Is there an analogue of projective spaces for proper schemes?Do compact complex manifolds fall into countably many families?Is there a Whitney theorem type theorem for projective schemes?Proper morphisms: Lie groups vs. group schemesEmbedding proper algebraic spacesProper morphism and irreducibility of schemesDoes there exist an algebraic space with large fundamental group but no finite etale covers by schemesEmbedding of a proper scheme into a smooth onePushouts of schemes along closed immersionsAre there smooth and proper schemes over $mathbb Z$ whose cohomology is not of Tate typeSmooth proper fibration of complex projective varietiesIrreducible Smooth Proper one-dimensional Schemes isomorphic to $mathbbP^1$
Is there an analogue of projective spaces for proper schemes?
Do compact complex manifolds fall into countably many families?Is there a Whitney theorem type theorem for projective schemes?Proper morphisms: Lie groups vs. group schemesEmbedding proper algebraic spacesProper morphism and irreducibility of schemesDoes there exist an algebraic space with large fundamental group but no finite etale covers by schemesEmbedding of a proper scheme into a smooth onePushouts of schemes along closed immersionsAre there smooth and proper schemes over $mathbb Z$ whose cohomology is not of Tate typeSmooth proper fibration of complex projective varietiesIrreducible Smooth Proper one-dimensional Schemes isomorphic to $mathbbP^1$
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Does there exist a countable set of connected proper smooth $mathbbC$-schemes such that any connected proper smooth $mathbbC$-scheme admits a $mathbbC$-immersion into one of them?
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atleatle
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Does there exist a countable set of connected proper smooth $mathbbC$-schemes such that any connected proper smooth $mathbbC$-scheme admits a $mathbbC$-immersion into one of them?
ag.algebraic-geometry complex-geometry schemes
asked Mar 29 at 17:36
atleatle
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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 $(pi_i:X_ito B_i)_i$ of smooth, separated $mathbbC$-schemes such that every proper smooth $mathbbC$-scheme is a fiber of (at least) one morphism $pi_i$. Thus, that scheme embeds in $X_i$.
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– Jason Starr
Mar 29 at 18:05
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Does there exist a countable set of connected proper smooth $mathbbC$-schemes such that any connected proper smooth $mathbbC$-scheme admits a $mathbbC$-immersion into one of them?
ag.algebraic-geometry complex-geometry schemes
asked Mar 29 at 17:36
atleatle
612
$endgroup$
Does there exist a countable set of connected proper smooth $mathbbC$-schemes such that any connected proper smooth $mathbbC$-scheme admits a $mathbbC$-immersion into one of them?
ag.algebraic-geometry complex-geometry schemes
ag.algebraic-geometry complex-geometry schemes
asked Mar 29 at 17:36
atleatle
612
asked Mar 29 at 17:36
atleatle
612
asked Mar 29 at 17:36
atleatle
612
asked Mar 29 at 17:36
atleatle
612
asked Mar 29 at 17:36
atleatle
612
612
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 $(pi_i:X_ito B_i)_i$ of smooth, separated $mathbbC$-schemes such that every proper smooth $mathbbC$-scheme is a fiber of (at least) one morphism $pi_i$. Thus, that scheme embeds in $X_i$.
$endgroup$
– Jason Starr
Mar 29 at 18:05
add a comment |
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 $(pi_i:X_ito B_i)_i$ of smooth, separated $mathbbC$-schemes such that every proper smooth $mathbbC$-scheme is a fiber of (at least) one morphism $pi_i$. Thus, that scheme embeds in $X_i$.
$endgroup$
– Jason Starr
Mar 29 at 18:05
2
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 $(pi_i:X_ito B_i)_i$ of smooth, separated $mathbbC$-schemes such that every proper smooth $mathbbC$-scheme is a fiber of (at least) one morphism $pi_i$. Thus, that scheme embeds in $X_i$.
$endgroup$
– Jason Starr
Mar 29 at 18:05
$begingroup$
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 $(pi_i:X_ito B_i)_i$ of smooth, separated $mathbbC$-schemes such that every proper smooth $mathbbC$-scheme is a fiber of (at least) one morphism $pi_i$. Thus, that scheme embeds in $X_i$.
$endgroup$
– Jason Starr
Mar 29 at 18:05
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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 $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetildeXto X$ such that $widetildeX$ 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, $$rho:mathcalXto B,$$ a Chow covering is an ordered $n$-tuple (for some integer $ngeq 0$) of pairs of $B$-morphisms, $$(nu_ell:widetildemathcalX_ell to mathcalX, e_ell:widetildemathcalX_ell to mathbbP^m_B),$$ 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 $mathcalX$.
