AFFINE THREEFOLDS ADMITTING Ga-ACTIONS R.V. Gurjar, M. Koras

AFFINE THREEFOLDS ADMITTING Ga-ACTIONS

R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

R.V. Gurjar, M. Koras, K. Masuda, M. Miyanishi, P. Russell had the wonderful opportunity to meet at BIRS during August 23-30, 2015, for the BIRS event Unipotent Geometry 15frg175. Some portion of our manuscript, including the Introduction of our manuscript below, describes some of the results we proved during our stay. Since we had only one week for discussions the Theorems 2.4 and 2.5 can be considered as the most complete results we proved during our stay. These results were previously not known and we believe that they will be appreciated by other experts interested in Ga actions. We also constructed several examples of Ga action on smooth aﬃne 3-folds. These examples are signiﬁcant since there is a general paucity of such examples in the literature.

Some of the results in the manuscript were proved during a Research In Pairs program at Oberwolfach, Germany, in August 2014. All the collaborators in this FRG, except Kayo Masuda, were also part of that program.

We still need to work out some details of proofs of results in other parts of the manuscript.

Date:September25,2015. 2000 Mathematics Subject Classification. Primary: 14R20; Secondary: 14R25. Key words and phrases. affine threefold, Ga-action, quotient surface, singularity. The second author is supported by Polish Grant MNiSW. The third author is supported by Grant-in-Aid for Scientific Research (B), No. 24340006, JSPS. The fourth author is supported by NSERC, Canada. 1 2R.V.GURJAR,M.KORAS,K.MASUDA,M.MIYANISHIANDP.RUSSELL

Abstract. Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by Ga-actions. We consider Ga-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers 4 of the quotient morphism. Relative Ga and Gm-actions on A or on an affine 4-fold having A3-fibrations over A1 are discussed when the actions leave one variable invariant or the fibration invariant.

Contents Introduction 2 1. The quotient space of smooth affine threefolds by Ga-actions 4 2. Singular fibers of the quotient morphism by a Ga-action 10 4 1 3. Relative fixpoint free Ga-actions on A over A 21 4 1 4. Relative effective Gm-actions on A over A 26 5. Danielewski surfaces of non-hypersurface type 31 6. Some results about A1-andP1-fibrations 39 7. Twisted additive groupschemes andtheiractions 42 References 48

Introduction Let X be a smooth affine threefold defined over an algebraically closed field k of characteristic zero. If X has a nontrivial Ga-action, there exists the quotient morphism q : X Y = X//G and Y has → a dimension two. Since the Ga-action is defined by a locally nilpotent k-derivation δ on the coordinate ring A of X, the quotient surface is defined by Y =SpecB with B =Kerδ.Thenthereexistsanopen 1 set U of Y such that U =SpecB[1/u]withu B and q− (U) = 1 ∈ ∼ U A . This implies that general fibers of q are Ga-orbits, but q has, in× general, singular fibers outside the open set U.Infact,topologicalor algebro-geometric properties of X are greatly affected by the singular fibers of q and the algebraic quotient surface Y ,andvice versa.In Theorem 1.1, we consider the case where X is, in addition, factorial and given two independent Ga-actions. Here we say that two Ga-actions on X are independent if the orbits by two Ga-actions have independent tangential directions at some point of X. Then it is shown that the quotient surface Y by one Ga-action is isomorphic to the affine plane A2 or an affine hypersurface x2 +y3 +z5 = 0 which is isomorphic to the quotient A2/Γ, where Γis the binary icosahedral subgroup of SL(2,k). AFFINE THREEFOLDS AND Ga-ACTIONS 3

Furthermore, Y is isomorphic to A2 if and only if X is simply connected. If one weakens the factoriality condition to the Q-factoriality, there are 2 examples for which Y ∼= A /Γ, where Γis now an arbitrary finite subgroup of SL(2,k). Some basic description of singular fibers of the quotient morphism q : X Y is given in [23]. Among other things, it is notable that if → asingularfiberF0 of q is one-dimensional, then every irreducible component of F0 is a contractible curve (k = C), and it is an interesting problem to ask whether it is always isomorphic to the affine line A1.In Theorem 2.1, it is shown that if an irreducible component has multiplicity one, i.e., it is reduced, then the component is isomorphic to A1. If the Ga-action is given by a homogeneous locally nilpotent derivation on the polynomial ring k[x, y, z] with a positive grading (quasi- homogeneous case), then the quotient morphism q : A3 A2,where → X = A3 and Y = A2 has only the affine line as the (unique) irreducible component of the (unique) singular fiber. This is shown by a detailed analysis of the singular fiber in Lemma 2.2 and Theorem 2.3. In the general case with only assumption that Y be smooth, every irreducible fiber component is isomorphic to A1 in Theorem 2.5. The singular locus Sing(q)ofthequotientmorphismq is also observed. It is shown in Theorem 2.8 that if X and Y are smooth and the Ga-action is analytically reduced everywhere on X then the union of multiple components of q coincide with the fixedpoint locus XGa . In the third section, we consider some of well-established results about Ga-actions acting on the affine space defined over k in the relative setting where we add one free variable to the polynomial ring and the Ga-action keeps the additional variable invariant. We consider 3 Kaliman’s theorem on a free Ga-action on A and Rentschler’s theo- 2 rem on the normalization of a Ga-action on A in the relative setting (see Kaliman’s Theorem and Example 3.5). Kaliman [27] has recently 4 proved that a fixed point free, proper Ga-action on A is a translation after a suitable change of variables. We analyzed this result from our point of view. In the fourth section, we consider a Gm-action on an affine 4-fold 1 3 fibered over A whose every fiber is isomorphic to A so that Gm acts on each fiber and the parameter space invariant. Theorem 4.1 shows that the 4-fold is actually trivial, i.e., isomorphic to A3 Y and the × Gm-action is diagonalized. In the fifth section, we treat a smooth affine surface V (m, 1) with 1 1 1 1 m 1whichhasanA -fibration ρ : V (m, 1) A such that ρ− (A ) = 1≥ 1 1 → ∗ ∼ A A and ρ− (0) consists of two reduced irreducible components ∗ × isomorphic to A1.Ifm =1,V (1, 1) is isomorphic to the hypersurface 4R.V.GURJAR,M.KORAS,K.MASUDA,M.MIYANISHIANDP.RUSSELL xy = z2 1, which is one of the Danielewski surfaces. However, if m − ≥ 2, V (m, 1) is not a hypersurface. In fact V (m, 1) = V (m#, 1) if m = '∼ 1 ' m#. Meanwhile, we have an isomorphism V (m, 1) A = V (m#, 1) × ∼ × A1.Hencethesesurfacesprovideanewkindofcounterexamplesto the cancellation problem (see Theorem 5.3). The surface V (m, 1) is constructed as the quotient variety of a Gm-action with Gm acting on the hypersurface threefold X(m, 1) = xy zmu =1 . In the sixth section, various generalizations{ − of the} Nori exact sequence of the fundamental groups of a fibration f : X Y with connected fibers such that each fiber contains at least one→reduced irreducible component (see [43]). This exact sequence is very effective when we use topological arguments for affine varieties. Hence its generalizations have been considered, and the section gathers together those generalizations. Applications of generalized results are also observed. In the seventh section, we defined an -twisted additive group scheme L Ga, over a scheme Y for an invertible sheaf on Y .Thisenablesus L L to treat A1-fibrations (even defined over a complete variety) from the viewpoint of the additive group scheme action. The idea is originally due to Dubouloz [11]. We denote by R[n] apolynomialringoveraringR in n variables whenever we do not specify variables of the polynomial ring. A locally nilpotent derivation is aften abbreviated as lnd. This article is an outcome of the joint research conducted as a RIP program of Mathematisches Forschungsinstitut Oberwolfach (MFO) during the period August 4 to 15, 2014 and as a workshop entitled unipotent geometry at the BanffInternational Research Station for Mathematical Innovation and Discovery (BIRS) during the period Au- gust 23 to 30, 2015. We are very grateful to MFO and BIRS for their splendid research environments and warm hospitality.

