Abstract. In this paper, we study the structure of isomorphisms of S-bundles over a cetain type of prevarieties, where S is a variety with vanishing logarithmic genera (for example, A

A CERTAIN TYPE OF AFFINE SURFACES WITH ISOMORPHIC CYLINDERS

RIKU KUDOU

Abstract. In this paper, we study the structure of isomorphisms of S-bundles over a cetain type of prevarieties, where S is a variety with vanishing logarithmic genera (for example, A

and P

).

As in the same way, for a certain type of affine surfaces W , we determine all varieties V such that V ×

A

≃ W ×

A

.

Introduction

Let V and W be varieties over an algebraically closed field k with characteristic 0. The Zariski cancellation problem asks when the existence of an isomorphism V × k A ¹ ≃ W × k A ¹ implies that V ≃ W . Following [6], we call a variety V a Zariski 1-factor if V × k A ¹ ≃ W × k A ¹ implies V ≃ W for any variety W. There are many examples of Zariski 1-factors, but in 1989, W. Danielewski found non-Zariski 1-factors by using the following property of principal G a -bundles.

Fact 0.1 ([2]). Let X be a k-scheme, and let V and W be affine k-schemes which are principal G a -bundles over X. Then V × k A ¹ ≃ W × k A ¹ .

By using the same arguments, many examples of non-Zariski 1-factors are constructed. From these examples, we can consider the following problem.

Problem 0.2. Let Y be a prevariety, and let W be an affine variety which is a principal G a -bundle over Y . Then for any variety V , does V × k A ¹ ≃ W × k A ¹ imply that V is affine and a principal G a -bundle over Y ?

In this paper, we will show the following main theorem (section 3).

Theorem 0.3. Let Y be a 1-dimensional nonsingular prevariety (a prevariety means an integral scheme of finite type over k), let Y ^′ be a nonsingular curve with nonnegative logarithmic kodaira dimension (that is, Y ^′ is neither A ¹ nor P ¹ ), let l : Y → Y ^′ be a dominant morphism, and let W be an affine variety which is a principal G a -bundle over Y . Then for any variety V , V × k A ¹ ≃ W × k A ¹ if and only if V is affine and a principal G a -bundle over Y .

From this theorem and counter examples for the cancellation problem by Dry lo ([3]), we find many non-Zariski 1-factors of the above type.

To show Theorem 0.3, we use the similar arguments as Fujita-Iitaka’s cancellation theorem ([8]) and Nishimura’s generalized version of it ([9]). In addition to Theorem 0.3, we will slightly generalize these theorems in section 2 as follows.

Theorem 0.4. Let X and Y be prevarieties, let Y ^′ be a variety with κ(Y ^′ ) ≥ 0 and dim Y ^′ = dim Y , let l : Y → Y ^′ be a dominant morphism, and let S ₁ and S ₂ be varieties with vanishing logarithmic genera and dim S ₁ = dim S ₂ . Let p : V → X be a S ₁ -bundle, q : W → Y a S ₂ -bundle. If Φ : V → W is an isomorphism, then there exists a unique isomorphism ϕ : X → Y such that the following

1

diagram is commutative.

V ^{Φ //}

p

W

q

X ∃ ϕ // Y

⟳

Using this theorem, we show 2 corollaries (Corollary 2.5, 2.6) useful for the cancellation problem.

Before the proof of Theorem 0.3 and 0.4, we will give a criterion for principal G a -bundles over some good schemes to be affine (Theorem 1.6). The proof of Theorem 1.6 is almost the same as the affine criterion for principal G a -bundles by A. Dubouloz ([4]).

From these theorems and Dry lo’s examples ([3]), we can find all varieties which are non-Zariski 1-factors of the type in Theorem 0.3.

1. Affine criterion for principal G a -bundles over a certain type of nonseparated schemes

The purpose of this section is to prove Theorem 1.6, which gives a computable way to determine principal G a -bundles to be affine or not.