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 $widetildemathcalX_ell$ 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 $widetildemathcalX$ the disjoint union of the $B$-schemes $widetildemathcalX_ell$. Denote by $nu:widetildemathcalXto mathcalX$ the unique $B$-morphism whose restriction to each component $widetildemathcalX_ell$ equals $nu_ell$. Denote by $mathcalY$ the disjoint union over all ordered pairs $(j,ell)$ with $1leq j,ellleq n$ of the closed subscheme of the product, $$mathcalY_j,ell:= widetildemathcalX_jtimes_mathcalX widetildemathcalX_ell subseteq widetildemathcalX_jtimes_B widetildemathcalX_ell,$$ together with its two projections, $$textpr_1,(j,ell):mathcalY_j,ellto widetildemathcalX_j, textpr_2,(j,ell):mathcalY_j,ellto widetildemathcalX_ell.$$ Denote the disjoint union of these morphisms by $$textpr_1:mathcalY to widetildemathcalX, textpr_2:mathcalYto widetildemathcalX.$$ For every $ell=1,dots,n$, denote the diagonal morphism by $$delta_ell:widetildemathcalX_ell to mathcalY_ell,ell.$$ Denote the disjoint union of these morphisms by $$delta:widetildemathcalXto mathcalY.$$ For every $(j,ell)$, denote the involution of $B$-schemes that transposes the factors by $$sigma_j,ell:mathcalY_j,ellto mathcalY_ell,j.$$ Denote the disjoint union of these involutions by the involution, $$sigma:mathcalYto mathcalY, textpr_2circ sigma = textpr_1, textpr_1circ sigma = textpr_2.$$
For every ordered triple $(j,ell,r)$ with $1leq j,ell,rleq n$, denote by $$c_j,ell,r:mathcalY_j,elltimes_widetildemathcalX_ell mathcalY_ell,r to mathcalY_j,r,$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:mathcalYtimes_textpr_2,widetildemathcalX,textpr_1 mathcalY to mathcalY.$$
Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((pi_ell:widetildemathcalX_ell to B, ((textpr_1,(j,ell),textpr_2,(j,ell)):mathcalY_j,ellhookrightarrow widetildemathcalX_jtimes_BwidetildemathcalX_ell)_j,ell, $$ $$(delta_ell:widetildemathcalX_ellto Y_ell,ell)_ell, (sigma_j,ell:Y_j,ellto Y_ell,j)_j,ell, (c_j,ell,r:Y_j,elltimes_textpr_2,widetildeX_ell,textpr_1 Y_ell,rto Y_j,r)_j,ell,r)$$ together with a collection of locally closed immersions of $B$-schemes, $$e_ell:widetildemathcalX_ellhookrightarrow mathbbP^m_B.$$ The isomorphism locus is the maximal open subscheme $U_ell$ of $widetildeX_ell$ 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 $textpr_1$-inverse image in $Y_j,ell$ of $U_j$ is an open subscheme whose closed complement in $mathcalY_j,ell$ equals its total inverse image under $textpr_2$ of its closed image in $widetildemathcalX_ell$. Denote by $widetildemathcalX_ell,j$ the open complement in $widetildemathcalX_ell$ of this closed image.
Constraint 2. For each $ell=1,dots,n$, the collection of open subschemes $$(widetildemathcalX_ell,j)_j=1,dots,n,$$
form an open covering of $widetildemathcalX_ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.
Constraint 3. The isomorphism locus of each projection $$textpr_2,(j,ell):mathcalY_j,ell to widetildemathcalX_ell,$$ contains $widetildemathcalX_ell,j$. Thus, the inverse image of $widetildemathcalX_ell,j$ under this isomorphism equals the graph of a unique $k$-morphism, $$nu_ell,j:widetildemathcalX_ell,j to U_j.$$
Constraint 4. Denote by $U_ell,j$ the intersection of $U_ell$ and the $nu_ell,j$-inverse image of $U_j$ in $widetildemathcalX_ell,j$. For each triple $(j,ell,r)$, the inverse image in $U_ell,j$ under $nu_ell,j$ of $U_j,r$ equals $U_ell,jcap U_ell,r$. Denote this open by $U_ell,j,r$. Also, the inverse image in $widetildemathcalX_ell,j$ under $nu_ell,j$ of $U_jcap widetildemathcalX_j,r$ equals the inverse image in $widetildemathcalX_ell,r$ under $nu_ell,r$ of $U_rcap widetildemathcalX_r,j$. Denote this open by $widetildemathcalX_ell,j,r$.