1. The quotient space of smooth affine threefolds by Ga-actions

We are interested in a smooth, factorial, aﬃne threefold X with Ga- actions and its quotient surface Y by one of the Ga-actions. Central problems are if Y is smooth or what kind of singularity it admits provided Y is singular. Given two Ga-actions on X,wesaythatthetwo actions are independent if two respective Ga-orbits passing a point P of X have independent tangential directions. If X admits two independent Ga-actions, then we have the following result. Theorem 1.1. Let X be a smooth aﬃne factorial threefold with two independent G -actions, say G ,G.AssumefurtherthatΓ(X, )∗ = a 1 2 OX AFFINE THREEFOLDS AND Ga-ACTIONS 5

2 k∗.ThenX//G1 is isomorphic to either A or an aﬃne hypersurface x2 + y3 + z5 =0.Furthermore,X is simply connected if and only if 2 X//G1 ∼= A .

Proof. Let q1 : X Z := X//G1 be the quotient morphism. For → 1 ageneralpointp Z, let Cp = q1− (p)andYp be the closure of ∈ 1 G2P .ThenCp = and Yp is an affine surface equipped with P Cp ∼ A ∈ 1 an A -fibration G2P P Cp .NotethatG2P = G2P # for distinct ! { | ∈ } points P, P # when the orbit G2P meets Cp in the point P #.Butnote that G2P G2P # = implies G2P = G2P #.Considerthefollowingtwo cases (i) and∩ (ii) separately.' ∅ (i) Suppose that q1 Y : Y Z is dominant. Choosing the point p | → 1 generally on Z,wemayassumethatq1 Y is quasi-finite .LetZ◦ be the | 1 smooth part of the normal surface Z.Thenq1− (Z Z◦) is a finite set and lies on a union of finitely many orbits G P with\ P C .Further, 2 ∈ p by eliminating finitely many G2-orbits, we have a dominant morphism 1 from a smooth affine surface with an A -fibration to Z◦.Henceκ(Z◦)= 0 1 .IfZ has an A -fibration, it extends to Z. Otherwise, Z◦ contains −∞ 2 2 an open set U of the form (A /Γ)◦, where Γis a finite group, (A /Γ)◦ 2 is the smooth part of A /ΓandZ◦ U is a disjoint union of the curves \ isomorphic to A1 which are called the half-point attachments.Hence the curves in Z◦ U give rise to the independent classes of Pic (Z◦). \ Meanwhile, since Z is factorial, it follows that Pic (Z◦) = 0. This 2 2 implies that Z◦ = U ∼= (A /Γ)◦.ThenZ = A /ΓbecauseZ Z◦ is a finite set. Since Z is factorial, Γmust be the binary icosahedral\ group of SL(2,k). Hence Z is an affine hypersurface x2 + y3 + z5 =0. Suppose that X is simply connected. Let ρ : A2 A2/Γbethe 2 → 2 2 quotient morphism by the Γ-action on A .Thenρ◦ : A := A 2 ∗ \ 0 (A /Γ)◦ is a universal covering with Galois group Γ. Let X◦ := { }→1 2 1 2 1 2 X q− (Sing(A /Γ)) = q− ((A /Γ)◦). Note that q− (Sing(A /Γ)) is a \ closed set of codimension 2 in X.HenceX◦ is simply connected. ≥ 2 0 π◦ 2 ρ◦ 0 Then q◦ := q X◦ : X◦ Z◦ is factored by A as q◦ : X A Z . | → ∗ −→ ∗ −→ Then a general fiber of q (and hence q◦) is a disjoint union of as many affine lines as the order of Γ. This is a contradiction. Hence the case 2 Z ∼= A /Γdoesnotoccur. 1 The actions of G1,G2 are associated with locally nilpotent derivations δ1,δ2. 1 1 We may replace δ1,δ2 by a1− δ1,a2− δ2 if necessary with a1 Ker δ1,a2 Ker δ2. ∈ 1 ∈ Thus we may assume that the fixed-point locus of G1 is contained in q1− (S1)with 1 afinitesetS1 of Z. Similarly, the fixed-point locus is contained in q2− (S2) with a finite set S of X//G ,whereq : X X//G is the quotient morphism. Then, for 2 2 2 → 2 ageneralpointp of Z, every point P of Cp has the one-dimensional orbit G2P .If G2P is contained in the orbit of q1, it must be Cp itself. This is impossible. 6R.V.GURJAR,M.KORAS,K.MASUDA,M.MIYANISHIANDP.RUSSELL

(ii) Suppose that q1 Y : Y Z is not dominant. Then the image of q is a rational curve| B →with one place at infinity which passes 1 |Y p through the point p.TheassumptionimpliesthatforanypointP Cp the orbit G P is mapped surjectively onto B .IfB B = ∈ for 2 p p ∩ p" ' ∅ distinct points p, p# of Z,thenG P G P # = for P C and P # C . 2 ∩ 2 ' ∅ ∈ p ∈ p" Since G2P, G2P # are the G2-orbits, it follows that G2P = G2P # and hence Bp = Bp" .Thusthefamily Bp p Z has no base points. This { } ∈ 1 implies that a general member Bp is smooth, hence isomorphic to A . 1 So, Z has an A -fibration. Since Z is factorial and Γ(Z, Z)∗ = k∗, it 2 O follows that Z ∼= A . By the above argument, we have shown that Z is isomorphic to 2 2 2 either A or A /Γ. Finally, we prove that if Z ∼= A then X is simply connected. Let S be the closed curve such that for each general point 1 1 p S the fiber q− (p) is not isomorphic to A in the scheme-theoretic ∈ sense. Let S1 be an irreducible component of S and let f be a prime [2] element of Γ(Z, Z) = k such that S1 = V (f). Since f is a prime O ∼ 1 element of Γ(X, X ), the surface T1 := q− (S1) is an irreducible surface. Considering theO Stein factorization of q : T S we know that the |T1 1 → 1 general fibers consist of reduced irreducible components. Now let Z◦ 1 be the open set of Z such that for every point p Z◦ the fiber q− (p) is ∈ 1 reduced. Then Z Z◦ is a finite set. Let X◦ = q− (Z◦)andq◦ = q X◦ . We apply Nori’s\ exact sequence of the fundamental groups [43] (see| Lemma 6.1 of the present article)

