PUSH-OUT OF SCHEMES AND SOME APPLICATIONS

Ke-Zheng Li

Graduate School of Academia Sinica/Max-Planck-Institut für Mathematik

0. Introduction.

Let p : Z → X and q : Z → Y be morphisms of S-schemes. A commutative diagram

\begin{matrix}Z & ⟶^{p} & X \\ ^{q}↓ & & ^{f}↓ \\ Y & ⟶^{g} & T \end{matrix}  (1)

over S is called a push-out if for any S-scheme T′ and any S-morphisms f′ : X → T′ and g′ : Y → T′ such that f′ ∘p = g′ ∘q, there is a unique morphism ϕ : T → T′ such that f′ = ϕ ∘ f and g′ = ϕ ∘ g. (in this case we also say T is a push-out of p and q). This is the dual concept of pull-back (i.e. fiber product) in the category of schemes. However, unlike pull-back, push-out usually does not exist.

But in many cases we need to see if a push-out exists. For example, let G be a group scheme over S, X be an S-scheme and ρ : G ×_{S} X → X be an action. Then the quotient X/G of ρ, if it exists, is just the push-out of ρ and pr_{2} : G×_{S} X → X. Another example comes from the construction of moduli schemes: a technical method is to construct a “catalogue scheme” first, and then descent to the moduli scheme by shrinking each locus corresponding to one equivalence class to one point. In other words, this is “dividing by an equivalence relation”, which is also a kind of push-out. One can also use push-out to show the existence of a “descent” of a morphism.

Of course the general answer to the existence of a push-out is hard. There have been many works about this, for example see [2], [3] or [20]. However, people have several quite different philoshphy along this line.

Typeset by AMS-TEX

In §1 we first give a criterion of push-out: for any two morphisms X → S and Y → S of finite type, if one of them is f.f. and the other is an epimorphism, then their pull-back is also a push-out. Then we give a necessary and sufficient condition for the exitence of push-out (1) when Z is a closed subscheme of X ×_{S} Y and p, q are both f.f. and relatively projective. We call this “flat projective case”. In this case the push-out is very “nice”, i.e. it is universal and geometric.

The non-flat or non-projevtive case is usually more complicated, and usually we can not hope the push-out to be very nice. We only have some partial results, which are included in §2. In particular, this gives a criterion of the existence of “rational push-out”.

As an application, in §3 we use the results of §1 to give a construction of Picard scheme in general case (i.e. for any flat family of projective varieties). In particular, when S = Spec(K) for a field K, this method does not need a field extension of K. We also give some other applications in §1 and §2.

Acknowledgements. I wish to thank A. Ogus and S. Mori for their very motivative discussions with me.

1. Push-out and some criteria.

Terminology: A morphism f : X → Y is called an epimorphism if f is settheoretically onto and O_{Y} → f_{∗}O_{X} is a monomorphism.

Definition. Let p : Z → X, q : Z → Y be two morphisms of schemes. If there is a scheme T together with two morphisms f : X → T and g : Y → T satisfying

a) The following diagram is commutative:

\begin{matrix}Z & ⟶^{p} & X \\ ^{q}↓ & & ^{f}↓ \\ Y & ⟶^{g} & T \end{matrix}  (1)

b) If there is another scheme T′ together with morphisms f′ : X → T′, g′ : Y → T′ such that

\begin{matrix}Z & ⟶^{p} & X \\ ^{q}↓ & ^{f′}↓ \\ Y & ⟶^{g′} & T′ \end{matrix}

is commutative, then there is a unique morphism ψ : T → T′ such that f′ = ψ ∘ f and g′ = ψ ∘ g,

then we say T (or (1), or (T, f, g)) is a (categorical) push-out of p and q. The push-out is called geometric if the induced morphism Z → X ×_{T} Y is epimorphic; called universal if for any morphism T′ → T, the following diagram

Z ×_{T} T′ ⟶^{p} X ×_{T} T′
q↓  ^{f}↓
Y ×_{T} T′ ⟶^{g} T′

is also a push-out.

Push-out is the dual concept of pull-back (i.e. fiber product) in the category of schemes. However, unlike pull-back, push-out usually does not exist.

Remark 1. Note that if (1) is a push-out, then

X ×_{T} Y ⟶^{pr_{1}} X
pr_{2}↓  ^{f}↓
Y  ⟶^{g} T

is also a push-out, hence we will often consider a push-out which is also a pull-back; furthermore, if ψ : Z′ → Z is an epimorphism, then

Z′ ⟶^{p∘ψ} X
q∘ψ↓  ^{f}↓
Y ⟶^{g} T

is also a push-out.

Example 1. Let G → S be a group scheme, X → S be a scheme and ρ : G×_{S}X → X be an action of G on X. An S-morphism p : X → Y is called a categorical quotient of ρ if the following diagram is a push-out

G ×_{S} X ⟶^{ρ} X
pr_{2}↓  ^{p}↓
X ⟶^{p} Y

In this case we denote Y = X/ρ (or Y = X/G if there is no confusion).

Example 2. More generally, let X → S be a scheme and W ⊂ X ×_{S} X be a locally closed subscheme. We say W gives an equivalence relation ∼_{W} of X over S if

i) (“reflexivity”) Δ(X) ⊂ W;

ii) (“symmetricity”) Let ι : X ×_{S} X → X ×_{S} X be the morphism by switching

factors, then ι(W) = W; iii) (“transitivity”) Let W_{ij} (i, j = 1 or 2) be the pull-back of pr_{i} : W → X and pr_{j} : W → X, viewed as a subscheme of X×_{S}X×_{S}X, then W_{11} = W_{12}.

In this case, an S-morphism p : X → Y is called a categorical equilizer of W (or a categorical quotient of ∼_{W}, denoted by Y = X/ ∼_{W}) if the following diagram is a push-out

\begin{matrix}W & ⟶^{pr_{1}} & X \\ ^{pr_{2}}↓ & & ^{p}↓ \\ X & ⟶^{p} & Y \end{matrix}  (2)

and Y (or p) is called a geometric equilizer of W (or a geometric quotient of ∼_{W}) if in addition that X ×_{Y} X = W as subschemes of X ×_{S} X (this is equivalent to that (2) is a geometric push-out; note that X ×_{Y} X is a closed subscheme of X ×_{S} X when Y is separated over S). A quotient Y = X/ ∼_{W} is called universal if for any morphism Y′ → Y, the equivalence relation of X′ = X ×_{Y} Y′ over S given by W′ = W ×_{Y} Y′ ⊂ X ×_{Y} X ×_{Y} Y′ ⊂ X′ ×_{S} X′ has a quotient X′/ ∼_{W′} ≅ (X/ ∼_{W}) ×_{Y} Y′ (this is equivalent to that (2) is a universal push-out).