Definition 1.1 (principal G a -bundle). Let X be a k-scheme, let V be a k-scheme with G a -action, and let p : V → X be a morphism of k-schemes. Then (V, p) is called a principal G a -bundle over X if the following two conditions are satisfied:

(1) p is G a -equivariant ( G a acts trivially on X);

(2) there exists an (Zariski) open covering U = { U _λ } λ ∈ Λ of X and a G a -equivariant isomorphism g _λ : p ^{− 1} U _λ → U _λ × k G a for each λ ∈ Λ such that the following diagram is commutative.

p ^{− 1} U _λ ^g

^//

p

U _λ × G a

zzvvv vvv vvv pr vv

U _λ

Remark 1.2. Our definition of principal G-bundles for a group variety G is slightly different from the ordinaly one. But for an affine group variety, those definitions coincide.

Remark 1.3. There exists a one-to-one correspondence between isomorphic classes of principal G a -bundles over X and H ¹ (X, O X ).

Definition 1.4. Let X be a variety, let Z be a closed subvariety of X, let r be a natural number, and let X 0 , . . . , X r be copies of X. Then

X ₊ rZ := X ⊔ X \ Z X ⊔ X \ Z · · · ⊔ X \ Z X

| {z }

r

= X ₀ ⊔ X \ Z X ₁ ⊔ X \ Z · · · ⊔ X \ Z X _r .

Namely, X + rZ is a nonseparated k-scheme which looks like X with r copies of Z. We fix an open covering X of X ₊ rZ to be X = { X ₀ , . . . , X _r } .

Lemma 1.5 ([7]). Let X be a scheme, let Y be an affine scheme, and let U = { U _λ } λ ∈ Λ be an open affine covering of X. Then for any morphism f : X → Y , f is separated if and only if

(1) U _µ ∩ U _λ is affine for any µ, λ ∈ Λ;

(2) Γ(U _µ ∩ U _λ , O X ) is generated by Γ(U _µ , O X ) and Γ(U _λ , O X ).

The following theorem is a generalization of the affine criterion for principal G a -bundles by A.

Dubouloz ([4]), but the proof is almost the same.

Theorem 1.6. Let X = SpecA be an affine variety, let Z ₁ , ..., Z _m be hypersurfaces of X defined by prime elements f ₁ , ..., f _m ∈ A, and let Z := ∪

Z _j . Let V be a principal G a -bundle over X ₊ rZ de- fined by a ˇ Cech cocycle [ { g _ij } ] ∈ H ¹ ( X , O X

rZ ) ≃ H ¹ (X ₊ rZ, O X

rZ ), where g _ij ∈ Γ(X _ij , O X

rZ ) = A _f

_{··· f}

and as an element of A _f

_{··· f}

, g _ij can be written as follows; g _ij = f ₁ ^{− k}

· · · f m ^{− k}

h _ij , where k _ij,l ∈ Z ≥ 0 and h _ij ∈ A such that h _ij can not be divided by f _l if k _ij,l > 0.

If (a) r = 1 or (b) r ≥ 2 and ∅ ̸ = Z _l

∩ Z _l

̸⊂ ∪

l ̸ =l

,l

Z _l for any l ₁ , l ₂ = 1, . . . , m, then the following conditions are equivalent.

(1) k _ij,l ≥ 1 and (h _ij , f ₁ · · · f _m ) = A for any i, j = 0, . . . , r and l = 1, . . . , m.

(2) V is separated.

(3) V is affine.

Proof. (3) ⇒ (2) is obvious. We show that (1) ⇔ (2) and (1) ⇒ (3).

Let us denote (k _ij,1 , . . . , k _ij,m ) ∈ Z ^m by [k _ij ], (1, . . . , 1) ∈ Z ^m by 1, and f ₁ ^k

· · · f m ^k

by f ^[k

^] . First of all, we give a necessaly and sufficient condition for V to be separated by using Lemma 1.5. By the definition of the open covering X of X + rZ , X i ∩ X j is affine for any i, j = 0, . . . r. Thus V is separated if and only if Γ(X _i ∩ X _j , O X

rZ ) is generated by Γ(X _i , O X

rZ ) and Γ(X _j , O X

rZ ) for any i, j = 0, . . . r. This condition equals to A _f

_{··· f}

[t] = A _g

[t] for any i, j = 0, . . . , r with i ̸ = j , where t is indeterminate. this implies that V is separated if and only if A f = A _g

for any i, j = 0, . . . , r with i ̸ = j.