π (F ) π (X◦) π (Z◦) (1), 1 → 1 → 1 → where F is a general ﬁber of q◦. Since X X◦ and Z Z◦ have codimen- \ \ sion larger than one, we have π1(X◦)=π1(X)andπ1(Z◦)=π1(Z)= (1). Hence π1(X)=(1). 2

2 We do not know if the case X//G1 ∼= A /ΓwithΓ =(1)canoccur. But if we drop the assumption that X is factorial, such' an example exists. Example 1.2. Let X be an affine quadric hypersurface defined by xz − yu =1in A4 =Speck[x, y, z, u].Thenthefollowingassertionshold. " (1) X is a smooth factorial affine threefold with a Ga-action defined by a locally nilpotent derivation δ such that " δ(x)=δ(y)=0, δ(z)=" y, δ(u)=x. (2) Let ι be the" involution" on X "defined by" ι(x)= x, ι(y)= y,ι(z)= z, ι(u)= u. − − " − − AFFINE THREEFOLDS AND Ga-ACTIONS 7

Then ι has no fixed point on X,andhencethequotientthree- fold X := X/ ι is a smooth affine threefold. Furthermore, δ + , commutes with ι,i.e.,δι = ιδ." (3) Let R := Γ("X, ) and let R =Γ(X, ).ThenR is the ι"- OX OX invariant subring and δ" defines" a locally nilpotent derivation δ ! on R". " (4) Let σ and σ be the Ga-actions" on X and X respectively defined 2 2 by δ and δ.ThenX//Ga ∼= A and X//Ga ∼= A /Γ,where Γ="Z/2Z. " (5) Pic" (X) ∼= Z/2Z. " Proof. The assertions (1), (2) and (3) are easy to show. As for the assertion (4), note that Ker δ =Kerδ R and Ker δ = k[x, y]. Hence ∩ Ker δ = k[x2,xy,y2]whichisthecoordinateringofA2/ΓwithΓ= Z/2Z. " " (5) Let p be a prime ideal of height 1 of R.ThenpR is a height 1 ideal of R.HenceitiswrittenaspR = fR with f R since R is ∈ factorial. Since ι∗(pR)=pR,wehaveι(f)=fh with h "R such that ∈ ι(h)h = 1." Since R∗ = k∗,wehaveh "k and"ι(h)=h.Then" h = " 1. If h =1,thenf "R and"p = fR.If∈ h = 1, then ι(f)=" f.In± ∈ − − particular, f 2 R", and if f ,f R satisfy ι(f )= f ,ι(f )= f ∈ 1 2 ∈ 1 − 1 2 − 2 then f1f2 R.ItfollowsfromtheseobservationsthatPic(X) = Z/2Z, ∈ ∼ and it is generated by the prime ideal" xR R. 2 ∩ The following example generalizes the above example. Namely, Ex- ample 1.2 is the case n =2intheexamplebelow."

ξ 0 n Example 1.3. Let Tn = 1 ξ =1 be a cyclic subgroup 0 ξ− #$ % & ' of SL(2) of order n.LetTn act on SL(2)& by the left multiplication 1& t and let X = T SL(2).LetU = t k be the additive n\ 01 ∈ #$ % & ' subgroup acting on SL(2) by the right multiplication.& Since the actions of Tn and U commute each other, there is a right& Ga-action on X.Let Y := X//Ga and let q : X Y be the quotient morphism. Then we have : → (1) X is a smooth aﬃne threefold with Pic (X) = Z/nZ. 2 ξ 1∼ 2 (2) Y = Tn A with the action (x, y)=(ξx,ξ− y) . ∼ \ 2 2 To avoid the notation Tn A , we can consider the left action of the lower tri- 10\ angular subgroup t k as a G -action and the right action of T . t 1 ∈ a n #$ % & ' & & 8R.V.GURJAR,M.KORAS,K.MASUDA,M.MIYANISHIANDP.RUSSELL

Proof. (1) Since X is a homogeneous space, it is clear that X is a smooth affine threefold. The computation of Pic (X) is the same as xu in Example 1.2. If we write a general matrix of SL(2) as , yz $ % ξ 0 xu x is transformed to ξx by the left action of 1 on . 0 ξ− yz $ % $ % Hence the prime ideal p = xR R gives rise to a generator of Pic (X), ∩ where R is the coordinate ring of SL(2) and R is the Tn-invariant subring of R. " (2) The" quotient morphism q : X Y is induced by the quotient → xu morphism q":SL(2) SL(2)/U which is given as (x, y). → yz -→ $ % (Take the quotients of SL(2) and SL(2)/U with respect to the left Tn- " 2 multiplications.) Hence Y = Tn A . 2 ∼ \ By considering other finite subgroups of SL(2), we can produce A2/Γ as the quotient surface of a smooth affine threefold by a Ga-action, where Γis any finite subgroup of SL(2). Furthermore, note that X = SL(2) admits other Ga-actions. For example, the right multiplication 10 by the lower triangular matrices t k gives another G - t 1 ∈ a #$ % & ' action. So, the assumption that X be factorial& seems to be crucial to conclude that X//G1 is smooth. &