Suppose W ⊂ X ×_{S} X gives an equivalence relation on X such that there is a geometric quotient X/ ∼_{W}. If X is separated over X/ ∼_{W}, then obviously W is a closed subscheme of X ×_{S} X. (In fact we mostly consider the case when W is a closed subscheme of X ×_{S} X.)

Lemma 1. Let S be a noetherian scheme, f : X → S and g : Y → S be morphisms of finite type, where g is f.f. and f is an epimorphism. Let p : X ×_{S} Y → X and q : X ×_{S} Y → Y be projections. Then S is a geometric push-out of p and q, and is universal if f is also flat.

Proof. Let f′ : X → S′ and g′ : Y → S′ be morphisms such that f′ ∘ p = g′ ∘ q. For any y ∈ Y, let s = g(y), then there exists x ∈ X such that f(x) = s (since f is an epimorphism); for any y′ ∈ g^{−1}(s) there exists z ∈ X ×_{S} Y such that p(z) = x and q(z) = y′, hence all of the points in g^{−1}(s) maps to the same point s′ = g′(y) ∈ S′; similarly all of the points in f^{−1}(s) maps to s′, therefore there is a (unique) induced map of sets ϕ : S → S′ satisfying set-theoretic equations ϕ ∘ g = g′ and ϕ ∘ f = f′. Since g is an open map (of topological spaces), we see ϕ is a continuous map of topological spaces.

Let U′ = SpecR′ ⊂ S′ be an affine open subscheme. For any s ∈ ϕ^{−1}(U′) and any x ∈ X, y ∈ Y such that f(x) = g(y) = s, there exists an affine open neighborhood U = SpecR ⊂ ϕ^{−1}(U′) of s, an affine open neighborhood V = SpecA ⊂ f^{−1}(U) of x and an affine open neighborhood W = SpecB ⊂ g^{−1}(U) of y such that g(W) = U and f^{∗} : R → A is a monomorphism (because f^{∗} : O_{s,S} → O_{x,X} is a monomorphism). Note that B is f.f. over R, i.e. g^{∗} : R → B is monomorphic and M = B/g^{∗}(R) is a flat R-module, hence we have a commutative diagram with exact rows

0 →R ⟶^{g^{∗}} B ⟶^{λ} M → 0
↓^{f^{∗}}  ↓^{q^{∗}}  ↓^{η}
0 →A ⟶^{p^{∗}} A ⊗_{R} B ⟶^{μ} A ⊗_{R} M→ 0

Since η is a monomorphism and η ∘ λ ∘ g^{′∗} = μ ∘ q^{∗} ∘ g^{′∗} = μ ∘ p^{∗} ∘ f^{′∗} = 0 : R′ → A ⊗_{R} M, we have λ ∘ g^{′∗} = 0 : R′ → M, hence there is a (unique) induced homomorphism ϕ^{∗} : R′ → R such that f^{∗} ∘ ϕ^{∗} = f^{′∗} and g^{∗} ∘ ϕ^{∗} = g^{′∗}. Therefore we can define ϕ as a morphism and we have equations of morphisms ϕ ∘ g = g′, ϕ ∘ g = g′. Q.E.D.

Example 3. Let S be a noetherian scheme, X be an S-scheme of finite type and G be a group scheme of finite type over S. An action ρ : G ×_{S} X → X of G on X is called transitive if (ρ, pr_{2}) : G ×_{S} X → X ×_{S} X is an epimorphism. If ρ has a geometric quotient X/G which is a locally closed subscheme of S, then ρ is transitive by definition; on the other hand, if X → S is f.f. and ρ is transitive, then there is a geometric quotient X/G ≅ S by Lemma 1 and Remark 1.

Proposition 1. Let S be a noetherian scheme, X → S be a relatively quasiprojective morphism and W ⊂ X ×_{S} X be a locally closed subscheme which gives an equivalence relation of X over S. Suppose that pr_{1} : W → X is relatively projective and flat. Then there is a universal geometric quotient Y = X/ ∼_{W} which is relatively quasi-projective (and relatively projective if X → S is so) over S. Furthermore, X → Y is f.f. and relatively projective.

Proof. Let X_{1} be a connected component of X and W_{1} = pr_{1}^{−1}(X_{1}) ⊂ W. Let X_{2} = pr_{2}(W_{1}) (this makes sense since pr_{2} is proper). Then by the definition of equivalence relation we see that pr_{1}^{−1}(X_{2}) = W ∩ X_{2} ×_{S} X_{2} which gives the restriction of ∼_{W} on X_{2}. Hence for simplicity we may assume X = X_{2}.

Take a relatively projective scheme X̄ → S which contains X as an open subscheme such that X → X̄ is dominant, and fix an OX̄(1). Then W is a closed subscheme of X̄ ×_{S} X since W → X is proper. The above assumption guarantees that the fibers of pr_{2} : W → X all have the same Hilbert polynomial χ. Let H = Hilb_{X̄/S}^{χ} be the Hilbert scheme representing the following functor:

((S-schemes)) → ((sets))

T ↦ {closed subschemes of X̄ ×_{S} T, flat over T with χ as the Hilbert polynomial of each fiber over T }

and Z ⊂ X̄ ×_{S} H be the universal subscheme. Then W induces an S-morphism ϕ : X → H such that W = Z ×_{H} X (as closed subschemes of X̄ ×_{S} X). Let Ȳ ⊂ H be the closed subscheme defined by the ideal sheaf ker(O_{H} → ϕ_{∗}O_{X}), and Z̄ = Z ×_{H} Ȳ ⊂ X̄ ×_{S} Ȳ. Then W = Z̄ ×Ȳ X. Denote by q : X → Ȳ, p_{1} : Z̄ → X̄, p_{2} : Z̄ → Ȳ and p : W → Z̄ the projections. Then p_{2} is f.f. and q is domoiant, hence p is dominant. Note that pr_{1} = p_{1} ∘ p : W → X̄, hence p_{1} is surjective (because it is dominant and proper).