Constraint 5. On the open $U_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ equals $nu_ell,r$. Similarly, on the open $widetildemathcalX_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ 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, $$((U_ell)_ell, (U_ell,jsubset U_ell)_ell,j, (nu_ell,j:U_ell,jto U_j)_ell,j)$$ together with the descent constraints for each Zariski descent datum of $B$-morphisms, $$(widetildeX_ell,jto U_j)_j.$$ 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 $widetildemathcalX_ell,jto U_ell$ is proper and surjective. Thus, the induced morphism $nu_j$ is proper and surjective. Since also $widetildemathcalX_j$ is proper over $B$, the $B$-scheme $mathcalX$ is proper. Since $mathcalX$ is covered by open $U_ell$ that are smooth $B$-schemes, also $mathcalX$ 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 $widetildeX_ell$, 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,$$(([widetildeX_ell])_ell,([Y_j,ell])_j,ell,([delta_ell])_ell,([sigma_j,ell])_j,ell, ([c_j,ell,r])_j,ell,r),$$ 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 $(n,m)in mathbbZ_geq 0times mathbbZ_geq 0$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetildeX_ell$ of $mathbbP^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetildeX_ell)_ell=1,dots,n$, 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 $widetildeX_jtimes_textSpec kwidetildeX_ell$. 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 $mathcalXto B$ 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 $iin I$, the Effectivity Proposition gives a smooth Chow covering. Denote this by $$(rho:mathcalX_i to B_i, ((nu_i,e_i):widetildemathcalX_i,ell hookrightarrow mathcalX_itimes_textSpec k mathbbP^m_mathbbC)_ell).$$
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 $mathcalX_i$ is itself a separated, quasi-compact, smooth $k$-scheme.
Certainly $mathcalX_i$ can be non-proper over $k$. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcalX_i$ as a dense open subscheme of a proper, smooth $k$-scheme $overlinemathcalX_i$. 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 $overlinemathcalX_i$.
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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 $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetildeXto X$ such that $widetildeX$ 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, $$rho:mathcalXto B,$$ a Chow covering is an ordered $n$-tuple (for some integer $ngeq 0$) of pairs of $B$-morphisms, $$(nu_ell:widetildemathcalX_ell to mathcalX, e_ell:widetildemathcalX_ell to mathbbP^m_B),$$ 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 $mathcalX$.
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 $widetildemathcalX_ell$ 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 $widetildemathcalX$ the disjoint union of the $B$-schemes $widetildemathcalX_ell$. Denote by $nu:widetildemathcalXto mathcalX$ the unique $B$-morphism whose restriction to each component $widetildemathcalX_ell$ equals $nu_ell$. Denote by $mathcalY$ the disjoint union over all ordered pairs $(j,ell)$ with $1leq j,ellleq n$ of the closed subscheme of the product, $$mathcalY_j,ell:= widetildemathcalX_jtimes_mathcalX widetildemathcalX_ell subseteq widetildemathcalX_jtimes_B widetildemathcalX_ell,$$ together with its two projections, $$textpr_1,(j,ell):mathcalY_j,ellto widetildemathcalX_j, textpr_2,(j,ell):mathcalY_j,ellto widetildemathcalX_ell.$$ Denote the disjoint union of these morphisms by $$textpr_1:mathcalY to widetildemathcalX, textpr_2:mathcalYto widetildemathcalX.$$ For every $ell=1,dots,n$, denote the diagonal morphism by $$delta_ell:widetildemathcalX_ell to mathcalY_ell,ell.$$ Denote the disjoint union of these morphisms by $$delta:widetildemathcalXto mathcalY.$$ For every $(j,ell)$, denote the involution of $B$-schemes that transposes the factors by $$sigma_j,ell:mathcalY_j,ellto mathcalY_ell,j.$$ Denote the disjoint union of these involutions by the involution, $$sigma:mathcalYto mathcalY, textpr_2circ sigma = textpr_1, textpr_1circ sigma = textpr_2.$$
For every ordered triple $(j,ell,r)$ with $1leq j,ell,rleq n$, denote by $$c_j,ell,r:mathcalY_j,elltimes_widetildemathcalX_ell mathcalY_ell,r to mathcalY_j,r,$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:mathcalYtimes_textpr_2,widetildemathcalX,textpr_1 mathcalY to mathcalY.$$
Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((pi_ell:widetildemathcalX_ell to B, ((textpr_1,(j,ell),textpr_2,(j,ell)):mathcalY_j,ellhookrightarrow widetildemathcalX_jtimes_BwidetildemathcalX_ell)_j,ell, $$ $$(delta_ell:widetildemathcalX_ellto Y_ell,ell)_ell, (sigma_j,ell:Y_j,ellto Y_ell,j)_j,ell, (c_j,ell,r:Y_j,elltimes_textpr_2,widetildeX_ell,textpr_1 Y_ell,rto Y_j,r)_j,ell,r)$$ together with a collection of locally closed immersions of $B$-schemes, $$e_ell:widetildemathcalX_ellhookrightarrow mathbbP^m_B.$$ The isomorphism locus is the maximal open subscheme $U_ell$ of $widetildeX_ell$ 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 $textpr_1$-inverse image in $Y_j,ell$ of $U_j$ is an open subscheme whose closed complement in $mathcalY_j,ell$ equals its total inverse image under $textpr_2$ of its closed image in $widetildemathcalX_ell$. Denote by $widetildemathcalX_ell,j$ the open complement in $widetildemathcalX_ell$ of this closed image.