Concerning the smoothness of the quotient surface X//Ga, the following problem is interesting, where the assumption that X be simply- connected and rational is new. Problem 1.4. Let X be a smooth, factorial, simply-connected, rational, affine threefold with a nontrivial Ga-action. Is Y := X//Ga then smooth? Proposition 1.5. With the notations in Problem 1.4, Y is a factorial, rational, affine surface such that (1) Y Sing(Y ) is simply-connected, and − (2) apointP Y has quotient singularity at worst of E8-type 1∈ provided q− (P ) = . ' ∅ Proof. Since X is factorial, the quotient morphism q : X Y does 1 → not contain codimension 1 fiber components. Hence q− (Sing(Y )) has 1 codimension larger than 1. This implies that π1(X q− (Sing(Y ))) = (1). Let p : Z Y Sing(Y )betheuniversalcovering.Thenthe− → − 1 restriction of q onto X q− (Sing(Y )) is factored by the mapping p. This implies that p is a− finite covering and p is the identity since the AFFINE THREEFOLDS AND Ga-ACTIONS 9 general fibers of q are connected. So, π1(Y Sing(Y )) = (1). This 1 − proves the assertion (1). If q− (P ) = ,thenthesingularityofY at P is at most quotient singularity. Since' ∅ is factorial, the singularity OY,P is at worst of E8-type. This verifies the assertion (2). 2 When we ask if a factorial, rational, affine surface Y is smooth provided π1(Y Sing(Y )) = (1), we have the following counterexample. Note that if− we assume additionally that Y is contractible, then Y is smooth by an affine Mumford theorem [21, Theorem 3.6]. Example 1.6. (1) Let V be an affine surface constructed in [21, Propo- sition 3.8].ThenV is a factorial, rational, affine surface with an E8- singularity and π (V Sing(V )) = (1).Thisimpliesthatafactorial, 1 − rational, affine surface Y with π1(Y Sing(Y )) = (1) is not necessarily smooth. − (2) Let a, b, c be mutually coprime positive integers. Then the affine hypersurface xa + yb + zc =0is factorial, though it has a non-quotient singularity for a suitable choice of a, b, c.Wereferto[41]. As a corollary of Theorem 1.1, we prove the following result. Theorem 1.7. Let R be a factorial affine domain of dimension two and let R[x] be a polynomial ring in one variable x over R.LetX = Spec R[x], Y =SpecR and p : X Y be the projection. Then the following three conditions are equivalent.→ (1) The Makar-Limanov invariant ML(X) is equal to k.Namely, there are three independent Ga-actions on X. (2) Y is isomorphic to A2. (3) X is isomorphic to A3. Proof. It suffices to show that the condition (1) implies the condition (2). The existence of three independent Ga-actions on X implies that there is a dominant morphism from G G G to X.Hencetheunit a × a × a group of R[x], which is equal to R∗, is k∗. Consider an lnd δ of R[x]such that δ(R)=0andδ(x)=1.Thenδ gives rise to a Ga-action along the fibers of the projection p. Namely, p is the quotient morphism under the Ga-action. Note that R[x] is factorial. By Theorem 1.1, the surface Y is isomorphic to the affine plane A2 or the hypersurface x2 +y3+z5 = 0 in A3.SupposethatY is isomorphic to the hypersurface x2 + y3 + z5 =0.SincethesingularityofY is not a cyclic singularity, ML(Y )=R by [39]. By Crachiola and Makar-Limanov [7], we then have ML(X)=ML(Y )=R.Thiscontradictstheassumption.Hence Y is isomorphic to A2. 2 Generalizing Theorem 1.7, we raise the following problem. 10 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

Problem 1.8. Let R be an aﬃne domain of dimension n over k and let R[x] be a polynomial ring in a variable x over R.Supposethat ML(R[x]) = k.IsML(R) then equal to k? But the problem has negative answers. We give two counterexam- ples. In the ﬁrst example, dim R =2butR is not factorial. In the second example, R is factorial but dim R = 3. This question was treated, for example, by Bandman-Makar-Limanov [5].

Example 1.9. For an integer n 1,letSn be the Danielewski surface n 2 ≥ x y = z 1.ThenML(Sn)=k if n =1and ML(Sn)=k[x] if − 1 1 1 n 2 [38].ThenS1 A = Sn A for n>1 and ML(S1 A )= ≥ 1 × ∼ × × ML(Sn A )=k,butML(Sn)=k[x] if n>1. × The following example is due to Dubouloz [13], which is also a counterexample to the conjecture in [5, p.209]. Example 1.10. Let X be Koras-Russell cubic threefold x2y + x + z2 + t3 =0,whichisahypersurfaceinA4.LetR be the coordinate ring of X.ThenR is factorial, ML(X A1)=k and ML(X)=k[x]. × 2. Singular fibers of the quotient morphism by a Ga-action In [23, Lemma 3.5], we proved that if X is a smooth factorial affine threefold with a G -action and the quotient morphism q : X Y := a → X//Ga,asingularfiberisadisjointunionofcontractiblecurves,and asked if the singular fiber is indeed a disjoint union of the affine lines. In the present section, we prove three results on the smoothness of fiber components of the morphism q,Theorem2.1,Theorem2.3andTheo- rem 2.5 among which the last result is the most general and contains the former two as partial cases. We think that the arguments used in the proofs of these results have independent interest. This is why we do not throw the first two results and retain only the last one. We prove first the following result.

Theorem 2.1. Let X be a smooth factorial affine threefold with a Ga- action. Assume that the quotient surface Y := X//Ga is isomorphic to the affine plane. Let F be a singular fiber of the quotient morphism q : X Y and let F be an irreducible component with multiplicity 1, → 0 i.e., a reduced component. Then F0 is isomorphic to the affine line. Proof. (1) Let B := Γ(Y, )=k[f,g], where f α, g β are prime OY − − elements in A := Γ(X, X )forallα,β k.LetQ = q(F ). We may O ∈ 1 assume that f =0atQ.LetL = f =0 in Y and let Z = q− (L). { } AFFINE THREEFOLDS AND Ga-ACTIONS 11

Then Z is an irreducible and reduced surface with the induced Ga- action. A general fiber of q Z : Z L is not necessarily irreducible. If reducible, by a linear change| of coordinates→ f,g on Y ,wemayassume that L passes through a point of Y which is the image of a general 1 fiber of q.Thenageneralfiberofq Z is isomorphic to A .Thenthe morphism q := q : Z L is the quotient| morphism of Z relative to Z |Z → the induced Ga-action. (2) We consider a singular fiber F on the affine surface Z.ThoughZ is not necessarily normal, it has the quotient curve L =Speck[g]with F defined by g =0.LetF0 be a reduced irreducible component of F . We then replace Z by (Z F ) F0,whichisanaffinesurfacewiththe \ ∪ 1 induced Ga-action. Suppose that F0 is not isomorphic to A .ThenF0 is a Ga-stable, singular, contractible curve. Hence F0 is contained in the fixed-point locus ZGa .Letδ be the locally nilpotent derivation on Ga the ring R := Γ(Z, Z). Since F0 is defined by g =0andF0 Z ,the induced locally nilpotentO derivation on R/gR is trivial. In⊆ fact, every maximal ideal m of R/gR is mapped into m itself. If we fix a maximal ideal m,everyelementa R/gR has some constant α k such that ∈ ∈ a α m.Thenδ(a)=δ(a α) m.So,δ(a) mm,wherem runs− through∈ over the set of all maximal− ∈ ideals of R/gR∈∩. Since R/gR is reduced by assumption, we have δ(a) mm = (0). This implies that 1 ∈∩ δ(R) gR. Since g Ker δ, g− δ is a locally nilpotent derivation of ⊂ ∈ R such that the associated Ga-action has the same quotient morphism Ga qZ : Z L.IfF0 is still contained in the fixed-point locus Z ,we repeat the→ same process. But it is impossible that δ is divisible by g infinitely many times. Hence we eventually reach to the situation Ga 1 F0 Z .ThenF0 must be isomorphic to A . This is a contradiction. '⊂ 1 Hence F0 has no singular points, and hence isomorphic to A . 2 We consider the following quasi-homogeneous case. We need some observations. Lemma 2.2. We assign weights a, b, c to the variables x, y, z of a polynomial ring k[x, y, z],wherea, b, c are pairwise coprime positive integers. Suppose that δ is a homogeneous locally nilpotent derivation with respect to this grading. Let F, G be the generators of Ker δ,i.e., Ker δ = k[F, G].Thenwehave: (1) We may assume that