(1) ⇒ (2) Suppose the condition (1). Then there exists a, b ∈ A such that 1 = ah _ij + bf, and this implies that f ^{− 1} = af ^[k

^{] − 1} g _ij + b and k _ij,l − 1 ≥ 0. Then it follows that A f = A _g

.

(2) ⇒ (1) Suppose the condition (2). Then f ^{− 1} ∈ A f = A _g

. Therefore f ^{− 1} can be written as follows;

f ^{− 1} = a ₀ + a ₁ g _ij + a ₂ g _ij ² + · · · + a _n g _ij ⁿ ,

where a ₀ , . . . a _s ∈ A and n is an integer. (If n = 0, then f ₁ , . . . , f _m should be units in A.) By multiplying both sides of the equation by f ^n[k

^] , we obtain the following equation,

f ^n[k

^{] − 1} = a ₀ f ^n[k

^] + h _ij s,

where s = a 1 f ^{(n − 1)[k}

^] + · · · + a n − 1 h ⁿ _ij ^{− 2} f ^[k

^] + a n h ⁿ _ij ^{− 1} . From this equation, we can deduce that k _ij,l ≥ 1 for any l = 1, . . . , m because f ₁ , . . . , f _m are prime elements and distinct up to units.

Moreover, s can be divided by f ^n[k

^{] − 1} because we take h _ij which is not in (f _l ) for all l = 1, . . . , m.

Then it follows that there exists s ^′ ∈ A such that 1 = a 0 f + h ij s. that is, A = (f, h ij ).

(1) ⇒ (3) Suppose the condition (1). We first observe that there exists an index j ^′ ∈ { 1, . . . , m } such that k _1j

,l = max _j { k _1j,l } for all l = 1, . . . , m. Assume that there exists indices j ₁ , j ₂ = 0, . . . r and l ₁ , l ₂ = 1, . . . , m such that j ₁ ̸ = j ₂ , l ₁ ̸ = l ₂ , k _1j

_,l

> k _1j

_,l

, and k _1j

_,l

< k _1j

_,l

for contradiction.

Put µ _l := max { k _1j

_,l , k _1j

_,l } and [µ] := (µ ₁ , . . . µ _m ) ∈ Z ^m _≥ 0 . It follows from the cocycle condition g _j

_j

= g _1j

− g _1j

that

f ^{[µ] − [k}

^] h j

j

= f ^{[µ] − [k}

^] h 1j

− f ^{[µ] − [k}

^] h 1j

.

By the definition of µ and h _ij , the right hand side of this equation is in (f _l

, f _l

) but not in (f _l

) and (f _l

). From this, it follows that µ _l

= k _j

_j

_,l

and µ _l

= k _j

_j

_,l

. On the other hand, f ^{[µ] − [k}

^] h j

j

∈ (f l

, f l

) implies f ^{[µ] − [k}

^] ∈ (f l

, f l

) because h j

j

is a nonzero function on Z and Z _l

∩ Z _l

̸ = ∅ . But this contradicts to the assumtion Z _l

∩ Z _l

̸⊂ ∪

l ̸ =l

,l

Z _l , and thus we can choose an index j ^′ ∈ { 1, . . . , m } such that k _1j

,l = max _j { k _1j,l } for all l = 1, . . . , m.

Next, we show that there exists an affine morphism ψ : V → A ¹ by induction on r. If r = 0,

then V should be isomorphic to X × k A ¹ . Then the first (or second) projection of X × k A ¹ is an

affine morphism. Suppose the statement holds for a natural number r − 1. By the assumption,

there exists s ij ∈ A such that h ij s ij = 1 in A/(f 1 · · · f m ). Define morphisms ϕ j : X j → A ¹ to be ϕ _j (x, t) = s _1j (f ^[k

^] t + f ^[k

^{] − [k}

^] h _1j ) and define morphisms ψ _j := ϕ ◦ g ⁻¹ _j : V _j ≃ X × k A ¹ → A ¹ . By the cocycle condition, { ψ _j } j=0,...,r glue to a morphism ψ : V → A ¹ . Define H _j := g _j (Z × k A ¹ ) ⊂ V _j . Then ψ(H ₁ ) = ϕ ₁ g ₁ ^{− 1} g ₁ (Z × k A ¹ ) = { 0 } and ψ(H _j

) = ϕ _j

g ⁻ _j

¹ g _j

(Z × k A ¹ ) = { 1 } . Thus ψ ^{− 1} ( A ¹ \ { 0 } ) ⊆ V \ H ₁ and ψ ^{− 1} ( A ¹ \ { 1 } ) ⊆ V \ H _j

. Moreover, V \ H ₁ is a principal G a -bundle defined by the cocycle { g _ij } i,j ̸ =1 and V \ H _j

is a principal G a -bundle defined by the cocycle { g _ij } i,j ̸ =j

therefore it follows from the induction hypothesis that V \ H ₁ and V \ H _j

are affine. Therefore the restriction maps ψ | V \ H

: V \ H ₁ → A ¹ and ψ | V \ H

: V \ H _j

→ A ¹ are affine morphisms. Then it follows that ψ ^{− 1} ( A ¹ \ { 0 } ) and ψ ^{− 1} ( A ¹ \ { 1 } ) are affine. Namely, ψ is affine. □

2. Generalization of Theorems of Fujita-Iitaka and Nishimura

The main purpose of this section is to prove Theorem 0.4. As corollaries, We obtain the unique- ness of base schemes of principal G a -bundles in the special case (Corollary 2.5) and a criterion for principal G a -bundles over a certain type of schemes to be isomorphic to each other (Corollary 2.6).

Definition 2.1.

• For a k-scheme X, we denote by X(k) the set of closed points (k-valued points) of X.

• A k-scheme X is called a prevariety if X is an integral scheme of finite type over k.

• A variety S is called a variety with vanishing logarithmic genera (or VLG variety for short) if S is a nonsingular variety with P _M (S) = 0 for all M ∈ Z ^⊕∞ _{≥ 0} , where P _M (S) = dim _k H ⁰ (S, Ω ^M _S (log ∂S)) is the logarithmic M -genus of S and (S, ∂S) is a smooth completion of S with boundary ∂S.

We will slightly generalize Fujita-Iitaka’s cancellation theorem [8] and the following Nishimura’s theorem.

Theorem 2.2 (Nishimura [9]). Let X and Y be varieties with κ(Y ) ≥ 0, and let S ₁ and S ₂ be VLG varieties with dim S ₁ = dim S ₂ . Let p : V → X be a S ₁ -bundle, q : W → Y a S ₂ -bundle. If Φ : V → W is an isomorphism, then there exists a unique isomorphism ϕ : X → Y such that the following diagram is commutative.

V ^{Φ //}

p

W

q

X ∃ ϕ

// Y

⟳

Lemma 2.3. Let X and Y be prevarieties, and let f, g : X → Y be morphisms of prevarieties. If f and g coincide on closed points of X, then f = g as a morphism of prevarieties.

The proof of Lemma 2.3 is the same as the one for varieties.

The next lemma is a part of the proof of Fujita-Iitaka’s Cancellation Theorem ([8]).

Lemma 2.4 ([8]). Let X be a variety of dimension n, let Y be a variety of dimension n + 1 with κ(X) ≥ 0, and let S be a VLG variety. If f : X × k S → Y is a morphism, then f is not dominant.

Proof of Theorem 0.4. We prove Theorem 0.4 with 4 steps as follows:

(1) For any prime divisor C on X, E = qΦp ^{− 1} C is a prime divisor on Y .

(2) We can show the same statement as (1) locally (this process is necessary to deal with closed points of the prevaries X and Y as an intersection of prime divisors on some open subset).

(3) We can construct a bijective map of sets ϕ ^′ : X(k) → Y (k) such that ϕ ◦ p = q ◦ Φ.

(4) We can construct a morphism of prevarieties ϕ : X → Y such that ϕ | X (k) = ϕ ^′ and ϕ ◦ p =

q ◦ Φ.

(1) and (4) are almost the same as [8] and [9], but (2) and (3) contain a new part to deal with closed points of non-separated schemes.

(1) Let C be a prime divisor on X. By the local triviality of p, there exists an open affine covering U = { U _λ } λ ∈ Λ of X such that p ^{− 1} U _λ ≃ U _λ × k S ₁ . Take λ ∈ Λ such that U _λ ∩ C ̸ = ∅ and put C _λ := C ∩ U _λ . Then we obtain the compositon f : C _λ × S ₁ → Y ^′ as the following diagram.

C _λ × S ₁ ^g

^{:iso //}

$$ J

J J J J J J J J J J

f

//

p ^{− 1} C _λ ^{ /}

V ^{Φ //}

p

W

q

C _λ ^{ /} X

⟳

Y

l

Y ^′

⟳

By Lemma 2.4, f is not dominant, and thus q ◦ Φ | p

C : p ^{− 1} C → Y is not dominant. Put E _C :=

qΦp ^{− 1} C ^Y , where the closure is taken in Y . Then E _C is irreducible closed subset of Y and q ^{− 1} E _C = q ^{− 1} (qΦp ^{− 1} C ^Y ) ⊇ Φp ^{− 1} C. Then it follows that q ^{− 1} E _C = Φp ^{− 1} C because q ^{− 1} E _C is irreducible and Φp ⁻¹ C is prime divisor of W . As a consequence, we obtain the equality E _C = qq ⁻¹ E _C = qΦp ⁻¹ C.

(2) Put V _λ := p ^{− 1} U _λ , W _λ := ΦV _λ , Y _λ := qW _λ , and q _λ := q | W

. Then Y _λ is an open subset of Y because q is locally trivial. Put E _C,λ := qΦp ^{− 1} C _λ ^Y

, where the closure is taken in Y _λ . Then q ^{− 1} E _C ∩ W _λ ⊇ q ⁻ _λ ¹ E _C,λ = q _λ ^{− 1} qΦp ^{− 1} C _λ ^Y

⊇ Φp ^{− 1} C _λ . Moreover, q ^{− 1} E _C ∩ W _λ and Φp ^{− 1} C _λ are prime divisors on W _λ . Thus q ^{− 1} E _C,λ = Φp ^{− 1} C _λ . As a consequence, we obtain the equality E _C,λ = qq ⁻ _λ ¹ E _C,λ = qΦp ^{− 1} C _λ .

(3) Let x ∈ X(k). Then there exists λ ∈ Λ such that x ∈ U _λ . In U _λ , x can be expressed as an intersection of prime divisors C _1,λ , · · · , C _m,λ on U _λ since U _λ is a variety. Put C _i := C _i,λ ^Y and E _C,i,λ := qΦp ^{− 1} C _i,λ . Then

S 1 ≃ p ^{− 1} (x) ≃ Φp ^{− 1} (x) = Φp ^{− 1} (

∩ m i=1

C j,λ ) =

∩ m i=1

Φp ^{− 1} (C i,λ )

=

∩ m i=1

q _λ ^{− 1} (E _C,i,λ ) = q ⁻ _λ ¹ (

∩ m i=1

E _C,i,λ ).

The fiber of q _λ is not necessarily equal to S ₂ , but is a nonempty open subset of S ₂ . Therefore dim ∩ _m

i=1 E _C,i,λ = dim S ₁ − dim S ₂ = 0. Moreover, ∩ _m

i=1 E _C,i,λ = qΦp ^{− 1} (x) is irreducible. Then it follows that ∩ _m

i=1 E _C,i,λ is a closed point of Y , denoted by y _x . In this way, we obtain a map ϕ ^′ : X(k) → Y (k); x 7→ y _x of sets. Moreover, ϕ ^′ satisfies

qΦ(v) ∈ qΦ(p ^{− 1} p(v)) = qΦ(Φ ^{− 1} q ^{− 1} (y _p(v) )) = { y _p(v) } = { ϕ ^′ (p(v)) } ,

that is, q ◦ Φ = ϕ ^′ ◦ p. The injectivity of ϕ ^′ also follows because x = pΦ ^{− 1} q ^{− 1} qΦp ^{− 1} (x) = pΦ ^{− 1} q ^{− 1} ϕ ^′ (x) for any closed point x ∈ X. Surjectivity of ϕ ^′ is obvious.

(4) For each λ ∈ Λ, we take a closed point a _λ ∈ S ₁ and define ϕ _λ,a

:= q ◦ Φ ◦ g _λ ◦ j _λ ◦ a _λ : U _λ → Y , where j _λ : U _λ × { a _λ } , → U _λ × S ₁ and a _λ : U _λ ≃ U _λ × { a _λ } . Then, for any closed point x ∈ U _λ , ϕ _λ,a

(x) = ϕ ^′ (x). Therefore by the Lemma 2.3, we can glue the morphisms { ϕ _λ,a

} to a morphism ϕ : X → Y such that q ◦ Φ = ϕ ◦ p. The converse morphism of ϕ can be constructed in the same way. Moreover, the uniqueness of ϕ follows from the equality ϕ(x) = ϕ(pp ^{− 1} (x)) = qΦ(p ^{− 1} (x)) for

any closed point x ∈ X and Lemma 2.3. □

Corollary 2.5. Let X and Y be prevarieties, let Y ^′ be a variety with κ(Y ^′ ) ≥ 0 and dim Y ^′ = dim Y , let l : Y → Y ^′ be a dominant morphism, and let V , W be principal G a -bundles over X, Y , respectively. Then

(1) V ≃ W ⇒ X ≃ Y .

(2) V × k A ¹ ≃ W × k A ¹ ⇒ X ≃ Y .

Corollary 2.6. Let X be an affine variety with κ(X) ≥ 0, let Z ₁ , ..., Z _m be hypersurfaces of X defined by f ₁ , ..., f _m ∈ A, respectively, let Z := ∪

Z _j , and for k = 1, 2, let V _k be a principal G a - bundle over X ₊ rZ . Then V ₁ and V ₂ are isomorphic if and only if they are in the same orbit of the action by Aut(X ₊ rZ ) × Γ(X, O _X ^× ).

Proof. The computation of this proof is almost the same as in [3]. Suppose that Φ : V ₁ → V ₂ is an isomorphism. Then by Theorem 0.4, there exists a unique automorphism ϕ : X ₊ rZ → X ₊ rZ which satisfies ϕ ◦ p 1 = p 2 ◦ Φ, where p k : V k → X + rZ is the canonical projection of principal G a -bundles for k = 1, 2. Put X _i ^′ := ϕ(X i ) and X ^′ := { X ₀ ^′ , . . . , X _r ^′ } which becomes an open covering of X + rZ . Suppose that the principal G a -bundle V ₁ is defined by a ˇ Cech cocycle { g _ij } ∈ Z ¹ ( X , O X

rZ ) and V ₂ is defined by a ˇ Cech cocycle { g _ij ^′ } ∈ Z ¹ ( X ^′ , O X

rZ ). Then the following diagram is commutative for each i, j = 0, . . . , r (i ̸ = j);

(X _i ∩ X _j ) × k A ^{1 g}

^//

α

p ⁻ ₁ ¹ (X _i ∩ X _j ) ^{Φ //}

id

p ⁻ ₂ ¹ (X _i ^′ ∩ X _j ^′ ) ^g

′−1 i

//

id

(X _i ^′ ∩ X _j ^′ ) × k A ¹

α

(X _i ∩ X _j ) × k A ^{1 g}

^//

pr

(( R

R R R R R R R R R R R R

R R p ⁻¹ ₁ (X _i ∩ X _j ) ^{Φ //}

p

p ⁻¹ ₂ (X _i ^′ ∩ X _j ^′ )

g

//

p

(X _i ^′ ∩ X _j ^′ ) × k A ¹

pr

vvllll llll llll ll

X i ∩ X j

ϕ // X _i ^′ ∩ X _j ^′

where α ij (x, t) = (x, t + g ij (x)), α ^′ _ij (x ^′ , t) = (x ^′ , t + g _ij ^′ (x ^′ )). Moreover, by the commutativity of this diagram, there exists a _i ∈ Γ(X _i , O X ^×