Constraint 2. For each $ell=1,dots,n$, the collection of open subschemes $$(widetildemathcalX_ell,j)_j=1,dots,n,$$
form an open covering of $widetildemathcalX_ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.
Constraint 3. The isomorphism locus of each projection $$textpr_2,(j,ell):mathcalY_j,ell to widetildemathcalX_ell,$$ contains $widetildemathcalX_ell,j$. Thus, the inverse image of $widetildemathcalX_ell,j$ under this isomorphism equals the graph of a unique $k$-morphism, $$nu_ell,j:widetildemathcalX_ell,j to U_j.$$
Constraint 4. Denote by $U_ell,j$ the intersection of $U_ell$ and the $nu_ell,j$-inverse image of $U_j$ in $widetildemathcalX_ell,j$. For each triple $(j,ell,r)$, the inverse image in $U_ell,j$ under $nu_ell,j$ of $U_j,r$ equals $U_ell,jcap U_ell,r$. Denote this open by $U_ell,j,r$. Also, the inverse image in $widetildemathcalX_ell,j$ under $nu_ell,j$ of $U_jcap widetildemathcalX_j,r$ equals the inverse image in $widetildemathcalX_ell,r$ under $nu_ell,r$ of $U_rcap widetildemathcalX_r,j$. Denote this open by $widetildemathcalX_ell,j,r$.
Constraint 5. On the open $U_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ equals $nu_ell,r$. Similarly, on the open $widetildemathcalX_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ 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, $$((U_ell)_ell, (U_ell,jsubset U_ell)_ell,j, (nu_ell,j:U_ell,jto U_j)_ell,j)$$ together with the descent constraints for each Zariski descent datum of $B$-morphisms, $$(widetildeX_ell,jto U_j)_j.$$ 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 $widetildemathcalX_ell,jto U_ell$ is proper and surjective. Thus, the induced morphism $nu_j$ is proper and surjective. Since also $widetildemathcalX_j$ is proper over $B$, the $B$-scheme $mathcalX$ is proper. Since $mathcalX$ is covered by open $U_ell$ that are smooth $B$-schemes, also $mathcalX$ 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 $widetildeX_ell$, 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,$$(([widetildeX_ell])_ell,([Y_j,ell])_j,ell,([delta_ell])_ell,([sigma_j,ell])_j,ell, ([c_j,ell,r])_j,ell,r),$$ 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 $(n,m)in mathbbZ_geq 0times mathbbZ_geq 0$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetildeX_ell$ of $mathbbP^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetildeX_ell)_ell=1,dots,n$, 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 $widetildeX_jtimes_textSpec kwidetildeX_ell$. 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 $mathcalXto B$ 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 $iin I$, the Effectivity Proposition gives a smooth Chow covering. Denote this by $$(rho:mathcalX_i to B_i, ((nu_i,e_i):widetildemathcalX_i,ell hookrightarrow mathcalX_itimes_textSpec k mathbbP^m_mathbbC)_ell).$$
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 $mathcalX_i$ is itself a separated, quasi-compact, smooth $k$-scheme.
Certainly $mathcalX_i$ can be non-proper over $k$. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcalX_i$ as a dense open subscheme of a proper, smooth $k$-scheme $overlinemathcalX_i$. 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 $overlinemathcalX_i$.
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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 $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetildeXto X$ such that $widetildeX$ 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, $$rho:mathcalXto B,$$ a Chow covering is an ordered $n$-tuple (for some integer $ngeq 0$) of pairs of $B$-morphisms, $$(nu_ell:widetildemathcalX_ell to mathcalX, e_ell:widetildemathcalX_ell to mathbbP^m_B),$$ 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 $mathcalX$.