Fy Fz ∂ Fz Fx ∂ Fx Fy ∂ δ =∆(F,G) := + + . Gy Gz ∂x Gz Gx ∂y Gx Gy ∂z & & & & & & & & & & & & Furthermore,& the coe&ﬃcients& of ∆(F,G)&has no& non-constant& common divisors& in k[x,& y, z]. & & & & 12 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

(2) The fixed-point locus (A3)Ga has dimension 1. (3) Let C be a singular curve which is a fiber≤ component of the 3 quotient morphism q : X Y = X//Ga,whereX = A and → Y =Speck[F, G].ThenC =(A3)Ga .HencesuchacurveC is uniquely determined. (4) Let π : X (0, 0, 0) Π be the quotient morphism by the Gm- action induced\{ by the}→ grading, where Π is the weighted projective plane. Let V+(F ) and V+(G) be the curves defined by F =0and G =0in the weighted projective plane Π.LetQ0 be the unique point of intersection V+(F ) V+(G) (see [9, Theorem] or [19, 1 ∩ 3 Ga Theorem 5.28]).Thenπ− (Q0) is contained in (A ) .IfC is 1 asingularcurveasintheassertion(3),thenC = π− (Q0) (0, 0, 0) . ∪ { } Proof. (1) In [8] or [19, Theorem 5.6], it is shown that the derivation ∆ is locally nilpotent, and δ = H∆ for H Ker δ.So,we (F,G) (F,G) ∈ can take δ to be ∆(F,G).IfA is a common divisor of the coefficients 3 1 of ∆(F,G),thenA Ker δ and A− δ is a locally nilpotent derivation giving rise to the same∈ kernel as Ker δ.Thenbythecitedresult[ibid.], 1 we have A− δ = Bδ with δ =∆(F,G) and B Ker δ.HenceAB =1 and A is a constant. ∈ (2) Suppose that dim(A3)Ga =2.Then(A3)Ga contains an irreducible component of codimension one which is defined by an element H of Ker δ.Sincethedefiningidealof(A3)Ga is the ideal generated by the coefficients of ∆(F,G),thepolynomialH is a common factor of the coefficients of ∆(F,G). This is a contradiction to the assertion (1). (3) We know that C is contractible by [23]. Let P0 be a singular 3 Ga point of C.ThenP0 (A ) because C is Ga-stable and the sin- ∈ gular point of C does not move under the Ga-action. If P is a point of C other than P0,theGa-orbit GaP contains P0 if dim GaP =1, 3 Ga whence P0 would be a smooth point. Hence C (A ) .Further- ⊆ more, the curve C is Gm-stable with respect to the Gm-action associated to the positive grading. In fact, the assumption that the locally nilpotent derivation δ is quasi-homogeneous derivation of degree d im- 1 plies that there is a Ga ! Gm-action on X,where(0,λ− )(t, 1)(0,λ)= d 3 Ga (λ t, 1) for λ Gm and t Ga.Hence,ifP (A ) ,wehave ∈ 1 ∈ ∈d (t, 1)(0,λ)P =(0,λ)(0,λ− )(t, 1)(0,λ)P =(0,λ)(λ t, 1)P =(0,λ)P . This implies that the point (0,λ)P is a G -fixed point. If (0,λ)P C, a '∈ then the Gm-translate GmC will give a surface after taking its closure.

3 Suppose that ∆(F,G) = AD for a derivation D.Thereexiststhenanelement ξ k[x, y, z]suchthat∆ (ξ) is a nonzero element of Ker∆ .Thenafactor ∈ (F,G) (F,G) A of an element of Ker ∆(F,G) is also in the same kernel. AFFINE THREEFOLDS AND Ga-ACTIONS 13

This leads to a common factor of the coefficients of∆ (F,G) which is a contradiction by the assertion (1). Hence (0,λ)P C.So,C is G - ∈ m stable. Since the closure of a Gm-orbit can have a singular point only 3 Ga at the origin (0, 0, 0), the point P0 is the origin (0, 0, 0). If (A ) contains another irreducible component, say C#,thenC# is a curve passing 1 through the origin and an irreducible component of the fiber q− (q(C)). But the irreducible components are connected components in the fiber 1 3 Ga q− (q(C)). Hence there are no other irreducible components in (A ) other than C. 1 (4) Let P π− (Q0). Then F ((t, 1)P )=(t, 1)∗F (P )=F (P )=0 ∈ 1 because F Kerδ.Similarly,G((t, 1)P )=0.Hence(t, 1)P π− (Q0). ∈ 1 1∈ 1 If (t, 1)P = P ,thenthereisadominantmorphismA π− (Q0) = A . ' → ∼ ∗ This is a contradiction. Hence (t, 1)P = P and P (A3)Ga . 2 ∈ Theorem 2.3. With the notations and assumptions as in Lemma 2.2, we further assume that the integers a, b, c in the triple (a, b, c) are greater than 1.Thentherearenosingularcurveswhicharefibercom- ponents of the quotient morphism q : X Y . → Proof. We make essential use of the following two facts. (1) Πis a projective normal surface at worst with cyclic quotient singularities (see [10, Proposition 1.3.3]). Let Q1,Q2,Q3 be the vertices of Π, i.e., Q1 = π(1 : 0 : 0),Q2 = π(0 : 1 : 0),Q3 = π(0 : 0 : 1). Then these three points are singular points of Π. 2 (2) Π (V+(F ) V+(G)) is isomorphic to P (,1 ,2) (see [9, Theorem]\ or∪ [19, Theorem 5.28]). \ ∪ We assume that there exists a singular curve C which is an irreducible component of the quotient morphism q as in Lemma 2.2 and show that this assumption leads to a contradiction. By (2) above, the singular points of Πlie on the curves V+(F ) V+(G). Let V = Π (V (F ) V (G)). By (2) above, V is an affi∪ne surface with an \ + ∪ + A1-fibration. In fact, V = A1 A1.NotethatF and G are irre- ∼ × ∗ ducible polynomial in k[x, y, z]. In fact, suppose that F = F1F2.Then F ,F k[F, G]becauseKerδ is factorially closed in k[x, y, z]. We can 1 2 ∈ write F1 =Φ1(F, G)andF2 =Φ2(F, G). Then F =Φ1(F, G)Φ2(F, G) with two non-constant polynomials Φ1(F, G), Φ2(F, G). This is a contradiction. Hence the curves V+(F )andV+(G)areirreducible.If Q0 Q1,Q2,Q3 ,thentwoofQ1,Q2,Q3 lie on one of V+(F ),V+(G). Suppose'∈{ Q ,Q } V (F ). Consider an affine surface W =Π V (G). 1 2 ∈ + \ + Then W is a normal affine surface with an A1-fibration whose fibers are all irreducible. But this is a contradiction because an irreducible fiber component of an A1-fibration on a normal affine surface can 14 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL carry at most one cyclic singular point of the surface by [39]. Hence 1 Q0 Q1,Q2,Q3 . This implies that C = π− (Q0) (0, 0, 0) is a ∈{ } 1 ∪{ } line and hence C ∼= A . This is a contradiction. 2 Now we will prove the following general result which has applications to the study of singular fibers of the quotient morphism X X//G , → a where X is a smooth affine threefold and X//Ga is a smooth affine surface. This result is much more general than Theorem 2.1 and Theorem 2.3.

Theorem 2.4. Let f : V Y be a projective morphism from a smooth threefold V onto a smooth→ surface Y .Assumethatageneralfiberoff is isomorphic to P1.ForapointP Y ,letC be a one-dimensional 1 ∈ 1 component of the fiber FP := f − (P ).ThenC is isomorphic to P . Proof. It is proved in Kollàr [33, p. 107, (2.8.6)] that with f,V,Y i as above we have R f ( V )=0fori>0. We can assume that Y is affine. Then, by a∗ standardO spectral sequence argument, we have i H (V, V )=0fori>0. NowO we will use the Theorem on Formal Functions [25, Chapter III, Theorem 11.1]. Let be the local ring of Y at P and let m be the maximal ideal of . As usualO let V := V Spec ( /mn)and := /mn, consideredO as n ×Y O On O the structure sheaf of Vn.Foreachn 0wehavenaturalmorphisms i n i ≥ R f ( V ) /m H (Vn, n). As n varies we have two inverse systems,∗ O inducing⊗O a→ natural morphiO sm which is an isomorphism by the Theorem of Formal functions

i ∼ i R f ( )P∧ lim H (Vn, n). ∗ O −→ ←− O For i>0 the LHS is 0, hence we get lim Hi(V , )=0. n On We write V = U U ,whereU ←is− a suitable open neighborhood n 1n ∪ 2n 1n in Vn of the union of all one-dimensional components of Vn and U2n is an open neighborhood in Vn of the union of all the irreducible components of dimension > 1. We can assume that U1n U2n is a disjoint union of connected (non-reduced) Stein spaces. ∩ We denote the sheaves of abelian groups , on U ,U resp. O1n O2n 1n 2n which are just restrictions of n to U1n,U2n resp. In this siuation there is a Mayer-VietorisO sequence (obtained by using sheaves of discontinuous sections to obtain cohomology) [1, p. 236]

H1(V , ) H1(U , ) H1(U , ) ···→ n On → 1n O1n ⊕ 2n O2n H1(U U , ) . → 1n ∩ 2n On →··· AFFINE THREEFOLDS AND Ga-ACTIONS 15

Since U1n U2n is a disjoint union of ﬁnitely many connected Stein spaces, the∩ last cohomology group is trivial. As n varies, we get Mayer- Vietoris sequences with maps from the groups in the (n+1) st sequence to the corresponding groups in the n th sequence making all the dia- grams commute. Since lim Hi(V , )=0,wededucethatlimHi(U , )=0.Let n On 1n O1n In be the←− ideal sheaf of U1n in U1(n+1). ←− ¿From the exact sequence 0 I 0, and → n →O1(n+1) →O1n → using the fact that U1n is a non-compact 2-dimensional complex space 2 without 2-dimensional compact components so that H (U1(n+1),In)= (0), we get that the natural maps H1(U , ) H1(U , ) 1(n+1) On+1 → 1n On are surjections for each n.SincelimHi(U , )=0wededucethat 1n O1n H1(U , )=0forn>0. ←− 1n On The reduced curve C Fp is a closed subscheme of U1n for each ⊂ 1 n.BythesameargumentasabovewededucethatH (U1n, n) 1 1 O → H (C, C) is a surjection. This means that H (C, C)=0,proving O 1 O that C ∼= P . 2 As a corollary, we obtain the following result.

Theorem 2.5. Let Ga act on a smooth affine threefold X.LetY be the affine variety corresponding to the ring of invariants. Assume that 4 Y is smooth .Letq : X Y be the induced morphism. If C0 is a → 1 one-dimensional component of a fiber of π,thenC0 ∼= A . Proof. We can embed X V as a Zariski-open subvariety where V is a smooth quasi-projective⊂ threefold such that π extends to a proper morphism f : V Y .ThentheclosureC of C in X is a one- → 0 dimensional component of the fiber containing C0.Asalreadymen- 1 tioned, C0 is contractible. By Theorem 2.4, we have C = P .Itfollows 1 ∼ that C0 ∼= A . 2 Remark 2.6. It seems reasonable to conjecture that Theorem 2.4 (Theorem 2.5) is true for P1-fibrations (resp. A1-fibrations) on n-folds for n>3. 2

Let q : X Y be an A1-fibration of normal affine varieties. Let Sing(q) be the→ set of points P Y such that the fiber F ,i.e.,the ∈ P 4 This condition is not a serious restriction. Since A := Γ(X, X ) is regular, Ker δ is normal. Hence Y has only isolated singular points. If oneO consider the fiber FP and its fiber component C0 over a smooth point P of Y ,wereplaceY by 1 asmoothaffineopenneighborhoodU of P and replace X by q− (U). Since q is an 1 affine morphism, the restriction q 1 : q− (U) U satisfies the conditions of |q− (U) → the theorem. 16 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL scheme-theoretic inverse image X Spec k(P ) is not isomorphic to ×Y A1,wherek(P ) is the residue field of P in Y . We call Sing(q)the singular locus or degeneracy locus of q. We do not know in general if Sing(q) is a closed set in Y . However we know the following partial result. Lemma 2.7. (1) Assume that q is a flat morphism and Y q(X) has codimension greater than one in Y .TheneitherSing(q)=− or has ∅ pure codimension one. If Sing(q)= then q : X q(X) is an A1- bundle. ∅ → (2) Assume that q is the quotient morphism of a smooth affine threefold X equipped with a Ga-action. Suppose that q is equi-dimensional. Then Sing(q) is a closed set. Furthermore, if the fiber FP contains a reduced irreducible component, then the point P is smooth in Y . Proof. For (1), see [23, Lemma 1.15], and for (2), see [22, Lemma 3.1]. 2 Hereafter until Proposition 2.9, we assume that X =SpecA is a smooth affine variety of dimension n equipped with a Ga-action σ which corresponds to an lnd δ of A and that q : X Y is the quotient → morphism by σ,henceB := Γ(Y, Y )=Kerδ is finitely generated over k.AssumethatthequotientmorphismO q : X Y := X//G has → a equi-dimension one. The fixed point locus XGa is then a union of fiber components of q by [23, Corollary 3.2], which is valid for all n 2. An irreducible fiber component E is a multiple component of q if the≥ Artin local ring FP ,ξ is not a field, where ξ is the generic point of E.The length of O is the multiplicity of E. OFP ,ξ Given the Ga-action σ on X =SpecA, let δ be the corresponding lnd of A.IfQ is a point of X then δ induces the k-derivation δQ of the local ring as well as the k-derivation δ of the completion . OX,Q Q OX,Q We say that σ (or δ) is analytically reducible at Q if δQ = h∆foran element h of the maximal ideal mX,Q and( a k-derivation ∆of (X,Q. Otherwise we call δ is analytically reduced at Q. The fixed( pointO locus XGa is defined by the ideal I of A generated by δ(A) := δ(a) a ( A . ( { | ∈ } Write δQ as ∂ ∂ δQ = f1 + + fn , ∂x1 ··· ∂xn where x1,...,xn is a regular system of parameters of X,Q.Then the following{ conditions} are equivalent. O (i) Q XGa . ∈ (ii) f1(Q)= = fn(Q)=0. (iii) f ,...,f··· I . 1 n ∈ OX,Q AFFINE THREEFOLDS AND Ga-ACTIONS 17

(iv) (f ,...,f ) = I . 1 n OX,Q OX,Q (v) (f ,...,f ) = I . 1 n OX,Q OX,Q We say that σ is reducible if δ = hδ# for a k-derivation δ# of A.If this is the case, h ( Ker δ (and δ# is an lnd (see [19, p.33]). Other- ∈ wise, δ is reduced.Ifδ = hδ# with h m and a k-derivation δ# of Q ∈ X,Q X,Q,wesaythatδQ is reducible. This is equivalent to saying that IO has a height one prime divisor since is factorial. Sim- OX,Q OX,Q ilarly, δ is analytically reducible if and only if I has a height Q OX,Q one prime divisor. Then δ is reducible at Q if and only if δQ is analytically( reducible. In fact, the “only if” part is clear.( For the “if” part, suppose that I has a height one prime divisor p(. Since OX,Q = k[[x ,...,x ]], where we may assume that x ,...,x m , OX,Q 1 n 1 n ∈ X,Q we have p =(h)with(h k[[x ,...,x ]]. By the Weierstrass( Prepa- ∈ 1 n ration( Theorem it follows that h = hu with h k[x ,...,x ]and ∈ 1 n aunitu( k[[(x1,...,x(n]]. Then h I X,Q X,Q = I X,Q and 1 ∈ 1 ∈ O ∩O O fih− =(fih− )u X,Q Q( X,Q( )= X,Q for every i.HenceI X,Q has a height one prime∈ O divisor∩ O and δ isO reducible( at Q. O Consider( the following( conditions. (1) δ is reduced. (2) δ is reduced at every closed point Q of X. (3) δ is analytically reduced at every closed point Q of X. The conditions (2) and (3) are equivalent by the previous argument, and they are equivalent to the condition that I has no height one prime divisors. Hence the conditions (2) and (3) imply the condition (1). If X is factorial, the three conditions are equivalent. Theorem 2.8. Every multiple component E of q is contained in the fixed point locus XGa .Conversely,assumethatn := dim X =3and Y is smooth. If σ is analytically reduced everywhere on X,thenXGa is the union of multiple components of q. Proof. Let Q X be a closed point and let ( , m)bethelocalring ∈ O Ga X,Q.ThenQ is a Ga-fixed point if and only if δ(m) m.IfQ X , Othen there exists an element x m such that δ(x) is⊂ a unit of '∈ .Let ∈ O be the m-adic completion of .Bythewell-knownresultofZariski, O O we have = [[x]]. This implies that the morphism q : X Y O O0 → is( complex-analytically a product Z C near the point Q,whereQ × corresponds( to( the point (Q0, 0) with Q0 Z and 0 C such that ∈ ∈ the local ring of Z at Q is .Hencethegerm(Z, Q ) is analytically 0 O0 0 isomorphic to (Y,q(Q)). This shows that the fiber FP with P := q(Q) is reduced at the point Q. Taking( the contrapositive of the assertion, 18 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL we conclude that if the fiber F is a multiple fiber then δ(m) m. P ⊂ Namely, Q XGa . ∈ Conversely, let Q XGa and let = .Thenδ (m) m, ∈ O OX,Q Q ⊂ where m is the maximal ideal of .Then is not smooth over := O O O0 Ker δQ.Indeed,otherwise = 0[[t]]( with a( fiber parameter( (t.Hence( O O 0 is a( complete regular local ring,( i.e., ( 0 = k[[u1,...,un 1]]( and O O − δQ =(h(∂/∂t)withh m.Then( (δ is analytically reducible at Q. This contradicts( the hypothesis.∈ So, the component( of q : X Y passing → through( Q is not reduced,( for otherwise the component is reduced and isomorphic to A1 by Theorem 2.5, hence q is smooth at Q.herewe used the assumption that n =3andY is smooth. This implies that Q is in a multiple component. 2 3 3 Let X = A .GivenaGa-action σ on A , it is known by [40] that 3 2 Y := A // G a is isomorphic to A .TheGa-action σ is said to be triangularizable if the corresponding lnd is written as ∂ ∂ δ = a(x) + b(x, y) ∂y ∂z with respect to a suitable system of variables x, y, z ,wherea(x) k[x]andb(x, y) k[x, y]. Assume that δ is nontrivial{ and} reduced. Let∈ q : X Y be the∈ quotient morphism, which is equi-dimensional and → surjective by [6]. If either a(x)orb(x, y) is zero then b(x, y) k∗ or 1 ∈ a(x) k∗ and f is a trivial A -bundle. Hence we exclude these cases in the following∈ result and assume that both a(x)andb(x, y)arenonzero polynomials. Proposition 2.9. With the above notations and assumptions we have the following assertions. (1) Ker δ = k[ξ,η],whereξ = x and η = a(x)z b(x, y)dy. (2) Sing(q) is defined by a(x)=0.HenceSing(q−) consists of parallel lines if it is not the empty set. ) (3) XGa is defined by a(x)=b(x, y)=0. Proof. It is clear that k[ξ,η] Ker δ.LetZ =Speck[ξ,η]andlet p : X Z (resp. π : Y Z)bethemorphismdefinedbytheinclusion⊆ k[ξ,η]→. A (resp. k[ξ,→η] . Ker δ). Then the quotient morphism q factors →p as p = π q.Let→U = a(x) =0 be the open set of Z. 1 ◦ 1 { ' } Then p : p− (U) U is an A -bundle because δ extends to an lnd of 1 → 1 1 A[a(x)− ] for which ya(x)− is a slice. This implies that π : π− (U) U is an isomorphism. Hence π is birational. Let α k be a root→ of ∈ a(x)=0andletLα be the line ξ = α on Z.Letb(x, y)= b(x, y)dy. 1 Let β k.Thenp− (α,β)= C(α,γ), where C(α,γ) is the line ∈ ) " * AFFINE THREEFOLDS AND Ga-ACTIONS 19

(α,γ, z) z k and γ runs over the roots of b(α, y)+β =0.Hence {the morphism| ∈ π} : Y Z is quasi-ﬁnite over L ,whereα runs over → α the roots of a(x)=0.ByZariskiMainTheorem," π is an isomorphism. Hence Ker δ = k[ξ,η]. The line C(α,γ) is a multiple* component if and only if γ is a multiple root of b(α, y)+β =0,i.e.,b(α,γ)=0.The above observations verify all the assertions. 2 The following result is somewhat" surprising.