rZ ) = A ^× and b _i ∈ Γ(X _i , O X

rZ ) for each i = 0, . . . , r such that g _i ^′−1 ◦ Φ ◦ g _i (x, t) = (ϕ(x), a _i t + b _i ), g _j ^′−1 ◦ Φ ◦ g _j (x, t) = (ϕ(x), a _j t + b _j ). This is because for an isomorphism f : A[t] → B[t] of domains such that f | ^B _A : A → B is an isomorphism, f (t) should be equals to at + b where a ∈ B ^× and b ∈ B by the computation of degree of t. By the commutativity of the above diagram once again, we obtain the following equation;

a _i (x)t + b _i (x) + g _ij ^′ (ϕ(x)) = a _j (x)(t + g _ij (x)) + b _j (x).

Thus, gluing { a i } , we obtain a ∈ Γ(X + rZ, O ^× _X

_rZ ) ≃ Γ(X, O _X ^× ). Moreover, we have g _ij ^′ (ϕ(x)) − g _ij (x)a(x) = b _j (x) − b _i (x),

that is, cocyles { g _ij ^′ (ϕ(x)) } and { g ij (x)a(x) } define principal G a -bundles isomorphic to each other.

□

3. The proof of main theorem The purpose of this section is to prove Theorem 0.3.

Lemma 3.1 ([5]). Let V be an affine G a -surface. Then the GIT quotient V // G a is a nonsingular affine curve and there exists an open affine subset C ⊆ V // G a such that p ^{− 1} C is G a -equivariantly isomorphic to C × k G a , where p : V → V // G a is the quotient morphism.

Theorem 3.2 ([6]). Let X be a nonsingular affine curve, let W be an A ¹ -fibered affine surface

over X. Then W is Zariski 1-factor if and only if W is a line bundle over X.

Proof of Theorem 0.3. It is easy to see that V is affine and nonsingular. Take a closed point a ∈ A ¹ and define a morphism p ^′ : V → Y to be the composition V ≃ V × k { a } , → V × k A ¹ → W × k A ¹ → W → Y . We show that p ^′ satisfies the condition of the structure morphism of a principal G a -bundle over Y .

Suppose that V has no nontrivial G a -action for contradiction. Then V ≃ W holds by the cancellation theorem for varieties with no nontrivial G a -action ([1]), but this contradicts to that V has no nontrivial G a -action. Thus V has a nontrivial G a -action. We fix a nontrivial G a -action on V , denoted by µ : G a × V → V .

Let B := V //µ be the GIT quotient of µ and p : V → B be its quotient morphism. By Lemma 3.1, B is a nonsingular affine curve and there exists a nonempty open subset C ⊆ B such that the following diagram is commutative.

C × A ^{1 iso //}

$$ J

J J J J J J J J

J J P ^{− 1} C

 / V

p

C ^{ /} B

⟳

.

Using the same arguments as Theorem 0.4, we obtain an injective morphism ϕ : C → Y such that the following diagram commutes.

p ^{− 1} C × k A ¹

pr

 / V × k A ¹

pr

Φ : iso // W × k A ¹

pr

C × A ^{1 iso //}

&&

M M M M M M M M M M M M

M p ^{− 1} C

p

 / V

p

⟳

W

q

⟳

C ^{ /}

∃ ϕ

55 B

⟳

Y

l

Y ^′

Next, we construct a morphism ψ : Y → B. At first, we show that there exists a map ψ ^′ : Y (k) → B(k) of sets such that ψ ^′ ◦ q ◦ pr _W = p ◦ pr _V ◦ Φ ^{− 1} . For any closed point y ∈ ϕ(C) ⊆ Y , there exists a unique closed point y ∈ C such that x = ϕ(y). On the other hand, for any closed point y ∈ Y \ ϕ(C), a morphism

τ _y := p ◦ pr _V ◦ Φ ^{− 1} ◦ j _y : pr ^{− 1} q ^{− 1} (y) , → W × k A ¹ ≃ V × k A ¹ → V → B

is not dominant. (If τ _y is dominant, then C ∩ τ _y pr ⁻ _V ¹ q ^{− 1} (y) ̸ = ∅ , but this contradicts to the construction of ϕ.) Then it follows that the image of τ _y is a closed point of B. we define a morphism ψ ^′ : Y (k) → B(k) of sets to be ψ(y) = Imτ _y for each y ∈ Y (k). Then we have the equality ψ ^′ ◦ q ◦ pr _W = p ◦ pr _V ◦ Φ ^{− 1} . Let { Y _λ } λ ∈ Λ be an open affine covering of Y . Put W _λ := q ^{− 1} Y _λ ( ≃ Y _λ × k A ¹ ), (V × k A ¹ ) _λ := Φ ^{− 1} (W _λ × k A ¹ ), V _λ := pr _V ((V × k A ¹ ) _λ ) and X _λ := pV _λ . Next we observe that (V × k A ¹ ) λ = V λ × k A ¹ . Put K y,λ := pr _V ◦ Φ ⁻¹ ◦ Q ⁻¹ (y) ^V

. Then (V × k

A ¹ ) _λ = ∪

y ∈ Y

Φ ^{− 1} Q ^{− 1} (y) and V _λ = ∪

y ∈ Y

K _y,λ . Moreover, we have pr ^{− 1} K _y,λ ⊇ Φ ^{− 1} Q ^{− 1} (y), and

thus pr ^{− 1} K _y,λ = Φ ^{− 1} Q ^{− 1} (y). As a consequence, we obtain the equality (V × k A ¹ ) _λ = V _λ × k A ¹ .

For each λ ∈ Λ, we take a closed point (a _1,λ , a _2,λ ) ∈ A ² and define

ψ _λ,a

: Y _λ −−→ ^a

Y × k { a _1,λ } , → Y _λ × A ¹ → W _λ −−→ ^a

W _λ × k { a _2,λ } , → W λ × k A ¹ → V λ × k A ¹ → V λ → B.

Then for any closed point y ∈ Y _λ , ψ _λ,a

(y) = ψ ^′ (y). Therefore by Lemma 2.3, we can glue the morphisms { ψ _λ,a

} to a morphism ψ : Y → B . Moreover, ψ is a surjective birational morphism and satisfies q ◦ pr _V ◦ Φ ^{− 1} = ψ ◦ p ◦ pr _W .

The variety W _λ is isomorphic to Y _λ × k A ¹ because W _λ is a principal G a -bundle over Y _λ and Y _λ is affine. By Theorem 3.2, we obtain an isomorphism F _λ : V _λ ≃ W _λ . In the same way as ψ, we obtain a surjective birational morphism f _λ : Y _λ → B _λ such that f _λ ◦ q = F _λ ^{− 1} ◦ p. Moreover Y _λ and B _λ are nonsingular curves. Thus f _λ (and ψ _λ := ψ | ^B Y

) should be an isomorphism. In conclusion, we obtain the following commutative diagram.

V _λ

p

F

// W _λ

q

g

// Y _λ × k A ¹

pr

zzvvv vvv vvv v

f

× id

// B _λ × k A ¹

pr

◦ (f

× id

)

ttiiiii iiiii iiiii iiiii iiii

B _λ Y _λ

f

oo

⟳

For a closed point x ∈ C λ = B λ ∩ C, the fiber p ⁻ _λ ¹ (x) is just the orbit of µ by the construction of p and p _λ . For a closed point x ∈ B _λ \ C _λ , we can not say that the fiber p ⁻ _λ ¹ (x) is just the orbit only by the construction. But thanks to the fact that each G a -orbits are closed ([10]) and are isomorphic to either A ¹ or a closed point, we can deduce that p ⁻ _λ ¹ (x) is just the µ-orbit. Therefore, the action µ on V can be restricted on V _λ , denoted by µ | V

, and its quotient morphism is just p _λ : V _λ → B _λ . By the commutativity of the above diagram, such a G a -action should be G a -equivariantly trivial, that is, p _λ : V _λ → B _λ is a G a -equivariantly trivial morphism.

By the construction of p ^′ and ψ, we have the equality p ^′ _λ = ψ _λ ^{− 1} ◦ p _λ . Then it follows that p ^′ satisfies the condition of the structure morphism of a principal G a -bundle over Y . □

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