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 $widetildemathcalX_ell$ 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 $widetildemathcalX$ the disjoint union of the $B$-schemes $widetildemathcalX_ell$. Denote by $nu:widetildemathcalXto mathcalX$ the unique $B$-morphism whose restriction to each component $widetildemathcalX_ell$ equals $nu_ell$. Denote by $mathcalY$ the disjoint union over all ordered pairs $(j,ell)$ with $1leq j,ellleq n$ of the closed subscheme of the product, $$mathcalY_j,ell:= widetildemathcalX_jtimes_mathcalX widetildemathcalX_ell subseteq widetildemathcalX_jtimes_B widetildemathcalX_ell,$$ together with its two projections, $$textpr_1,(j,ell):mathcalY_j,ellto widetildemathcalX_j, textpr_2,(j,ell):mathcalY_j,ellto widetildemathcalX_ell.$$ Denote the disjoint union of these morphisms by $$textpr_1:mathcalY to widetildemathcalX, textpr_2:mathcalYto widetildemathcalX.$$ For every $ell=1,dots,n$, denote the diagonal morphism by $$delta_ell:widetildemathcalX_ell to mathcalY_ell,ell.$$ Denote the disjoint union of these morphisms by $$delta:widetildemathcalXto mathcalY.$$ For every $(j,ell)$, denote the involution of $B$-schemes that transposes the factors by $$sigma_j,ell:mathcalY_j,ellto mathcalY_ell,j.$$ Denote the disjoint union of these involutions by the involution, $$sigma:mathcalYto mathcalY, textpr_2circ sigma = textpr_1, textpr_1circ sigma = textpr_2.$$
For every ordered triple $(j,ell,r)$ with $1leq j,ell,rleq n$, denote by $$c_j,ell,r:mathcalY_j,elltimes_widetildemathcalX_ell mathcalY_ell,r to mathcalY_j,r,$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:mathcalYtimes_textpr_2,widetildemathcalX,textpr_1 mathcalY to mathcalY.$$
Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((pi_ell:widetildemathcalX_ell to B, ((textpr_1,(j,ell),textpr_2,(j,ell)):mathcalY_j,ellhookrightarrow widetildemathcalX_jtimes_BwidetildemathcalX_ell)_j,ell, $$ $$(delta_ell:widetildemathcalX_ellto Y_ell,ell)_ell, (sigma_j,ell:Y_j,ellto Y_ell,j)_j,ell, (c_j,ell,r:Y_j,elltimes_textpr_2,widetildeX_ell,textpr_1 Y_ell,rto Y_j,r)_j,ell,r)$$ together with a collection of locally closed immersions of $B$-schemes, $$e_ell:widetildemathcalX_ellhookrightarrow mathbbP^m_B.$$ The isomorphism locus is the maximal open subscheme $U_ell$ of $widetildeX_ell$ 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 $textpr_1$-inverse image in $Y_j,ell$ of $U_j$ is an open subscheme whose closed complement in $mathcalY_j,ell$ equals its total inverse image under $textpr_2$ of its closed image in $widetildemathcalX_ell$. Denote by $widetildemathcalX_ell,j$ the open complement in $widetildemathcalX_ell$ of this closed image.
Constraint 2. For each $ell=1,dots,n$, the collection of open subschemes $$(widetildemathcalX_ell,j)_j=1,dots,n,$$
form an open covering of $widetildemathcalX_ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.
Constraint 3. The isomorphism locus of each projection $$textpr_2,(j,ell):mathcalY_j,ell to widetildemathcalX_ell,$$ contains $widetildemathcalX_ell,j$. Thus, the inverse image of $widetildemathcalX_ell,j$ under this isomorphism equals the graph of a unique $k$-morphism, $$nu_ell,j:widetildemathcalX_ell,j to U_j.$$
Constraint 4. Denote by $U_ell,j$ the intersection of $U_ell$ and the $nu_ell,j$-inverse image of $U_j$ in $widetildemathcalX_ell,j$. For each triple $(j,ell,r)$, the inverse image in $U_ell,j$ under $nu_ell,j$ of $U_j,r$ equals $U_ell,jcap U_ell,r$. Denote this open by $U_ell,j,r$. Also, the inverse image in $widetildemathcalX_ell,j$ under $nu_ell,j$ of $U_jcap widetildemathcalX_j,r$ equals the inverse image in $widetildemathcalX_ell,r$ under $nu_ell,r$ of $U_rcap widetildemathcalX_r,j$. Denote this open by $widetildemathcalX_ell,j,r$.
Constraint 5. On the open $U_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ equals $nu_ell,r$. Similarly, on the open $widetildemathcalX_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ 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, $$((U_ell)_ell, (U_ell,jsubset U_ell)_ell,j, (nu_ell,j:U_ell,jto U_j)_ell,j)$$ together with the descent constraints for each Zariski descent datum of $B$-morphisms, $$(widetildeX_ell,jto U_j)_j.$$ 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 $widetildemathcalX_ell,jto U_ell$ is proper and surjective. Thus, the induced morphism $nu_j$ is proper and surjective. Since also $widetildemathcalX_j$ is proper over $B$, the $B$-scheme $mathcalX$ is proper. Since $mathcalX$ is covered by open $U_ell$ that are smooth $B$-schemes, also $mathcalX$ 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 $widetildeX_ell$, 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,$$(([widetildeX_ell])_ell,([Y_j,ell])_j,ell,([delta_ell])_ell,([sigma_j,ell])_j,ell, ([c_j,ell,r])_j,ell,r),$$ 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 $(n,m)in mathbbZ_geq 0times mathbbZ_geq 0$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetildeX_ell$ of $mathbbP^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetildeX_ell)_ell=1,dots,n$, 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 $widetildeX_jtimes_textSpec kwidetildeX_ell$. 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 $mathcalXto B$ 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 $iin I$, the Effectivity Proposition gives a smooth Chow covering. Denote this by $$(rho:mathcalX_i to B_i, ((nu_i,e_i):widetildemathcalX_i,ell hookrightarrow mathcalX_itimes_textSpec k mathbbP^m_mathbbC)_ell).$$
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 $mathcalX_i$ is itself a separated, quasi-compact, smooth $k$-scheme.
