mathcal{Y}_{j,\ell}\to \mathcal{Y}_{\ell,j}.$$ Denote the disjoint union of these involutions by the involution, $$\sigma:\mathcal{Y}\to \mathcal{Y}, \ \ \text{pr}_2\circ \sigma = \text{pr}_1, \ \ \text{pr}_1\circ \sigma = \text{pr}_2.$$ For every ordered triple $(j,\ell,r)$ with $1\leq j,\ell,r\leq n$, denote by $$c_{j,\ell,r}:\mathcal{Y}_{j,\ell}\times_{\widetilde{\mathcal{X}}_\ell} \mathcal{Y}_{\ell,r} \to \mathcal{Y}_{j,r},$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:\mathcal{Y}\times_{\text{pr}_2,\widetilde{\mathcal{X}},\text{pr}_1} \mathcal{Y} \to \mathcal{Y}.$$

Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((\pi_\ell:\widetilde{\mathcal{X}}_\ell \to B, ((\text{pr}_{1,(j,\ell)},\text{pr}_{2,(j,\ell)}):\mathcal{Y}_{j,\ell}\hookrightarrow \widetilde{\mathcal{X}}_j\times_{B}\widetilde{\mathcal{X}}_\ell)_{j,\ell}, $$ $$(\delta_\ell:\widetilde{\mathcal{X}}_\ell\to Y_{\ell,\ell})_\ell, (\sigma_{j,\ell}:Y_{j,\ell}\to Y_{\ell,j})_{j,\ell}, (c_{j,\ell,r}:Y_{j,\ell}\times_{\text{pr}_2,\widetilde{X}_\ell,\text{pr}_1} Y_{\ell,r}\to Y_{j,r})_{j,\ell,r})$$ together with a collection of locally closed immersions of $B$-schemes, $$e_\ell:\widetilde{\mathcal{X}}_\ell\hookrightarrow \mathbb{P}^m_B.$$ The isomorphism locus is the maximal open subscheme $U_\ell$ of $\widetilde{X}_\ell$ on which the closed immersion $\delta_\ell$ is an open immersion.

Constraint axioms for a Chow datum. As the $1$-truncation of the simplicial scheme induced by $\nu$, there are many constraints satisfied by the Chow descent predatum of a Chow covering. Taken together, these constraints define a Chow descent datum.

Constraint 1. For every $(j,\ell)$, the $\text{pr}_1$-inverse image in $Y_{j,\ell}$ of $U_j$ is an open subscheme whose closed complement in $\mathcal{Y}_{j,\ell}$ equals its total inverse image under $\text{pr}_2$ of its closed image in $\widetilde{\mathcal{X}}_\ell$. Denote by $\widetilde{\mathcal{X}}_{\ell,j}$ the open complement in $\widetilde{\mathcal{X}}_\ell$ of this closed image.

Constraint 2. For each $\ell=1,\dots,n$, the collection of open subschemes $$(\widetilde{\mathcal{X}}_{\ell,j})_{j=1,\dots,n},$$ form an open covering of $\widetilde{\mathcal{X}}_\ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.

Constraint 3. The isomorphism locus of each projection $$\text{pr}_{2,(j,\ell)}:\mathcal{Y}_{j,\ell} \to \widetilde{\mathcal{X}}_\ell,$$ contains $\widetilde{\mathcal{X}}_{\ell,j}$. Thus, the inverse image of $\widetilde{\mathcal{X}}_{\ell,j}$ under this isomorphism equals the graph of a unique $k$-morphism, $$\nu_{\ell,j}:\widetilde{\mathcal{X}}_{\ell,j} \to U_j.$$

Constraint 4. Denote by $U_{\ell,j}$ the intersection of $U_\ell$ and the $\nu_{\ell,j}$-inverse image of $U_j$ in $\widetilde{\mathcal{X}}_{\ell,j}$. For each triple $(j,\ell,r)$, the inverse image in $U_{\ell,j}$ under $\nu_{\ell,j}$ of $U_{j,r}$ equals $U_{\ell,j}\cap U_{\ell,r}$. Denote this open by $U_{\ell,j,r}$. Also, the inverse image in $\widetilde{\mathcal{X}}_{\ell,j}$ under $\nu_{\ell,j}$ of $U_j\cap \widetilde{\mathcal{X}}_{j,r}$ equals the inverse image in $\widetilde{\mathcal{X}}_{\ell,r}$ under $\nu_{\ell,r}$ of $U_r\cap \widetilde{\mathcal{X}}_{r,j}$. Denote this open by $\widetilde{\mathcal{X}}_{\ell,j,r}$.

Constraint 5. On the open $U_{\ell,j,r}$, the composition $\nu_{j,r}\circ \nu_{\ell,j}$ equals $\nu_{\ell,r}$. Similarly, on the open $\widetilde{\mathcal{X}}_{\ell,j,r}$, the composition $\nu_{j,r}\circ \nu_{\ell,j}$ equals $\nu_{\ell,r}$.

Effectivity Proposition. Every Chow descent datum is effective, i.e., it is isomorphic to the Chow descent datum of a Chow covering.

Proof Altogether, these constraints are the descent constraints for a Zariski descent datum of $B$-schemes, $$((U_\ell)_\ell, (U_{\ell,j}\subset U_\ell)_{\ell,j}, (\nu_{\ell,j}:U_{\ell,j}\to U_j)_{\ell,j})$$ together with the descent constraints for each Zariski descent datum of $B$-morphisms, $$(\widetilde{X}_{\ell,j}\to U_j)_j.$$ By Zariski gluing for schemes, this descent datum for a $B$-scheme is effective. By Zariski gluing for morphisms, each of these descent data for a $B$-morphism is effective.