The pull-back of Z̄ → Ȳ and q ∘ pr_{2} : W → Ȳ is equal to the pull-back of pr_{2} and pr_{2} (i.e. W_{22}, see Example 2), while the pull-back of Z̄ → Ȳ and q ∘ pr_{1} : W → Ȳ is equal to the pull-back of pr_{1} and pr_{2} (i.e. W_{12}). Since W_{12} = W_{22}, by the universality of H we have q ∘ pr_{1} = q ∘ pr_{2} : W → Ȳ. Hence

q ∘ p_{1} ∘ p = q ∘ pr_{1} = q ∘ pr_{2} = p_{2} ∘ p  (3)

Let Z = p_{1}^{−1}(X). By abuse of notation we still denote by p_{1} : Z → X, p_{2} : Z → Ȳ and p : W → Z the projections. Since p : W → Z is an epimorphism (because it is proper and dominant), (3) gives

q ∘ p_{1} = p_{2} : Z → Ȳ  (4)

Hence p : W ≅ Z ×Ȳ X → Z has a section. Since p_{2} : Z̄ → Ȳ is an open map we can define Y = p_{2}(Z) = q(X) which is an open dense subscheme of Ȳ. Furthermore we have W ⊂ X ×_{Y} X, hence in X ×_{S} X ×_{S} X we have

\begin{aligned}Z ×_{Y} W & ⊂ Z ×_{Y} (X ×_{Y} X) = W ×_{Y} X = W ×_{Z} (Z ×_{Y} X) \\ & = W ×_{Z} W ⊂ W ×_{X} W = Z ×_{Y} W \end{aligned}

i.e. Z ×_{Y} W = Z ×_{Y} (X ×_{Y} X), thus W = X ×_{Y} X because Z is f.f. over Y. Therefore

Z ×_{Y} W = W_{21} = W_{12} = W ×_{Y} Z = X ×_{Y} X ×_{Y} Z = X ×_{Y} W  (5)

Now (5)×_{W} Z (via the section Z → W of p) gives

Z ×_{Y} Z ≅ X ×_{Y} Z  (6)

hence p_{1} : Z → X is an isomorphism because Z is f.f. over Y.

Finally, let Z′ = p_{2}^{−1}(Y), then W = Z′ ×_{Y} X, hence W is f.f. over Z′. This shows Z′ = Z ≅ X, hence W = X ×_{Y} X. We see that Y is a universal geometric quotient of ∼_{W} by Lemma 1. Q.E.D.

Let X, Y be S-schemes and W ⊂ X ×_{S} Y be a locally closed subscheme. If pr_{1} : W → X and pr_{2} : W → Y has a geometric push-out Z, then by definition

W →^{≃} X ×_{Z} Y, hence

W ×_{X} W ×_{Y} W = W ×_{Y} W ×_{X} W ⊂ X ×_{S} X ×_{S} Y ×_{S} Y  (7)

because both sides are equal to X ×_{Z} X ×_{Z} Y ×_{Z} Y.

Proposition 2. Let S be a noetherian scheme, X → S and Y → S be relatively quasi-projective morphisms and W ⊂ X ×_{S} Y be a locally closed subscheme satisfying (7). Suppose that pr_{1} : W → X pr_{2} : W → Y are f.f. and relatively projective. Then there is a universal geometric push-out Z of pr_{1} and pr_{2} which is relatively quasi-projective (and relatively projective if X → S and Y → S are so) over S. Furthermore, X → Z and Y → Z are f.f. and relatively projective.

Proof. In the language of presheaves (for simplicity denote \underline{X}(T) = Mor_{S}(T, X) etc.), (7) can be expressed as: For any S-scheme T and any x, x′ ∈ \underline{X}(T) and y, y′ ∈ \underline{Y} (T), if (x, y), (x, y′), (x′, y′) ∈ \underline{W} (T), then (x′, y) ∈ \underline{W} (T). We will use this language in the following.

Let X_{1} = W ×_{X} W, viewed as a (relatively projective) scheme over Y_{2} = Y ×_{S} Y. Let W_{1} = X_{1}×_{Y} W. Then it is easy to see (by the language of presheaves) that

W_{1} = X_{1} ×_{Y_{2}} X_{1}  (8)

hence gives an equivalence relation on X_{1} over Y_{2}. Furthermore, pr_{1} : W_{1} → X_{1} is f.f. and relatively projective, hence by Proposition 1 there is a universal geometric quotient Y_{3} = X_{1}/ ∼_{W_{1}}, and X_{1} → Y_{3} is f.f. and relatively projective. By (8) we have

X_{1} ×_{Y_{3}} X_{1} = W_{1} = X_{1} ×_{Y_{3}} (Y_{3} ×_{Y_{2}} X_{1})  (9)

hence X_{1} ≅ Y_{3} ×_{Y_{2}} X_{1} because X_{1} is f.f. over Y_{3}, by the same reason we see Δ : Y_{3} → Y_{3} ×_{Y_{2}} Y_{3} is an isomorphism. This shows Y_{3} → Y_{2} is a closed immersion (a proper morphism T → T′ is a closed immersion iff Δ : T → T ×_{T′} T is an isomorphism). Furthermore, the following commutative diagram

\begin{matrix}X_{1} & ⟶^{pr_{2}} & W \\ ↓ & & ↓ \\ Y_{3} & ⟶^{pr_{2}} & Y \end{matrix}  (10)

is a pull-back, i.e. X_{1} ≅ Y_{3} ×_{Y} W, this comes from

X_{1} ×_{Y_{3}} X_{1} ≅ W_{1} = X_{1} ×_{Y} W ≅ X_{1} ×_{Y_{3}} (Y_{3} ×_{Y} W)  (11)

(and that X_{1} → Y_{3} is f.f.). Hence pr_{2} : Y_{3} → Y is f.f. and relatively projective (since the composition Y_{3} ↪ Y_{2} → Y is quasi-projective and proper).

We now show that Y_{3} gives an equivalence relation of Y over S. The commutativity is obvious. For the reflexivity, note that the composition

W →^{Δ} W ×_{X} W = X_{1} → Y_{3} → Y_{2}

is equal to the composition W → Y →^{Δ} Y_{2}, where W → Y is an epimorphism. For the transitivity, we need to show Y_{311} = Y_{312} ⊂ Y ×_{S} Y ×_{S} Y, where Y_{3ij} (i, j = 1 or 2) is the pull-back of pr_{i} : Y_{3} → Y and pr_{j} : Y_{3} → Y. It is enough to show that X_{1} ×_{Y_{3}} Y_{311} = X_{1} ×_{Y_{3}} Y_{312} in X ×_{S} Y ×_{S} Y ×_{S} Y. But by (10) we have

X_{1} ×_{Y_{3}} Y_{311} ≅ X_{1} ×_{Y} Y_{3} ≅ W ×_{X} W ×_{Y} Y_{3} ≅ W ×_{X} X_{1} ≅ W ×_{X} W ×_{X} W

similarly we have X_{1} ×_{Y_{3}} Y_{311} ≅ W ×_{X} W ×_{X} W, hence we get an isomorphism X_{1} ×_{Y_{3}} Y_{311} → X_{1} ×_{Y_{3}} Y_{312} compatible with the inclusions to X ×_{S} Y ×_{S} Y ×_{S} Y.

By Proposition 1 there is a universal geometric quotient Z = Y/ ∼_{Y_{3}} which is relatively quasi-projective over S, and Y → Z is f.f. and relatively projective. By (10) we see that W → Z equilizes pr_{1} and pr_{2} : X_{1} → W, hence W → Z factors through W/ ∼_{X_{1}} ≅ X. Since W → X and W → Z are both f.f. and relatively projective, we see that X → Z is f.f. and relatively projective. Furthermore,

\begin{aligned}W ×_{X} (X ×_{Z} Y) & ≅ W ×_{Z} Y ≅ W ×_{Y} (Y ×_{Z} Y) \\ & ≅ W ×_{Y} Y_{3} ≅ X_{1} ≅ W ×_{X} W \end{aligned}  (12)

Hence W → X ×_{Z} Y is an isomorphism. Q.E.D.

2. Some results for non-flat case.

Proposition 1. Let S be a noetherian scheme, X → S and Y → S be relatively quasi-projective morphisms and W ⊂ X ×_{S} Y be a locally closed subscheme satisfying (7) of §1. Suppose that pr_{1} : W → X and pr_{2} : W → Y are relatively projective, pr_{1} : W → X is f.f. and pr_{2} : W → Y is an epimorphism. Then pr_{2} : W → Y factors through an epimorphism W → Z, where Z → Y is finite and set-theoretically one to one, such that there is a geometric push-out

\begin{matrix}W & ⟶^{pr_{2}} & Z \\ ↓^{pr_{1}} & ↓ \\ X & ⟶ & T \end{matrix}  (1)

where T is relatively quasi-projective (and relatively projective if X → S and Y → S are so) over S, X → T and Z → T are relatively projective, Z → T is f.f., and X → T is an epimorphism.

Proof. Take a relatively projective scheme Ȳ → S which contains Y as an open subscheme such that Y → Ȳ is dominant, and fix an OȲ (1). Then W is a closed subscheme of X ×_{S} Ȳ since W → X is proper. Let H = Hilb_{Ȳ /S} be the Hilbert scheme representing the following functor:

((S-schemes)) → ((sets)) S′ ↦ {closed subschemes of Ȳ ×_{S} S′, flat over S′}

and Z ⊂ Ȳ ×_{S} H be the universal subscheme. Then W induces an S-morphism ϕ : X → H such that W = X ×_{H} Z (as closed subschemes of X ×_{S} Ȳ). Let T̄ ⊂ H be the closed subscheme defined by the ideal sheaf ker(O_{H} → ϕ_{∗}O_{X}), and Z̄ = Z ×_{H} T̄ ⊂ Ȳ ×_{S} T̄. Then W = X ×T̄ Z̄. Denote by q : X → T̄, p_{1} : Z̄ → Ȳ, p_{2} : Z̄ → T̄ and p : W → Z̄ the projections. Then p_{2} is f.f. and q is dominant, hence p is dominant. Note that pr_{2} = p_{1} ∘ p : W → Ȳ, hence p_{1} is surjective (because it is dominant and proper). Let Z = p_{1}^{−1}(Y), then Z → Y is relatively projective (because Z̄ → Ȳ is so), and W → Z is an epimorphism (because it is proper and dominant).

The pull-back of Z̄ → T̄ and q ∘ pr_{1} : W → T̄ is equal to Z ×T̄ W ≅ (Z ×T̄ X) ×_{X} W ≅ W ×_{X} W, hence the pull-back of Z̄ → T̄ and q ∘ pr_{1} ∘ pr_{1} : W ×_{Y} W → T̄ is equal to W ×_{X} W ×_{Y} W, and the pull-back of Z̄ → T̄ and q ∘ pr_{1} ∘ pr_{2} : W ×_{Y} W → T̄ is equal to W ×_{Y} W ×_{X} W. By abstract nonsense (and the assumption that W ×_{X} W ×_{Y} W = W ×_{Y} W ×_{X} W), we see the following diagram is commutative:

\begin{matrix}W ×_{Y} W & ⟶^{pr_{2}} & W \\ ↓^{pr_{1}} & & ↓ \\ W & ⟶ & T̄ \end{matrix}  (2)

This induces a closed immersion W ×_{Y} W → W ×T̄ W.

We now show that W ×_{Z} W → W ×_{Y} W is an isomorphism. Since W → X is f.f., it is enough to show that W ×_{Z} W ×_{X} W ≅ W ×_{Y} W ×_{X} W. Note that W ×_{Z} W ×_{X} W ≅ W ×T̄ W, hence we have a closed immersion

η : W ×_{Y} W ×_{X} W ↪ W ×T̄ W ×_{X} W →^{≃} W ×_{Z} W ×_{X} W ×_{X} W  (3)

and pr_{1} ∘η = pr_{1}, pr_{3} ∘η = pr_{2}, pr_{4} ∘η = pr_{3}: W ×_{Y} W ×_{X} W → W. Furthermore, we have pr_{2} ∘ η = pr_{2} because pr_{1} ∘ pr_{2} ∘ η = pr_{1} ∘ pr_{2}: W ×_{Y} W ×_{X} W → X and pr_{2} ∘ pr_{2} ∘ η = pr_{2} ∘ pr_{1} = pr_{2} ∘ pr_{2}: W ×_{Y} W ×_{X} W → Y. Thus we have a commutative diagram

\begin{matrix}W ×_{Y} W ×_{X} W & ⟶^{η} & W ×_{Z} W ×_{X} W ×_{X} W \\ ↓^{ζ} & & ↓ \\ W ×_{Z} W ×_{X} W & ⟶^{id_{W} ×_{Z}Δ×_{X}id_{W}} & W ×_{Z} W ×_{X} W ×_{X} W \end{matrix}  (4)

where Δ : W → W ×_{X} W is the diagonal morphism. Hence ζ is a closed immersion. It is easy to see that the composition of ζ with the closed immersion W ×_{Z} W ×_{X} W → W ×_{Y} W ×_{X} W is equal to the identity morphism of W ×_{Z} W ×_{X} W, hence ζ is an isomorphism.

For a closed point t : Spec(K) ↪ T (where K is a field), denote by W_{t} (resp. X_{t}, Z_{t}) the fiber of W (resp. X, Z) over t. Then W_{t} ≅ X_{t} ×_{K} Z_{t}, and W_{t} ×_{Y} W_{t} ⊂ W ×_{Y} W_{t} = W ×_{Z} W_{t} = W_{t} ×_{Z} W_{t}, hence W_{t} ×_{Y} W_{t} = W_{t} ×_{Z} W_{t} = W_{t} ×_{Z_{t}} W_{t}. Note that the following commutative diagram

\begin{matrix}W_{t} ×_{Z_{t}} W_{t} & ⟶^{≃} & W_{t} ×_{Y} W_{t} \\ ↓ & & ↓ \\ Z_{t} & ⟶^{Δ} & Z_{t} ×_{Y} Z_{t} \end{matrix}  (5)

is a pull-back, and W_{t} ×_{Y} W_{t} → Z_{t} ×_{Y} Z_{t} is f.f., we see Δ : Z_{t} → Z_{t} ×_{Y} Z_{t} is an isomorphism, hence Z_{t} → Y is a closed immersion. This shows that p_{1} : Z → Y is set-theoretically one to one, hence finite.

The other statements hold by Lemma 1.1. Q.E.D.

Remark 1. In many cases we can take Z = Y, for example when W is flat over Y (see Proposition 2). This is also the case when W is reduced and Y is normal, as shown in the following corollary.

Corollary 1. Under the conditions of Proposition 1, if in addition that W is reduced and Y is normal, then Z ≅ Y.

Proof. The reducedness of W implies that Z and T are reduced. Let t ∈ T be a generic point. Then by the argument in the proof of Proposition 1 we see that the diagonal morphism Z_{t} → Z_{t} ×_{Y} Z_{t} is an isomorphism. Let I be the ideal sheaf of Δ : Z → Z ×_{Y} Z, then I_{t} = 0, hence there is an open neighborhood U_{t} ⊂ T of t such that I|_{Z′×_{Y} Z′} = 0, where Z′ = p_{2}^{−1}(U_{t}) (because I is coherent). Let Y′ = Y − p_{1}(Z − Z′) (where p_{1} : Z → Y is as in the proof of Proposition 1), then Y′ is an open subscheme of Y and p_{1}^{−1}(Y′) = Z′. Since the diagonal morphism Z′ → Z′ ×_{Y′} Z′ is an isomorphism, we have Z′ ≅ Y′.

Therefore for each irreducible component Z_{0} of Z, we have K(Z_{0})≅K(p_{1}(Z_{0})). But Z → Y is finite and Y is normal by assumption, we see Z_{0} → p_{1}(Z_{0}) is an isomorphism, hence Z ≅ Y. Q.E.D.

Remark 2. In Proposition 1.1, it is essential to assume that W → X is both relatively projective and flat. However, without these assumptions, at least ∼_{W} has a “rational quotient” when X is reduced, as shown in the following corollary.

Corollary 2. Let S be a noetherian scheme, X be a reduced scheme, X → S be a relatively quasi-projective morphism and W ⊂ X ×_{S} X be a locally closed subscheme defining an equivalence relation. Then there is an open dense subscheme X′ ⊂ X such that the equivalence relation of X′ given by W ∩ X′ ×_{S} X′ has a universal geometric quotient. Furthermore, X′/ ∼_{W′}→ S is relatively quasiprojective and X′ → X′/ ∼_{W′} is f.f. and relatively quasi-projective.

Proof. For simplicity we may assume X → S is relatively projective and W is closed. Hence there is an open dense subscheme X′ ⊂ X such that W |_{X′} is relatively projective and flat over X′. As in the proof of Proposition 1.1, we may assume that under a choice of O_{X}(1), this induces a morphism of finite type ϕ : X′ → H = Hilb_{X/S}^{χ} for some Hilbert polynomial χ. By further shrinking X′ we may assume ϕ maps X′ onto a locally closed subscheme Y′ ⊂ H. Let W′ = W ∩ X′ ×_{S} X′. Then by the proof of Proposition 1 we have W′ = X′ ×_{Y′} X′ and X′ → Y′ is f.f., hence Y′ is a universal geometric quotient of ∼_{W′} by Lemma 1.1. Q.E.D.

Example 1. Let k be an algebraically closed field, A be an abelian variety over k, X be a variety over k, and ρ : A ×_{k} X → X be an action. If ρ is free, then it has a universal geometric quotient by Proposition 1.1. More generally, let H be the pull-back of

H ⟶ X
↓  ↓^{Δ}
A ×_{k} X ⟶^{(ρ,pr_{2})} X ×_{k} X

which is a closed subgroup scheme of A ×_{k} X over X (called the stabilizer of ρ). If H is flat over X, then ρ has a universal geometric quotient by Proposition 1.1.

If H is not flat over X, at least ρ has a rational quotient by Corollary 2.

3. An application: A construction of Picard schemes

Let S be a noetherian scheme and π : X → S be a morphism of finite type. A closed subscheme D ⊂ X is called an S-effective divisor if D is flat over S and the ideal sheaf I of D is an invertible sheaf. In this case denote L(D) = I^{−1}.

Suppose π is f.f. and relatively projective with a fixed O_{X}(1), D is an Seffective divisor and L = L(D). If S is connected, then the fibers of D over S all have the same Hilbert polynomial χ. By Castelnuovo and Mumford (see [12]), there exists an integer N, depending only on χ, such that for any m > N, π_{∗}L(m) is locally free of rank χ(m) and generate L(m).

For any Hilbert polynomial χ, there is an open subscheme Div_{X/S}^{χ} of Hilb_{X/S}^{χ} representing the following functor:

((locally noetherian S-schemes))→((sets))

T ↦ {T -effective divisors D ⊂ X ×_{S} T whose fibers have Hilbert polynomial χ}

Let S = Spec(R), where R is a noetherian local ring, and X → S be a f.f. relatively projective morphism with geometrically integral fibers. Let X_{0} be the fiber of X over the closed point s_{0} ∈ S. Fix an O_{X}(1). Let ξ be a numerical class of X_{0}. For simplicity we use the following terminology: For any S-scheme T whose closed points are all over s_{0} and any invertible sheaf L on X ×_{S} T, if for any closed point t ∈ T the fiber L_{t} is in the numerical class ξ, then we say L is in the numerical class ξ.

Lemma 1. Let S be a noetherian local scheme, X → S be a f.f. relatively projective morphism with geometrically integral fibers, and X_{0} be the fiber of X over the closed point of S. Then for any numerical class ξ_{0} of X_{0}, there exists n > 0 such that for ξ = ξ_{0}(n),

(∗) for any noetherian S-scheme T and any invertible sheaf L on X ×_{S} T in the numerical class ξ, E = pr_{2∗}L is finitely generated locally free sheaf of O_{T} - modules which generates L, and for any t ∈ T we have E_{t} ≅ Γ(X ×_{S} {t}, L_{t}).

We omit the proof which needs some criterion of ampleness and semicontinuity theory (cf. [20]).

Suppose π : X → S is f.f. and relatively projective with geometrically integral fibers, E = π_{∗}L is locally free generating L, and E_{s} ≅ π_{∗}L_{s} for any s ∈ S. Let O_{S} → E be an O_{S}-monomorphism such that E/f(O_{S}) is locally free, then f induces an O_{X}-monomorphism O_{X}L, hence induces an O_{X}-monomorphism f′ : L^{−1} → O_{X}. By local criterion of flatness, we see f′ is a monomorphism and O_{X}/im(f′) is O_{S}-flat, hence defines an S-effective divisor D such that L(D) ≅ L. Conversely, if D ⊂ X is an S-effective divisor satisfying L(D) ≅ L, then D gives an O_{X}-monomorphism f′ : L^{−1} → O_{X} which is equivalent to an O_{X}-monomorphism O_{X} → L, hence induces an O_{S}-monomorphism f : O_{S} → E. By local criterion of flatness, we see that E/f(O_{S}) is O_{S}-locally free. Note that such an f is equivalent to a section of ℙ(E^{∨}) → S, we get:

Lemma 2. Let π and L be as above. Then ℙ(E^{∨}) represents the following functor:

((locally noetherian S-schemes))→((sets))

(ϕ : T → S) ↦ {T -effective divisors D ⊂ X ×_{S} T

such that L(D) ≅ (id_{X} ×_{S} ϕ)^{∗}L}

Lemma 3 ([13, III.10. Proposition]). Let S be a noetherian scheme and π : X → S be an f.f. relatively projective morphism with geometrically reduced and connected fibers. Let L be an invertible sheaf on X. Then there exists a closed subscheme S_{0} ⊂ S representing the following functor:

((locally noetherian schemes))→((sets))

T ↦ {morphisms T → S such that (id_{X} ×_{S} f)^{∗}L is locally trivial over T }

Proof. Let X_{0} be a reduced connected projective scheme over a field K, and L_{0} be an invertible sheaf on X_{0}. Then it is easy to see that L_{0} ≅ O_{X_{0}} iff Γ(X_{0}, L_{0}) and Γ(X_{0}, L_{0}^{−1}) are both non-zero.

For simplicity we may assume S = SpecR. Let

S̃_{0} = {s ∈ S|Γ(X_{s}, L_{s}) ≠ 0, Γ(X_{s}, L_{s}^{−1}) ≠ 0}

Then S̃_{0} is a closed subset of S by semicontinuity theory. Furthermore, there is an R-homomorphism of finitely generated free R-modules ϕ : M_{1} → M_{2} such that for any morphism f : T = SpecA → S we have

Γ(X ×_{S} T, (id_{X} ×_{S} f)^{∗}L) ≅ ker(ϕ ⊗_{R} id_{A} : M_{1} ⊗_{R} A → M_{2} ⊗_{R} A)

Let M = coker(ϕ^{∨} : M_{2}^{∨} → M_{1}^{∨}), then ker(ϕ ⊗_{R} id_{A}) ≅ Hom_{R}(M, A). In particular, for any maximal ideal P ⊂ R in S̃_{0}, we have M/P M ≅ R/P, hence there is an open neighborhood U ⊂ S of S̃_{0} such that M is locally generated by one element on U. Note that if (id_{X} ×_{S} f)^{∗}L is locally trivial over T, then f maps the points of T into S̃_{0}. Hence by localization we may assume M ≅ R/I for some ideal I ⊂ R.

We check that S_{0} = Spec(R/I) = V (I) ⊂ S satisfies our requirement. Indeed, for any f as above, let J = {ψ(1)|ψ ∈ Hom_{R}(R/I, A)}, then J is an ideal of A. If J ≠ A, take a maximal ideal P ⊃ J in A, then Hom_{R}(R/I, A) → Hom_{R}(R/I, A/P) is zero homomorphism, hence (id_{X} ×_{S} f)^{∗}L is not trivial near P. Note that J = A iff f^{∗}(I) = 0, and in this case Hom_{R}(R/I, A) ≅ A, which induces (id_{X} ×_{S} f)^{∗}L ≅ O_{X×_{S}T}. Q.E.D.

Lemma 4. Let S be a noetherian scheme, π : X → S be a smooth relatively projective morphism and D ⊂ X be a closed subscheme which is flat over S. Suppose there is an open dense subscheme U ⊂ S such that D ×_{S} U ⊂ X ×_{S} U is a U-effective divisor. Then D ⊂ X is an S-effective divisor.

Proof. Case 1: S is regular. For simplicity assume S = SpecR. Let x ∈ X be a closed point and s = π(x) corresponding to maximal ideal m ⊂ R. Then A = O_{x,X} is a regular local ring, hence a UFD. Let I be the stalk of the ideal sheaf of D ⊂ X at x, and a be the greatest common divisor of the elements in I. Then a ∉ mA since I ⊄ mA. Hence there is an open neighborhood U′ ⊂ X of x such that (a) defines a divisor D_{1} in U′, flat over S. Let D_{2} ⊂ U′ be the closed subscheme defined by J = (I : a). By the exact sequence

0 → J → A →^{a·} A/I → A/(a) → 0

we see A/J is R-flat, i.e. D_{2} is flat over S. By assumption it is easy to see that D_{2} ×_{S} U is a divisor of U′ ×_{S} U, hence any non-empty fiber of D_{2} over U has dimension equal to the relative dimension of π minus 1; On the other hand, the fiber of D_{2} over π(x) has dimension less than the relative dimension of π minus 1. Since the fiber dimension of a flat morphism equals the relative dimension, the only possibility is D_{2} = ∅, i.e. I = (a). Hence D is a divisor in X.

Case 2: S is of finite type over ℤ. Again assume S = SpecR. By local criterion of flatness, we need only show that for any s ∈ S, the fiber D_{s} is a divisor in X_{s}. Let K be the residue field at s. We can take an integral one-dimensional subscheme V ⊂ S containing s such that V ∩ U ≠ ∅. Let V′ be the normalization of V, then V′ is regular. By Case 1 we see that D ×_{S} V′ is a divisor in X ×_{S} V′. Hence there exists a finite extension K′ ⊃ K such that D_{s} ⊗_{K} K′ is a divisor in X_{s} ⊗_{K} K′. Therefore the ideal sheaf of D_{s} ⊂ X_{s} is a flat O_{X_{s}}-module, i.e. D_{s} is a divisor in X_{s}.

Case 3: General case. We may assume S = SpecR and S is connected. Take a closed immersion X ⊂ ℙ_{S}^{n} for some n. Let χ_{0} and χ be the Hilbert polynomials of the fibers of D and X over S respectively. Then there is an induced morphism from S to the closed subscheme of Hilb_{ℙ^{n}}^{χ_{0}} × Hilb_{ℙ^{n}}^{χ} representing the following functor

((locally noetherian schemes))→((sets))

T ↦ {pairs of closed subschemes D_{0} ⊂ X_{0} ⊂ ℙ_{T}^{n}, flat over T, with χ_{0} and χ as Hilbert polynomials of fibers respectively}

This gives a finitely generated ℤ-subalgebra R′ ⊂ R such that D ⊂ X is defined over R′, i.e. there are projective schemes D′ ⊂ X′ over S′ = SpecR′ whose pullback to S is isomorphic to D ⊂ X. We may choose R′ such that X′ → S′ is smooth and D′ is a divisor over an open dense subscheme of S′. Hence D′ ⊂ X′ is a divisor by Case 2. Therefore D ⊂ X is a divisor. Q.E.D.