Certainly $mathcalX_i$ can be non-proper over $k$. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcalX_i$ as a dense open subscheme of a proper, smooth $k$-scheme $overlinemathcalX_i$. 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 $overlinemathcalX_i$.
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add a comment |
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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 $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetildeXto X$ such that $widetildeX$ 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, $$rho:mathcalXto B,$$ a Chow covering is an ordered $n$-tuple (for some integer $ngeq 0$) of pairs of $B$-morphisms, $$(nu_ell:widetildemathcalX_ell to mathcalX, e_ell:widetildemathcalX_ell to mathbbP^m_B),$$ 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 $mathcalX$.
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 $widetildemathcalX_ell$ 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 $widetildemathcalX$ the disjoint union of the $B$-schemes $widetildemathcalX_ell$. Denote by $nu:widetildemathcalXto mathcalX$ the unique $B$-morphism whose restriction to each component $widetildemathcalX_ell$ equals $nu_ell$. Denote by $mathcalY$ the disjoint union over all ordered pairs $(j,ell)$ with $1leq j,ellleq n$ of the closed subscheme of the product, $$mathcalY_j,ell:= widetildemathcalX_jtimes_mathcalX widetildemathcalX_ell subseteq widetildemathcalX_jtimes_B widetildemathcalX_ell,$$ together with its two projections, $$textpr_1,(j,ell):mathcalY_j,ellto widetildemathcalX_j, textpr_2,(j,ell):mathcalY_j,ellto widetildemathcalX_ell.$$ Denote the disjoint union of these morphisms by $$textpr_1:mathcalY to widetildemathcalX, textpr_2:mathcalYto widetildemathcalX.$$ For every $ell=1,dots,n$, denote the diagonal morphism by $$delta_ell:widetildemathcalX_ell to mathcalY_ell,ell.$$ Denote the disjoint union of these morphisms by $$delta:widetildemathcalXto mathcalY.$$ For every $(j,ell)$, denote the involution of $B$-schemes that transposes the factors by $$sigma_j,ell:mathcalY_j,ellto mathcalY_ell,j.$$ Denote the disjoint union of these involutions by the involution, $$sigma:mathcalYto mathcalY, textpr_2circ sigma = textpr_1, textpr_1circ sigma = textpr_2.$$
For every ordered triple $(j,ell,r)$ with $1leq j,ell,rleq n$, denote by $$c_j,ell,r:mathcalY_j,elltimes_widetildemathcalX_ell mathcalY_ell,r to mathcalY_j,r,$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:mathcalYtimes_textpr_2,widetildemathcalX,textpr_1 mathcalY to mathcalY.$$
Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((pi_ell:widetildemathcalX_ell to B, ((textpr_1,(j,ell),textpr_2,(j,ell)):mathcalY_j,ellhookrightarrow widetildemathcalX_jtimes_BwidetildemathcalX_ell)_j,ell, $$ $$(delta_ell:widetildemathcalX_ellto Y_ell,ell)_ell, (sigma_j,ell:Y_j,ellto Y_ell,j)_j,ell, (c_j,ell,r:Y_j,elltimes_textpr_2,widetildeX_ell,textpr_1 Y_ell,rto Y_j,r)_j,ell,r)$$ together with a collection of locally closed immersions of $B$-schemes, $$e_ell:widetildemathcalX_ellhookrightarrow mathbbP^m_B.$$ The isomorphism locus is the maximal open subscheme $U_ell$ of $widetildeX_ell$ 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 $textpr_1$-inverse image in $Y_j,ell$ of $U_j$ is an open subscheme whose closed complement in $mathcalY_j,ell$ equals its total inverse image under $textpr_2$ of its closed image in $widetildemathcalX_ell$. Denote by $widetildemathcalX_ell,j$ the open complement in $widetildemathcalX_ell$ of this closed image.
Constraint 2. For each $ell=1,dots,n$, the collection of open subschemes $$(widetildemathcalX_ell,j)_j=1,dots,n,$$
form an open covering of $widetildemathcalX_ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.
Constraint 3. The isomorphism locus of each projection $$textpr_2,(j,ell):mathcalY_j,ell to widetildemathcalX_ell,$$ contains $widetildemathcalX_ell,j$. Thus, the inverse image of $widetildemathcalX_ell,j$ under this isomorphism equals the graph of a unique $k$-morphism, $$nu_ell,j:widetildemathcalX_ell,j to U_j.$$
Constraint 4. Denote by $U_ell,j$ the intersection of $U_ell$ and the $nu_ell,j$-inverse image of $U_j$ in $widetildemathcalX_ell,j$. For each triple $(j,ell,r)$, the inverse image in $U_ell,j$ under $nu_ell,j$ of $U_j,r$ equals $U_ell,jcap U_ell,r$. Denote this open by $U_ell,j,r$. Also, the inverse image in $widetildemathcalX_ell,j$ under $nu_ell,j$ of $U_jcap widetildemathcalX_j,r$ equals the inverse image in $widetildemathcalX_ell,r$ under $nu_ell,r$ of $U_rcap widetildemathcalX_r,j$. Denote this open by $widetildemathcalX_ell,j,r$.
Constraint 5. On the open $U_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ equals $nu_ell,r$. Similarly, on the open $widetildemathcalX_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ 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, $$((U_ell)_ell, (U_ell,jsubset U_ell)_ell,j, (nu_ell,j:U_ell,jto U_j)_ell,j)$$ together with the descent constraints for each Zariski descent datum of $B$-morphisms, $$(widetildeX_ell,jto U_j)_j.$$ 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 $widetildemathcalX_ell,jto U_ell$ is proper and surjective. Thus, the induced morphism $nu_j$ is proper and surjective. Since also $widetildemathcalX_j$ is proper over $B$, the $B$-scheme $mathcalX$ is proper. Since $mathcalX$ is covered by open $U_ell$ that are smooth $B$-schemes, also $mathcalX$ 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 $widetildeX_ell$, 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,$$(([widetildeX_ell])_ell,([Y_j,ell])_j,ell,([delta_ell])_ell,([sigma_j,ell])_j,ell, ([c_j,ell,r])_j,ell,r),$$ 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 $(n,m)in mathbbZ_geq 0times mathbbZ_geq 0$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetildeX_ell$ of $mathbbP^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetildeX_ell)_ell=1,dots,n$, 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 $widetildeX_jtimes_textSpec kwidetildeX_ell$. 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 $mathcalXto B$ 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 $iin I$, the Effectivity Proposition gives a smooth Chow covering. Denote this by $$(rho:mathcalX_i to B_i, ((nu_i,e_i):widetildemathcalX_i,ell hookrightarrow mathcalX_itimes_textSpec k mathbbP^m_mathbbC)_ell).$$
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 $mathcalX_i$ is itself a separated, quasi-compact, smooth $k$-scheme.
Certainly $mathcalX_i$ can be non-proper over $k$. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcalX_i$ as a dense open subscheme of a proper, smooth $k$-scheme $overlinemathcalX_i$. 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 $overlinemathcalX_i$.
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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 $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetildeXto X$ such that $widetildeX$ 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, $$rho:mathcalXto B,$$ a Chow covering is an ordered $n$-tuple (for some integer $ngeq 0$) of pairs of $B$-morphisms, $$(nu_ell:widetildemathcalX_ell to mathcalX, e_ell:widetildemathcalX_ell to mathbbP^m_B),$$ 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 $mathcalX$.
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 $widetildemathcalX_ell$ 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 $widetildemathcalX$ the disjoint union of the $B$-schemes $widetildemathcalX_ell$. Denote by $nu:widetildemathcalXto mathcalX$ the unique $B$-morphism whose restriction to each component $widetildemathcalX_ell$ equals $nu_ell$. Denote by $mathcalY$ the disjoint union over all ordered pairs $(j,ell)$ with $1leq j,ellleq n$ of the closed subscheme of the product, $$mathcalY_j,ell:= widetildemathcalX_jtimes_mathcalX widetildemathcalX_ell subseteq widetildemathcalX_jtimes_B widetildemathcalX_ell,$$ together with its two projections, $$textpr_1,(j,ell):mathcalY_j,ellto widetildemathcalX_j, textpr_2,(j,ell):mathcalY_j,ellto widetildemathcalX_ell.$$ Denote the disjoint union of these morphisms by $$textpr_1:mathcalY to widetildemathcalX, textpr_2:mathcalYto widetildemathcalX.$$ For every $ell=1,dots,n$, denote the diagonal morphism by $$delta_ell:widetildemathcalX_ell to mathcalY_ell,ell.$$ Denote the disjoint union of these morphisms by $$delta:widetildemathcalXto mathcalY.$$ For every $(j,ell)$, denote the involution of $B$-schemes that transposes the factors by $$sigma_j,ell:mathcalY_j,ellto mathcalY_ell,j.$$ Denote the disjoint union of these involutions by the involution, $$sigma:mathcalYto mathcalY, textpr_2circ sigma = textpr_1, textpr_1circ sigma = textpr_2.$$
For every ordered triple $(j,ell,r)$ with $1leq j,ell,rleq n$, denote by $$c_j,ell,r:mathcalY_j,elltimes_widetildemathcalX_ell mathcalY_ell,r to mathcalY_j,r,$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:mathcalYtimes_textpr_2,widetildemathcalX,textpr_1 mathcalY to mathcalY.$$
Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((pi_ell:widetildemathcalX_ell to B, ((textpr_1,(j,ell),textpr_2,(j,ell)):mathcalY_j,ellhookrightarrow widetildemathcalX_jtimes_BwidetildemathcalX_ell)_j,ell, $$ $$(delta_ell:widetildemathcalX_ellto Y_ell,ell)_ell, (sigma_j,ell:Y_j,ellto Y_ell,j)_j,ell, (c_j,ell,r:Y_j,elltimes_textpr_2,widetildeX_ell,textpr_1 Y_ell,rto Y_j,r)_j,ell,r)$$ together with a collection of locally closed immersions of $B$-schemes, $$e_ell:widetildemathcalX_ellhookrightarrow mathbbP^m_B.$$ The isomorphism locus is the maximal open subscheme $U_ell$ of $widetildeX_ell$ 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 $textpr_1$-inverse image in $Y_j,ell$ of $U_j$ is an open subscheme whose closed complement in $mathcalY_j,ell$ equals its total inverse image under $textpr_2$ of its closed image in $widetildemathcalX_ell$. Denote by $widetildemathcalX_ell,j$ the open complement in $widetildemathcalX_ell$ of this closed image.
Constraint 2. For each $ell=1,dots,n$, the collection of open subschemes $$(widetildemathcalX_ell,j)_j=1,dots,n,$$
form an open covering of $widetildemathcalX_ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.
Constraint 3. The isomorphism locus of each projection $$textpr_2,(j,ell):mathcalY_j,ell to widetildemathcalX_ell,$$ contains $widetildemathcalX_ell,j$. Thus, the inverse image of $widetildemathcalX_ell,j$ under this isomorphism equals the graph of a unique $k$-morphism, $$nu_ell,j:widetildemathcalX_ell,j to U_j.$$
Constraint 4. Denote by $U_ell,j$ the intersection of $U_ell$ and the $nu_ell,j$-inverse image of $U_j$ in $widetildemathcalX_ell,j$. For each triple $(j,ell,r)$, the inverse image in $U_ell,j$ under $nu_ell,j$ of $U_j,r$ equals $U_ell,jcap U_ell,r$. Denote this open by $U_ell,j,r$. Also, the inverse image in $widetildemathcalX_ell,j$ under $nu_ell,j$ of $U_jcap widetildemathcalX_j,r$ equals the inverse image in $widetildemathcalX_ell,r$ under $nu_ell,r$ of $U_rcap widetildemathcalX_r,j$. Denote this open by $widetildemathcalX_ell,j,r$.
Constraint 5. On the open $U_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ equals $nu_ell,r$. Similarly, on the open $widetildemathcalX_ell,j,r$, the composition $nu_j,rcirc nu_ell,j$ 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, $$((U_ell)_ell, (U_ell,jsubset U_ell)_ell,j, (nu_ell,j:U_ell,jto U_j)_ell,j)$$ together with the descent constraints for each Zariski descent datum of $B$-morphisms, $$(widetildeX_ell,jto U_j)_j.$$ 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 $widetildemathcalX_ell,jto U_ell$ is proper and surjective. Thus, the induced morphism $nu_j$ is proper and surjective. Since also $widetildemathcalX_j$ is proper over $B$, the $B$-scheme $mathcalX$ is proper. Since $mathcalX$ is covered by open $U_ell$ that are smooth $B$-schemes, also $mathcalX$ 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 $widetildeX_ell$, 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,$$(([widetildeX_ell])_ell,([Y_j,ell])_j,ell,([delta_ell])_ell,([sigma_j,ell])_j,ell, ([c_j,ell,r])_j,ell,r),$$ 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 $(n,m)in mathbbZ_geq 0times mathbbZ_geq 0$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetildeX_ell$ of $mathbbP^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetildeX_ell)_ell=1,dots,n$, 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 $widetildeX_jtimes_textSpec kwidetildeX_ell$. 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 $mathcalXto B$ 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 $iin I$, the Effectivity Proposition gives a smooth Chow covering. Denote this by $$(rho:mathcalX_i to B_i, ((nu_i,e_i):widetildemathcalX_i,ell hookrightarrow mathcalX_itimes_textSpec k mathbbP^m_mathbbC)_ell).$$
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 $mathcalX_i$ is itself a separated, quasi-compact, smooth $k$-scheme.
Certainly $mathcalX_i$ can be non-proper over $k$. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcalX_i$ as a dense open subscheme of a proper, smooth $k$-scheme $overlinemathcalX_i$. 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 $overlinemathcalX_i$.
edited Mar 30 at 9:21
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Jason Starr
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$begingroup$
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 $(pi_i:X_ito B_i)_i$ of smooth, separated $mathbbC$-schemes such that every proper smooth $mathbbC$-scheme is a fiber of (at least) one morphism $pi_i$. Thus, that scheme embeds in $X_i$.
$endgroup$
– Jason Starr
Mar 29 at 18:05