For each $U_\ell$, each $\widetilde{\mathcal{X}}_{\ell,j}\to U_\ell$ is proper and surjective. Thus, the induced morphism $\nu_j$ is proper and surjective. Since also $\widetilde{\mathcal{X}}_j$ is proper over $B$, the $B$-scheme $\mathcal{X}$ is proper. Since $\mathcal{X}$ is covered by open $U_\ell$ that are smooth $B$-schemes, also $\mathcal{X}$ is a smooth $B$-scheme. QED

Constraint 6. The morphisms $\sigma$ and $c$ equal the natural morphisms induced by this effective descent datum.

Nota bene. Since we have already incorporated the glueing constraints above, Constraint 6, and even the data of the morphisms $\sigma$ and $c$, are extraneous. However, from the point of view of a "groupoid scheme" associated to the Chow covering, it seems best to include this data as part of the definition.

Constructibility Proposition. For every separated, finite type $k$-scheme $B$, for every Chow descent predatum over $B$, there exists a finite collection of locally closed subschemes of $B$ (with the induced reduced structure) whose set of geometric points are precisely those geometric points of $B$ where the Chow descent predatum is a Chow descent datum.

Proof sketch. Each constraint condition involves equalities or inclusions of closed subsets of $\widetilde{X}_\ell$, or, equivalently, equalities or inclusions of open subsets, it involves a morphism being an isomorphism, or it involves certain associated morphisms to a self-fiber product factoring through the diagonal.

Pass to flattening stratifications of the base $B$ for the relevant closed subschemes with their reduced structure. By separatedness of the Hilbert scheme, the locus in $B$ where two $B$-flat closed subschemes of a projective $B$-scheme are equal is a closed subset of $B$. Similarly, using properness of the Hilbert scheme and quasi-compactness of $B$, the locus where one $B$-flat closed subscheme factors through a second $B$-flat closed subscheme is also a closed subset of $B$.

The isomorphism locus is open. By taking set complements, inclusion or equality of open subsets is equivalent to inclusion or equality of the closed complements. This defines finitely many locally closed subsets of $B$ by the previous paragraph.

Finally, since all $B$-schemes in this setup are separated, factorization of a morphism through the diagonal is equivalent to equality of the domain of the morphism with the closed inverse image of the diagonal. Up to taking closures of each of these in the appropriate ambient projective $B$-scheme, this again reduces to an inclusion of closed subschemes. Thus each of these conditions defines a finite union of locally closed subscheme of $B$. QED

By the proposition, we can form a countable collection of families of Chow descent data such that every Chow descent datum occurs as a fiber of one of these families.. For every Chow descent predatum,$$(([\widetilde{X}_\ell])_{\ell},([Y_{j,\ell}])_{j,\ell},([\delta_\ell])_{\ell},([\sigma_{j,\ell}])_{j,\ell}, ([c_{j,\ell,r}])_{j,\ell,r}),$$ the first two parts of the datum give points in appropriate Hilbert schemes. The last three parts give points in appropriate Hom schemes.

Altogether, there are countably many ordered pairs $(n,m)\in \mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $\widetilde{X}_\ell$ of $\mathbb{P}^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(\widetilde{X}_\ell)_{\ell=1,\dots,n}$, there are countably many $n^2$-tuples of Hilbert polynomials for the closed subschemes $Y_{j,\ell}$. For each such tuple of Hilbert polynomials, there is a further relative Hilbert scheme parameterizing closed subschemes $Y_{j,\ell}$ in $\widetilde{X}_j\times_{\text{Spec}\ k}\widetilde{X}_\ell$. Then there are countably many components of each Hom scheme for the morphisms $\delta_\ell$, $\sigma_{j,\ell}$ and $c_{j,\ell,r}$. For each of these countably many quasi-projective $k$-schemes parameterizing a Chow descent predatum, by the Constructibility Proposition, there are finitely many locally closed subschemes (with reduced structures) in the base parameterizing those Chow descent predata that are Chow descent data.

Over this countable disjoint union of quasi-projective $k$-schemes, we glue together the isomorphism loci to form a proper flat morphism $\mathcal{X}\to B$ and the strongly projective $B$-morphisms $\nu_\ell$. By the existence of Chow coverings, every proper $k$-scheme occurs as a fiber over a $k$-point of one of the schemes $B$ in the countable collection.

Denote by $I$ the countable set of irreducible components for all bases of all of these countably many families. For each $i\in I$, the Effectivity Proposition gives a smooth Chow covering. Denote this by $$(\rho:\mathcal{X}_i \to B_i, ((\nu_i,e_i):\widetilde{\mathcal{X}}_{i,\ell} \hookrightarrow \mathcal{X}_i\times_{\text{Spec}\ k} \mathbb{P}^m_{\mathbb{C}})_{\ell}).$$

Finally, by applying Hironaka's Theorem to each irreducible, quasi-projective $k$-scheme $B_i$, we can assume that each $B_i$ is a smooth, connected quasi-projective $k$-scheme. The effect is that every $\mathcal{X}_i$ is itself a separated, quasi-compact, smooth $k$-scheme.

Certainly $\mathcal{X}_i$ can be non-proper over $k$. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $\mathcal{X}_i$ as a dense open subscheme of a proper, smooth $k$-scheme $\overline{\mathcal{X}}_i$. Since every smooth proper $k$-scheme arises as a fiber of some $\pi_i$, every smooth proper $k$-scheme admits a closed immersion in one of the countably many smooth, proper $k$-schemes $\overline{\mathcal{X}}_i$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy