THE MAXIMAL FREE RATIONAL QUOTIENT

JASON MICHAEL STARR

Abstract. This short, expository note proves existence of the maximal quotient of a variety by free rational curves.

1. Definition of a maximal free rational quotient

Definition 1.1. Let V be a Deligne-Mumford stack over a field k, and denote the smooth locus by V^{sm} ⊂ V. A 1-morphism f : ℙ_{k}^{1} → V^{sm} is a free rational curve to V if f^{∗}T_{V} is generated by global sections and has positive degree.

Let S be an irreducible algebraic space, let π\overline{X} : \overline{X} → S be a proper, locally finitely presented 1-morphism of Deligne-Mumford stacks with integral geometric generic fiber, and let X ⊂ \overline{X} be a normal dense open substack. Denote by π_{X} : X → S the restriction of π\overline{X}.

Definition 1.2. A free rational quotient of π_{X} is a triple (X^{∗}, Q^{∗}, ϕ) where X^{∗} ⊂ X is a dense open substack, where Q^{∗} is a normal algebraic space, finitely presented over S with integral geometric generic fiber, and where ϕ : X^{∗} → Q^{∗} is a dominant 1-morphism of S-stacks satisfying,

(i) the geometric generic fiber F of ϕ is integral, and (ii) a general pair of distinct points of F is contained in the image of a free

rational curve. A free rational quotient is trivial if ϕ : X^{∗} → Q^{∗} is birational, and nontrivial otherwise. A free rational quotient (X^{∗}, Q^{∗}, ϕ : X^{∗} → Q^{∗}) is maximal if for every free rational quotient (X_{1}^{∗}, Q_{1}^{∗}, ϕ_{1} : X_{1}^{∗} → Q_{1}^{∗}) there exists a dense open subset U ⊂ Q_{1}^{∗} and a smooth morphism ψ : U → Q^{∗} such that ϕ|_{ϕ_{1}^{−1}(U)} = ψ ∘ ϕ_{1}.

Theorem 1.3. There exists a maximal free rational quotient.

It is not true that a maximal free rational quotient is unique, but it is unique up to unique birational equivalence of Q^{∗}.

2. Proof of Theorem 1.3

The proof is very similar to the proofs of existence of the rational quotient in [1] and [2]. Existence of the maximal free rational quotient can be deduced from theorems there. However there are 2 special features of this case: The relation of containment in a free rational curve is already a rational equivalence relation, so existence of the quotient is less technical than the general case. Also, unlike the general case, there is no need to perform a purely inseparable base-change of S to define the quotient. Of course if O_{S} contains ℚ and if π is smooth and proper, [3, 1.1] implies the free rational quotient is the rational quotient.

Date: February 27, 2006.

If there is no free rational curve in the geometric generic fiber of π_{X}, then the trivial rational quotient (X, X, Id_{X} : X → X) is a maximal free rational quotient. Therefore assume there is a free rational curve to X whose image is contained in the smooth locus of π_{X} (equivalently, there is a free rational curve in the geometric generic fiber of π_{X}). Use will be made of the flat, proper morphism C → S obtained from 2 copies of ℙ_{S}^{1} by identifying the 0 section in the first copy to the 0 section in the second copy.

Denote by Hom_{S}(ℙ_{S}^{1}, X) the Deligne-Mumford stack constructed in [4]. Define H ⊂ Hom_{S}(ℙ_{S}^{1}, X) to be the open substack parametrizing free rational curves in fibers of the smooth locus of π_{X}. By hypothesis, H is nonempty.

Let H_{i} ⊂ H be a connected component. Denote by,

u_{i} : H_{i} ×_{S} ℙ_{S}^{1} → \overline{X},

and by,

u_{i}^{(2)} : H_{i} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1} → \overline{X} ×_{S} \overline{X},

the obvious 1-morphisms.

Lemma 2.1. The 1-morphism u_{i} is smooth.

Proof. The proof is the same as in the case that π_{X} is a projective morphism, [2, Cor. II.3.5.4]. □

Denote by W_{i} ⊂ \overline{X} ×_{S} \overline{X} the closed image of u_{i}^{(2)}, i.e., W_{i} is the minimal closed substack such that u_{i}^{(2)} factors through W_{i}. For any geometric point x of \overline{X} with residue field κ(x), denote by (W_{i})_{x} ⊂ X ⊗_{O_{S}} κ(x) the scheme pr_{2}(pr_{1}^{−1}(x) ∩ W_{i}).

Because H_{i} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1} is irreducible and reduced, also W_{i} is irreducible and reduced. Because u_{i} is smooth, pr_{1} : W_{i} → \overline{X} is surjective on geometric points. The geometric generic fiber has pure dimension

d_{i} = dim(W_{i} ⊗_{O_{S}} K(S)) − dim(X ⊗_{O_{S}} K(S)).

Of course d_{i} is bounded by dim(X ⊗_{O_{S}} K(S)). Let H_{i} be a connected component such that d_{i} is maximal.

Let H_{j} be any connected component of H. Denote by V_{i,j} the 2-fiber product,

V_{i,j} = (H_{i} ×_{S} ℙ_{S}^{1}) ×_{u_{i},X,u_{j}} (H_{j} ×_{S} ℙ_{S}^{1}).

In other words V_{i,j} parametrizes data (([f_{i}], t_{i}), ([f_{j}], t_{j}), θ) where f_{i}, resp. f_{j}, is an object of H_{i}, resp. H_{j}, where t_{i}, t_{j} are points of ℙ^{1}, and where θ : f_{i}(t_{i}) → f_{j}(t_{j}) is an equivalence of objects. Denote by F_{i,j} the 1-morphism of S-stacks,

F_{i,j} : V_{i,j} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1} → \overline{X} ×_{S} \overline{X},

that sends a datum ((([f_{i}], t_{i}), ([f_{j}], t_{j}), θ), t′_{i}, t′_{j}) to (f_{i}(t′_{i}), f_{j}(t′_{j})). Alternatively F_{i,j} is the 1-morphism whose domain is,

(H_{i} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{2}^{1}) ×_{u_{i}∘pr_{1,3},X,u_{j}∘pr_{1,2}} (H_{j} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1}),

such that pr_{1} ∘ F_{i,j} is u_{i} ∘ pr_{1,2} and such that pr_{2} ∘ F_{i,j} is u_{j} ∘ pr_{1,3}.

Proposition 2.2. The image of F_{i,j} is contained in W_{i}.

Proof. Note that V_{i,j} is smooth over S. Moreover, because u_{i} and u_{j} are smooth, V_{i,j} is nonempty. Let ((f_{i}, t_{i}), (f_{j}, t_{j}), θ) be a point of V_{i,j}. There is a reducible, connected genus 0 curve C obtained by identifying t_{i} in one copy of ℙ^{1} to t_{j} in a second copy of ℙ^{1}. The morphisms f_{i}, f_{j} and the equivalence θ induce a 1morphism f : C → X whose restriction to the first irreducible component is f_{i} and whose restriction to the second irreducible component is f_{j}. In a suitable sense, f : C → X is still a free rational curve, and it deforms to free rational curves f′ : ℙ^{1} → X. After examining these deformations, the proposition easily follows.

Let 0 : S → 𝔸_{S}^{1} ×_{S} ℙ_{S}^{1} be the morphism whose projection to each factor is the zero section. Denote by Z ⊂ 𝔸_{S}^{1} ×_{S} ℙ_{S}^{1} the image of 0. Denote by P the blowing up of 𝔸_{S}^{1} ×_{S} ℙ_{S}^{1} along Z. The projection morphism pr_{𝔸^{1}} : P → 𝔸_{S}^{1} is flat and projective. Moreover, the restriction of P over 𝔾_{m,S} ⊂ 𝔸_{S}^{1} is canonically isomorphic to 𝔾_{m,S} ×_{S} ℙ_{S}^{1}. And the restriction of P over the zero section of 𝔸_{S}^{1} is canonically isomorphic to the curve C over S obtained by identifying 0 in one copy of ℙ^{1} to 0 in a second copy of ℙ^{1}.

Denote S′ = 𝔸_{S}^{1}. Denote by Hom_{S′}(P, X_{S′}) the Deligne-Mumford stack constructed in [4]. Denote by H′ ⊂ Hom_{S′}(P, X_{S′}) the open substack parametrizing morphisms to the smooth locus of (π_{X})′ such that the pullback of T_{(π_{X})′} restricts to a globally generated sheaf of positive degree on every irreducible component of every geometric fiber. The morphism H′ → S′ is smooth for reasons similar to [2, Cor. II.3.5.4].

Let V_{i,j,k} be a connected component of V_{i,j}. Define f_{V,i} : V_{i,j,k} ×_{S} ℙ_{S}^{1} → X to be the composition

u_{i} ∘ (pr_{H_{i}} ∘ pr_{H_{i}×_{S}ℙ_{S}^{1}}, Id_{ℙ_{S}^{1}}) : V_{i,j,k} ×_{S} ℙ_{S}^{1} → H_{i} ×_{S} ℙ_{S}^{1} → X.

Define f_{V,j} : V_{i,j,k} ×_{S} ℙ_{S}^{1} → X to be the composition u_{j} ∘ (pr_{H_{j}} ∘ pr_{H_{j}×_{S}ℙ_{S}^{1}}, Id_{ℙ_{S}^{1}}). Define s_{i} : V_{i,j,k} → V_{i,j,k}×_{S}ℙ_{S}^{1} to be the unique V_{i,j,k}-morphism such that pr_{ℙ_{S}^{1}} ∘s_{i} = pr_{ℙ_{S}^{1}} ∘ pr_{H_{i}×_{S}ℙ_{S}^{1}}. Define s_{j} : V_{i,j,k} → V_{i,j,k} ×_{S} ℙ_{S}^{1} similarly.

Replacing V_{i,j,k} by a dense open subset, there exist 2 isomorphisms of V_{i,j,k}-schemes,

α_{i}, α_{j} : V_{i,j,k} ×_{S} ℙ_{S}^{1} → V_{i,j,k} ×_{S} ℙ_{S}^{1},

such that s_{i} = α_{i}∘0 and s_{j} = α_{j}∘0 where 0 is the zero section of V_{i,j,k}×_{S}ℙ_{S}^{1} → V_{i,j,k}. There is a unique 1-morphism of S-stacks,

f_{i,j,k} : V_{i,j,k} ×_{S} C → X,

such that the restriction of f_{i,j,k} to the first irreducible component of V_{i,j,k} ×_{S} C is f_{V,i} ∘ α_{i}, and the restriction to the second irreducible component is f_{V,j} ∘ α_{j}. The image of f_{i,j,k} is contained in the smooth locus of π_{X}, and the restriction of f_{i,j,k}^{∗}T_{π_{X}} to each irreducible component is generated by global sections relative to V_{i,j,k}. Denote by,

f_{i,j,k}^{(2)} : V_{i,j,k} ×_{S} C ×_{S} C → \overline{X} ×_{S} \overline{X},

the obvious 1-morphism.

By definition of H′ there is a 1-morphism,

q : V_{i,j,k} → S ×_{0,S′} H′,

such that the pullback by q of the universal morphism is 2-equivalent to f_{i,j,k}. Because V_{i,j,k} is connected, the image of q is contained in a connected component H′_{l} of H′. By definition of H there is a 1-morphism of S-stacks,

r : 𝔾_{m,S} ×_{S′} H′_{l} → H,

such that the restriction to 𝔾_{m,S} ×_{S′} H′_{l} of the universal morphism over H′_{l} is 2-equivalent to the pullback by r of the universal morphism over H. Denote by H_{l} the connected component of H dominated by r. The morphism r dominates a connected component H_{l} ⊂ H. There exists a 1-isomorphism,

i : 𝔾_{m,S} ×_{S′} H′_{l} → 𝔾_{m,S} ×_{S} H_{l},

unique up to unique 2-equivalence, such that pr_{𝔾_{m}} ∘i = pr_{𝔾_{m}} and such that pr_{H_{l}} ∘i is 2-equivalent to r.

Denote by,

v_{l} : H′_{l} ×_{S′} P → \overline{X},

and by,

v_{l}^{(2)} : H′_{l} ×_{S′} P ×_{S′} P → \overline{X} ×_{S} \overline{X},

the obvious morphisms. Denote by W′_{l} ⊂ \overline{X} ×_{S} \overline{X} the minimal closed substack through which v_{l}^{(2)} factors. Because pr_{S′} : H′_{l} ×_{S′} P ×_{S′} P → S′ is flat, the preimage of 𝔾_{m,S} ⊂ S′ is dense. Thus W′_{l} is the image of the restriction of v_{l}^{(2)} over 𝔾_{m,S}. The restriction of v_{l}^{(2)} is 2-equivalent to u_{l}^{(2)} ∘ (r, Id_{ℙ^{1}}, Id_{ℙ^{1}}). Therefore the image of v_{l}^{(2)} equals the image of u_{l}^{(2)}, i.e., W′_{l} = W_{l}.

On the other hand, the pullback of v_{l}^{(2)} to V_{i,j,k} ×_{S} C ×_{S} C is 2-equivalent to f_{i,j,k}^{(2)}. There are 2 irreducible components of C, and thus 4 irreducible components of C ×_{S} C. Restrict f_{i,j,k}^{(2)} to the irreducible component of C ×_{S} C that is the product of the first irreducible component of C and the first irreducible component of C. This is 2-equivalent to the pullback of u_{i}^{(2)}, hence W_{l} contains W_{i}. Because W_{l} is an integral stack of dimension at most d_{i} containing the d_{i}-dimensional stack W_{i}, W_{l} equals W_{i}.

Finally, restrict f_{i,j,k}^{(2)} to the irreducible component of C ×_{S} C that is the product of the first irreducible component of C and the second irreducible component of C. This is 2-equivalent to the pullback of F_{(i,j)}, hence W_{i} = W_{l} contains the image of F_{(i,j)}. □

Lemma 2.3. The geometric generic fiber of pr_{1} : W_{i} → \overline{X} is integral.

Proof. Denote by K the algebraic closure of the function field of \overline{X}. Since u_{i} : H_{i} ×_{S} ℙ_{S}^{1} → \overline{X} is smooth, the geometric generic fiber (H_{i} ×_{S} ℙ_{S}^{1}) ⊗_{O\overline{X}} K is smooth over K. There is an induced 1-morphism,

(u_{i}^{(2)})_{K} : (H_{i} ×_{S} ℙ_{S}^{1} ⊗_{O\overline{X}} K)×_{Spec (K)} ℙ_{K}^{1} → (\overline{X} ×_{S} \overline{X})⊗_{pr_{1},\overline{X}} Spec (K) ≅ \overline{X} ⊗_{O_{S}} K

Formation of the closed image is compatible with flat base change. Therefore the closed image of (u_{i}^{(2)})_{K} is W_{i} ⊗_{pr_{1},\overline{X}} Spec (K). In particular, W_{i} ⊗_{pr_{1},\overline{X}} Spec (K) is reduced since the closed image of a reduced stack is reduced.

The proof that the geometric generic fiber is irreducible is essentially the same as the proof of Proposition 2.2. Let W′ ⊂ W_{i} ⊗_{pr_{1},\overline{X}} Spec (K) be an irreducible component. To prove that W′ = W_{i}⊗_{pr_{1},\overline{X}} Spec (K), it suffices to prove that it contains

the image of (u_{i}^{(2)})_{K}. Let x ∈ \overline{X} ⊗_{O_{S}} K be the K-point corresponding to the diagonal; x is contained in Image(u_{i}^{(2)})_{K}. Let y_{1} be a K-point of W′ ∩ Image(u_{i}^{(2)})_{K} and let y_{2} be a K-point of Image(u_{i}^{(2)})_{K}. There are free K-morphisms, f_{1}, f_{2} : ℙ_{K}^{1} → X ⊗_{O_{S}} K such that f_{1}(0) = f_{2}(0) = x and f_{1}(∞) = y_{1}, f_{2}(∞) = y_{2}. This defines a morphism from C ⊗_{O_{S}} K to X ⊗_{O_{S}} K. As in the proof of Proposition 2.2, deformations of this morphism are free rational curves that come from a connected component H_{l} of H. By construction, there is an irreducible component of W_{l} ⊗_{pr_{1},X} Spec (K) that contains W and y_{2}. Since the dimension of W_{l} is at most d_{i}, this irreducible component equals W. Therefore y_{2} ∈ W, i.e., W = W_{i} ⊗_{pr_{1},X} Spec (K). □

Consider the projection pr_{1} : W_{i} → \overline{X}. By [5, Thm. 3.2], there exists a dense open subset X^{flat} ⊂ X over which W_{i} is flat. Denote W_{i}^{flat} = W_{i} ×_{pr_{1},X} X^{flat}. By Lemma 2.3, there is a dense open substack X^{0} ⊂ X^{flat} such that every geometric fiber of W_{i} ×_{pr_{1},X} X^{0} → X^{0} is integral. Denote W_{i}^{0} = W_{i} ×_{pr_{1},\overline{X}} X^{0}.

Let H_{j} ⊂ H be a connected component and denote by G_{i,j} the unique 1-morphism,

G_{i,j} : (H_{j} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1}) ×_{u_{j}∘pr_{1,3},\overline{X},pr_{1}} W_{i} → \overline{X} ×_{S} \overline{X},

such that,

pr_{1} ∘ G_{i,j} = u_{j} ∘ pr_{1,2} ∘ pr_{H_{j}×ℙ^{1}×ℙ^{1}},

and such that,

pr_{2} ∘ G_{i,j} = pr_{2} ∘ pr_{W_{i}}.

Corollary 2.4. (i) The image of G_{i,j} is contained in W_{i}.

(ii) For every geometric point s ∈ S, for every free morphism f : ℙ_{s}^{1} → X_{s} and for every point x ∈ X_{s} such that f(ℙ^{1}) ∩ (W_{i})_{x} is nonempty, f(ℙ^{1}) is contained in (W_{i})_{x}.

(iii) The image of the “composition morphism”,

c : W_{i} ×_{pr_{2},\overline{X},pr_{1}} W_{i}^{0} → \overline{X} ×_{S} \overline{X},

is contained in W_{i}. (iv) For every geometric point s ∈ S, for every pair of closed points (x, y) ∈

\overline{X}_{s} × X_{s}^{0}, if y ∈ (W_{i})_{x} then (W_{i})_{y} ⊂ (W_{i})_{x}.

(v) For every geometric point s ∈ S, for every pair of closed points (x, y) ∈ X_{s}^{0} × X_{s}^{0}, y_{i} ∈ (W_{i})_{x} iff x ∈ (W_{i})_{y} iff (W_{i})_{x} = (W_{i})_{y}.

Proof. (i): First of all, the projection morphism,

pr_{W_{i}} : (H_{j} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1}) ×\overline{X} W_{i} → W_{i},

is smooth. Therefore every connected component of the domain is integral and dominates W_{i}. So to prove W_{i} contains the image of G_{i,j} it suffices to first basechange by,

u_{i}^{(2)} : H_{i} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1} → W_{i}.

The base-change of G_{i,j} by u_{i}^{(2)} is 2-equivalent to F_{i,j}. By Proposition 2.2 the W_{i} contains the image of F_{i,j}. Therefore W_{i} contains the image of G_{i,j}.

(ii): By construction, W_{i} ⊂ \overline{X} ×_{S} \overline{X} is symmetric with respect to permuting the factors. Let t′ ∈ ℙ^{1} be a point such that x′ = f(t′) is in (W_{i})_{x}. Let H_{j} be the connected component of H that contains [f]. Then the subset,

{(([f], t, t′), (x′, x)) ∈ (H_{j} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1}) × W_{i}|t ∈ ℙ_{s}^{1}} ,

is contained in,

(H_{j} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1}) ×\overline{X} W_{i}.

Therefore by (i), W_{i} contains the image under G_{i,j}. Because W_{i} is symmetric, this implies that (W_{i})_{x} contains f(ℙ^{1}),

(iii): The 1-morphism c satisfies pr_{1} ∘ c = pr_{1} ∘ pr_{1} and pr_{2} ∘ c = pr_{2} ∘ pr_{2}, up to 2-equivalence. Because pr_{1} : W_{i}^{flat} → \overline{X} is flat and the geometric fibers are integral also the projection,

pr_{1} : W_{i} ×_{pr_{2},\overline{X},pr_{1}} W_{i}^{flat} → W_{i},

is flat and the geometric fibers are integral. In particular the domain is integral. Hence to prove W_{i} contains the image of c it suffices to first base-change by,

u_{i}^{(2)} : H_{i} ×_{S} ℙ_{S}^{1} ×_{S} ℙ_{S}^{1} → W_{i}.

After base-change, this morphism factors through G_{i,i}. By (i), the W_{i} contains the image of G_{i,i}. Therefore W_{i} contains the image of c.

(iv) and (v): Item (iv) follows immediately from (iii), and Item (v) follows from (iv) and symmetry of W_{i}. □

Lemma 2.5. Let k be a field, let g : Y → Z be a morphism of smooth Deligne-Mumford stacks over k, and let f : ℙ_{k}^{1} → Y be a free morphism such that f(ℙ^{1}) is contained in a fiber of g. Denote by Y^{sm} the smooth locus of g. If f(ℙ^{1}) ∩ Y^{sm} is nonempty, then f(ℙ^{1}) ⊂ Y^{sm}.

Proof. There is a morphism of locally free sheaves on ℙ_{k}^{1},

dg : f^{∗}T_{Y} → g^{∗}f^{∗}T_{Z}.

Because f(ℙ^{1}) ∩ Y^{sm} is nonempty, the cokernel of dg is torsion. Because f(ℙ^{1}) is contained in a fiber of g, g^{∗}f^{∗}T_{Z} ≅ O_{ℙ_{k}^{1}}^{r} for some nonnegative integer r. Because f^{∗}T_{Y} is generated by global sections, also the image of dg is generated by global sections. But the only coherent subsheaf of O_{ℙ_{k}^{1}}^{r} whose cokernel is torsion and that is generated by global sections is all of O_{ℙ_{k}^{1}}^{r}. Therefore dg is surjective, i.e., f(ℙ^{1}) ⊂ Y^{sm}. □

By [5, Thm. 1.1], the Hilbert functor of \overline{X} → S is represented by an algebraic space that is separated and locally finitely presented, Hilb_{\overline{X}/S}. And W_{i}^{flat} ⊂ X^{flat} ×_{S} \overline{X} is a closed substack that is proper, flat and finitely presented over X^{flat}. Therefore there is a 1-morphism of S-stacks,

ϕ^{flat} : X^{flat} → Hilb_{\overline{X}/S},

such that W_{i}^{flat} is the pullback by ϕ^{flat} of the universal closed substack. Denote by X^{∗} ⊂ X the maximal open substack over which ϕ^{flat} extends to a morphism. Denote by Q^{∗} → Hilb_{X/S} the Stein factorization of X^{∗} → Hilb_{X/S}, i.e., the integral closure of the image in the function field of the coarse moduli space |X^{∗}|. Denote by ϕ : X^{∗} → Q^{∗} the induced morphism.

Proposition 2.6. The morphism ϕ : X^{∗} → Q^{∗} is a free rational quotient.

Proof. By construction, Q^{∗} is normal, Q^{∗} → S is a finitely presented morphism whose geometric generic fiber is integral, and ϕ is a dominant 1-morphism whose geometric generic fiber is integral. It remains to prove Definition 1.2 (ii).

By [5], Hilb_{\overline{X}/S} satisfies the valuative criterion of properness (but it is not necessarily proper since it is not necessarily quasi-compact). And X is normal. Therefore every irreducible component of X − X^{∗} has codimension ≥ 2 in X; more precisely, the geometric generic fiber over S has codimension ≥ 2 in the geometric generic fiber of X over S. For reasons similar to [2, Prop.II.3.7], X^{∗} contains f(ℙ^{1}) for every H_{j} and general [f] ∈ H_{j}.

There is a dense open subspace Q^{0} ⊂ Q^{∗} over which ϕ is flat and the geometric fibers are integral. Replace X^{0} by X^{0} ∩ ϕ^{−1}(Q^{0}). By Corollary 2.4 (v), the subscheme X^{0} × Q^{∗}X^{0} equals W_{i} ∩ (X^{0} ×_{S} X^{0}). In particular, for every x ∈ X^{0} the fiber of ϕ containing x is (W_{i})_{x}. Therefore for a general fiber of ϕ, for a general pair of points in the fiber, there is a free rational curve in H_{i} whose image is contained in the fiber and contains the two points. By Lemma 2.5, the image is contained in the smooth locus of ϕ, i.e. this is a free rational curve in the fiber. This proves Definition 1.2 (ii). □

Proposition 2.7. The morphism ϕ : X^{∗} → Q^{∗} is a maximal free rational quotient.

Proof. Let ϕ_{1} : X_{1}^{∗} → Q_{1}^{∗} be a free rational quotient. If this is a trivial free rational quotient, the morphism ψ is trivial. Therefore assume it is a nontrivial free rational quotient.

There exists a dense open U ⊂ Q_{1}^{∗} such that,

χ : U′ → U,

is faithfully flat and quasi-compact and the geometric fibers are integral, where U′ = X^{0} ∩ ϕ_{1}^{∗}(U) and where χ is the restriction of ϕ_{1}. By faithfully flat descent, to construct ψ : U → Q^{∗}, it is equivalent to construct a morphism ψ′ : U′ → Q^{∗} satisfying a cocycle condition: indeed, the morphism ψ is equivalent to the graph of ψ, which is equivalent to a certain kind of quasi-coherent sheaf on U × Q^{∗}, so faithfully flat descent for quasi-coherent sheaves applies to (χ, 1) : U′×Q^{∗} → U×Q^{∗}.

Define ψ′ to be the restriction of ϕ. For each geometric point x in U′, define Y_{x} to be the fiber of χ containing x. The cocycle condition for ψ′ is that the fiber product U′′ = U′ ×_{χ,U,χ} U′ is contained in U′ ×_{ψ′,Q^{∗},ψ′} U′. Now U′′ → U′ is a flat morphism whose geometric fibers are integral. Thus U′′ is integral. So it suffices to prove it is set-theoretically contained in W_{i}, i.e., for a general geometric point x of U′, U′′_{x} ⊂ (W_{i})_{x}.

By hypothesis, there is a dense subset of U′′_{x} consisting of points y contained in a free morphism f : ℙ^{1} → X such that f(0) = x and f(∞) = y. By Corollary 2.4 (ii), f(ℙ^{1}) ⊂ (W_{i})_{x}, in particular y ∈ (W_{i})_{x}. Therefore U′′_{x} ⊂ (W_{i})_{x}. □

References

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

   E-mail address: jstarr@math.mit.edu