Note that Hilb_{X/S}^{χ} is relatively projective over S, we get:

Corollary 1. If π : X → S is relatively projective and smooth, then Div_{X/S}^{χ} is relatively projective over S.

For any scheme X denote by Pic(X) the Picard group of X (i.e. the group of isomorphism classes of invertible sheaves on X). For any morphism π : X → S there is an induced homomorphism π^{∗} : Pic(S) → Pic(X). If π is f.f. and relatively projective, then Pic(S) can be viewed as a subgroup of Pic(X) via π^{∗}. We call Pic(X/S) := Pic(X)/Pic(S) the relative Picard group of π. Furthermore, if in addition that S is local and X → S has geometrically integral fibers, letting X_{0} be the fiber of X over the closed point of S, then for any numerical class ξ of X_{0} we denote by Pic^{ξ}(X/S) the subset of Pic(X/S) consisting of invertible sheaves with numerical class ξ. Similarly we can also use the notation Div^{ξ}(X/S).

Theorem. Let π : X → S be a f.f. relatively projective morphism with geometrically integral fibers, and with a section ζ : S → X. Then there exists a locally quasi-projective scheme Pic(X/S) over S representing the following functor:

𝔓𝔦𝔠(X/S): ((locally noetherian S-schemes))→((sets))

T ↦ Pic(X ×_{S} T/T)

Furthermore, if π is smooth, then Pic(X/S) is locally projective over S.

Proof. For any S-scheme T and any invertible sheaf L_{0} on X ×_{S} T, the invertible sheaf L = L_{0} ⊗_{O_{X×_{S} T}} ((ζ ∘ π) ×_{S} id_{T})^{∗}L_{0}^{−1} satisfies (ζ ×_{S} id_{T})^{∗}L ≅ O_{T}, Hence every element of Pic(X ×_{S} T/T) can be represented by an invertible sheaf L on X ×_{S} T satisfying (ζ ×_{S} id_{T})^{∗}L ≅ O_{T}. In this way Pic(X ×_{S} T/T) is embedded into Pic(X ×_{S} T) as a subgroup. In the following Pic(X ×_{S} T/T) will be understood as this.

First consider the case when S is local, with closed point s_{0}. Let X_{0} be the fiber of X over the closed point of S. Let ξ be a numerical class of X_{0} satisfying (∗). Let Y_{1} = Div_{X/S}^{ξ} representing divisors with numerical class ξ (which is an open and closed subscheme of some Div_{X/S}^{χ}). By Lemma 3 there is a closed subscheme Y_{2} ⊂ Y_{1} ×_{S} Y_{1} representing the following functor:

( ( \begin{matrix}locally noetherian S-schemes \\ whose closed fibers are over s_{0} \end{matrix} ) ) →((sets))

(ϕ : T → S) ↦ {pairs D_{1}, D_{2} ∈ Div^{ξ}(X ×_{S} T/T)

such that L(D_{1}) ≅ L(D_{2})}

and pr_{1} : Y_{2} → Y_{1} is a projective space bundle by Lemma 2. Clearly Y_{2} gives an equivalence relation of Y_{1} over S. Hence by Proposition 1.1, pr_{1} and pr_{2} : Y_{2} → Y_{1} have an equilizer p : Y_{1} → Y = Y_{1}/ ∼_{Y_{2}} which is f.f. and relatively projective, where Y is relatively quasi-projective over S, and Y_{2} = Y_{1} ×_{Y} Y_{1}. We are going to show that Y represents the following functor:

𝔓𝔦𝔠^{ξ}(X/S): ( ( \begin{matrix}locally noetherian S-schemes \\ whose closed fibers are over s_{0} \end{matrix} ) ) →((sets))
T ↦ Pic^{ξ}(X ×_{S} T/T)

Let Y_{3} = Y_{2} ×_{Y_{1}} Y_{2} ≅ Y_{1} ×_{Y} Y_{1} ×_{Y} Y_{1} (each copy of Y_{2} is viewed as a Y_{1}- scheme via pr_{2}). Let D_{1} ⊂ X ×_{S} Y_{1} be the universal divisor, ζ_{1} = ζ ×_{S} id_{Y_{1}} : Y_{1} → X ×_{S} Y_{1}, and L_{1} = L(D_{1}) ⊗_{O_{Y_{1}}} (ζ_{1} ∘ pr_{2})^{∗}L(D_{1})^{−1}. Then ζ_{1}^{∗}L_{1} ≅ O_{Y_{1}} (i.e. L_{1} ∈ Pic^{ξ}(X ×_{S} Y_{1}/Y_{1})) and L_{2} := pr_{12}^{∗}L_{1} ≅ pr_{13}^{∗}L_{1}. Let E_{1} = pr_{2∗}L_{1} and E_{2} = pr_{2∗}L_{2}.

Then ζ induces a canonical epimorphism η_{1} : E_{1} → ζ_{1}^{∗}L_{1} ≅ O_{Y_{1}} and similarly a canonical epimorphism η_{2} : E_{2} → O_{Y_{2}}. By Lemma 2 and (∗) Y_{2} ≅ ℙ(E_{1}^{∨}), we see η_{1} gives a canonical hyperplane bundle H_{2} ⊂ Y_{2} in the projective space bundle Y_{2} → Y_{1}. Note that the pull-backs of η_{1} via pr_{1} and pr_{2} : Y_{2} → Y_{1} both equal to η_{2} : E_{2} → O_{Y_{2}}, we have H_{3} := pr_{12}^{−1}(H_{2}) = pr_{13}^{−1}(H_{2}) ⊂ Y_{3}. Obviously pr_{12} and pr_{13} induce projective space bundles q_{1} : H_{3} → H_{2} and q_{2} : H_{3} → H_{2} respectively, and H_{3} ⊂ H_{2} ×_{Y_{1}} H_{2}, it is easy to check that H_{3} gives an equivalence relation on H_{2} (induced by the equivalence relation on Y_{2} given by Y_{3}), hence by Proposition 1.1 q_{1} and q_{2} has a geometric equilizer H_{2} → H_{1} = H_{2}/ ∼_{H_{3}} over Y_{1}, and we have an induced morphism H_{2} → H_{1} ×_{Y_{1}} Y_{2}. By

H_{2} ×_{H_{1}} H_{2} = H_{3} = H_{2} ×_{Y_{1}} Y_{2} ≅ H_{2} ×_{H_{1}} (H_{1} ×_{Y_{1}} Y_{2})

we have H_{2} ≅ H_{1} ×_{Y_{1}} Y_{2}, hence H_{1} → Y_{1} is a closed immersion because its pullback via the f.f. morphism pr_{2} is H_{2} → Y_{2}. Since Y_{1} → Y is f.f. we see H_{1} ⊂ Y_{1} is a Y -effective divisor, and L(H_{1}) gives p : Y_{1} → Y a projective space bundle structure.

For any open subscheme U ⊂ Y and any local section μ : U → Y_{1} of p, the invertible sheaf L_{μ} = (id_{X} ×_{S} μ)^{∗}L_{1} on X ×_{S} U satisfies

(id_{X} ×_{S} p|_{U})^{∗}L_{μ} ≅ L_{1}|_{X×_{S}p^{−1}(U)}

Hence up to isomorphism L_{μ} is independent of the choice of μ. Furthermore, since End_{O_{X×_{S} U}} (L_{μ}) ≅ O_{X×_{S}U} and pr_{2∗}O_{X×_{S}U} ≅ O_{U} (because π has geometrically integral fibers), for any other section μ′ : U → Y_{1} there is a unique isomorphism L_{μ} ≅ L_{μ′} whose pull-back via ζ ×_{S} id_{U} is the identity isomorphism of O_{U}. Thus all of such L_{μ}’s glue to give an invertible sheaf L on Y such that (id_{X} ×_{S} p)^{∗}L ≅ L_{1}.

We now show that (Y, L) represents 𝔓𝔦𝔠^{ξ}(X/S). For any S-scheme T and any L′ ∈ Pic^{ξ}(X ×_{S} T/T), by (∗) E′ = pr_{2∗}L′ is locally free. Over any open subscheme U ⊂ T, any local section μ′ : E^{′∨}|_{U} ↠ O_{U} of E′ gives a U-effective divisor D ⊂ X ×_{S} U such that L(D) ≅ L′_{U} := L′|_{X×_{S}U}, this induces an Smorphism ϕ : U → Y_{1} such that (id_{X} ×_{S} ψ)^{∗}L ≅ L′_{U}, where ψ = p ∘ ϕ. If there is another S-mrphism ψ′ : U → Y such that (id_{X} ×_{S} ψ′)^{∗}L ≅ L′_{U}, by taking a local section μ of p (for simplicity we take U small enough so that some μ is defined on ψ′(U)), we get a divisor D′ ⊂ X ×_{S} U such that L(D′) ≅ L′_{U}, this gives a morphism λ : U → Y_{2} such that pr_{1} ∘ λ = ϕ and pr_{2} ∘ λ = μ ∘ ψ′, hence

ψ′ = p ∘ μ ∘ ψ′ = p ∘ pr_{2} ∘ λ = p ∘ pr_{1} ∘ λ = p ∘ ϕ = ψ

This shows that Y represents 𝔓𝔦𝔠^{ξ}(X/S), thus we can denote Y = Pic^{ξ}(X/S).

For any numerical class ξ_{0}, one can take ξ = ξ_{0}(n) for some n ≫ 0 such that ξ satisfies (∗) by Lemma 1. If (Y, L) represents 𝔓𝔦𝔠^{ξ}(X/S), then clearly (Y, L(−n)) represents 𝔓𝔦𝔠^{ξ_{0}}(X/S).

For general S and each point s ∈ S, denote by S_{(s)} and X_{(s)} the localizations at s. The by abstract nonsense, all of the Pic^{ξ}(X_{(s)}/S_{(s)}) (for all s and all ξ) glue together to give a locally quasi-projective scheme Pic^{ξ}(X/S) representing 𝔓𝔦𝔠(X/S). Also by abstract nonsense, Pic(X/S) has a group scheme structure.

Finally, if π is smooth, then by Corollary 1 and Proposition 1.1 we see that Pic(X/S) → S is locally projective. Q.E.D.

Remark 1. From the above proof we see that without the given section ζ, we can still construct Pic(X/S) (but may not construct a universal invertible sheaf), and there is a natural transformation θ : 𝔓𝔦𝔠(X/S) → \underline{Pic(X/S)}. Note that for any morphism T → S we have Pic(X ×_{S} T/T) ≅ Pic(X/S) ×_{S} T, hence θ(T) : Pic(X ×_{S} T/T) → Mor_{S}(T, Pic(X/S)) is a monomorphism for any Sscheme T, and is a one to one correspondence when pr_{2} : X ×_{S} T → T has a section (at least pr_{2} : X ×_{S} X → X has a section). Therefore Pic(X/S) is a coarse moduli scheme of 𝔓𝔦𝔠(X/S) if π has no section.

References

[1] Hélène Esnault, V. Srinivas & Eckart Viehweg: The universal regular quotient of the Chow group of points on projective varieties. (to appear in Invent. Math.)

[2] Seán Keel and Shigefumi Mori: Quotient by groupoids. Annals of Math. 145 (1997), 193-213

[3] János Kollár: Quotient spaces modulo algebraic groups. Annals of Math. 145 (1997), 33-79

[4] János Kollár: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113 (1993), 117-215

[5] Ke-Zheng Li: Classification of supersingular abelian varieties. Math. Ann. 283, 333-351 (1989)

[6] Ke-Zheng Li: Actions of group schemes (I). Compositio Math. 80, 55-74 (1991)

[7] Ke-Zheng Li: Commutative Algebra and Homological Algebra (in Chinese). Sci. Pub. (1998)

[8] Ke-Zheng Li: Push-out of schemes. (preprint)

[9] Ke-Zheng Li: Lectures on group schemes and their actions. (in Chinese)

[10] Ke-Zheng Li & Frans Oort: Moduli of Supersingular Abelian Varieties, LNM 1680. Springer (1998)

[11] Matsumura & Oort: Representations of group functors and automorphisms of algebraic schemes. Invent. Math. 4, 1-25 (1967)

[12] D. Mumford: Lectures on Curves on an Algebraic Surface, Annals of Math. Studies 59, Princeton University Press, Princeton (1966)

[13] D. Mumford - Abelian varieties, 2nd print, Tata Inst. Fund. Res. & Oxford Univ. Press (1974)

[14] D. Mumford: Geometric Invariant Theory, 2nd ed., Ergebnisse, Springer-Verlag, Berlin-Heidelberg-New York (1982)

[15] D. Mumford - The structure of the moduli spaces of curves and abelian varieties. In: Actes, Congrès international math. 1970, tome 1, 457-465. Paris: Gauthier-Villars 1971

[16] P. Norman & F. Oort - Moduli of abelian varieties. Ann. Math. 112 (1980), 413-439.

[17] F. Oort - Subvarieties of moduli spaces. Invent. Math. 24 (1974), 95-119.

[18] F. Oort - Moduli of abelian varieties in positive characteristic. In: Barsotti Symposium in Algebraic Geometry (1991) (Eds.: W. Messing, V. Cristante), Perspect. Math. 15, Acad. Press (1994), 253-276.

[19] M. Rosenlicht: Some basic theorems on algebraic groups. Amer. J. Math. 78 (1956), 401-443

[20] Eckart Viehweg: Quasi-projective Moduli for Polarized Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge·Band 30. Springer-Verlag (1995)