Lemma 2.10. Let X be a smooth affine threefold with a Ga-action 2 such that Y := X//Ga is isomorphic to a quotient A /Γ,whereΓ is anon-trivialfinitegroupoflinearautomorphismsofA2 without nontrivial pseudo-reflections. Let q : X Y be the quotient morphism. → Let P be the unique singular point of A2/Γ.Assumethatq is an A1- 1 bundle over Y P .Thenq− (P ) cannot contain a one-dimensional component. \{ } Proof. Only finitely many fibers of q can contain divisorial components. If X0 is the complement in X of the union of these two- dimensional fiber components then X0 is again affine, Ga-stable and X0// G a = Y .Inviewofthiswecanassumethatnofiberofq contains atwo-dimensionalcomponent.ByTheorem2.5thefiberoverP is a 1 1 disjoint union of curves isomorphic to A .Weassumethatq− (P ) = . ' ∅ Consider the natural morphism A2 A2/Γwhichisunramified → 2 outside P .LetX# be the normalized fiber product X Y A .Then × 2 the Ga-action extends to a Ga-action on X# such that X#// G a = A . 2 The quotient morphism q# : X# A has the property that outside one 2 1 → point P # in A it is an A -bundle. By a result of A.K. Dutta [16], q# is 1 an A -bundle. This is a contradiction since the inverse image of P # has at least two irreducible components since A1 is simply-connected. 2 We will observe below various examples of the quotient morphism q : X Y := X//G ,whereX is a smooth affine threefold with a → a Ga-action. Example 2.11. Let X be as in Example 1.2. Then the quotient surface Y is isomorphic to A2/ΓwithΓ=Z/2Z,andthequotientmorphism q is induced by the projection q :SL(2) A2 defined by → xu (x, y) yz" -→ $ % which is the quotient morphism SL(2) SL(2)/Ga with Ga acting on SL(2) from the right via the upper triangular→ unipotent matrices. 1 Then q− (0, 0) is the empty set. Hence the fiber of q over the singular

" 20 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL point of Y is the empty set. The other ﬁbers of q are all isomorphic to A1. 2

Example 2.12. Let X be a smooth hypersurface in A4 =Speck[x, y, z, u] defined by xu y2z = y.ThenX has a G -action defined by an lnd − a ∂ ∂ δ = x + y2 . ∂z ∂u Then Ker δ = k[x, y]andthequotientmorphismq : X A2 is 1 2 → given by (x, y, z, u) (x, y). Hence q− (0, 0) = A =Speck[z, u] 1 1 -→ and q− (α,β) = A if (α,β) =(0, 0). The ideal I =(y,u)A is Ga- ∼ ' invariant, where A =Γ(X, X ), and SL(2) is obtained from an affine transformation of X with respectO to the ideal I and an element y. 2

Example 2.13. Let X be the Koras-Russell threefold x + x2y + z2 + 3 t =0.SinceeveryGa-action on X makes x invariant, there are two independent Ga-actions which correspond to the following lnds: ∂ ∂ δ = 2z + x2 1 − ∂y ∂z ∂ ∂ δ = 3t3 + x2 . 2 − ∂y ∂t

2 The quotient morphism is given by q1 : X A =Speck[x, t]and 2 → q2 : X A =Speck[x, z]. Then we have → A1 if α =0 1 1 1 ' q1− (α,β)= A + A if α =0,β=0  1 '  2A if α = β =0 A1 if α =0 1  1 1 1 ' q2− (α,β)= A + A + A if α =0,β=0  1 '  3A if α = β =0 2 

We give one example in the case dim X = 4. It is due to Winkelmann [47].

4 Example 2.14. Let X = A =Speck[x1,x2,x3,x4]equippedwitha Ga-action deﬁned by

∂ ∂ 2 ∂ δ = x1 + x2 +(x2 2x1x3 1) . ∂x2 ∂x3 − − ∂x4 AFFINE THREEFOLDS AND Ga-ACTIONS 21

Then Ker δ = k[ξ1,ξ2,ξ3,ξ4], where

ξ1 = x1 ξ = x2 2x x 2 2 − 1 3 ξ = x x x (x2 2x x 1) 3 1 4 − 2 2 − 1 3 − ξ = x x2 2x x (x2 2x x 1) + 2x (x2 2x x 1)2 4 1 4 − 2 4 2 − 1 3 − 3 2 − 1 3 − 2 2 4 and Y := X//Ga is a hypersurface ξ1ξ4 = ξ3 ξ2(ξ2 1) in A = − − Speck[ξ1,ξ2,ξ3,ξ4]. Then Y has a unique singular point (ξ1,ξ2,ξ3,ξ4)= 1 (0, 1, 0, 0), and the ﬁber q− (α1,α2,α3,α4) is given as follows: 1 A if (α1,α2,α3,α4) =(0, 1, 0, 0) 2 2 ' A + A if (α1,α2,α3,α4)=(0, 1, 0, 0) # 2

4 1 3. Relative fixpoint free Ga-actions on A over A Let X =SpecA be an affine variety defined over a field k of characteristic zero which we assume to be not necessarily algebraically closed in this section. We say that X is a k-form of A3 if there exists a finite algebraic extension K/k such that XK := SpecA k K is K-isomorphic 3 ⊗ to AK.Letσ : Ga X X be a Ga-action which corresponds to an lnd δ of A. Then the× action→ σ is fixpoint free if and only if the ideal

a A δ(a)A is the unit ideal. Now we prove the following result. ∈ .Lemma 3.1. Let X =SpecA be a k-form of A3 equipped with a ﬁxed 3 point free Ga-action. Then X is k-isomorphic to A .

Proof. With the above notations, let B =Kerδ.ThenAK := A k K has an lnd δ := δ K such that δ (a)A = A . Since⊗ K k a AK K K K 3 ⊗ ∈ XK = AK, it follows that BK := B k K =KerδK is isomorphic to ∼ ⊗. K[x, y]. In fact, this is so over an algebraic closure K of K by [40]. Let 2 2 Y =SpecB.ThenY is a k-form of A .ThenY ∼= Ak by Kambayashi [31]. Let q : X Y be the morphism deﬁned by the inclusion B. A, → 3 → which is the quotient morphism by the action σ. Since qK : AK YK 1 → is an A -bundle as the action σK is a translation by Kaliman [26], the 1 2 1 morphism q : X Y is an A -bundle. Since Y = Ak and an A -bundle 2 → 3 ∼ over A is trivial, we have X ∼= A and the action σ is a translation. 2 Let X =SpecA be an aﬃne variety over k equipped with a Ga- action σ.Wesaythatσ is a translation if X = Y A1,i.e.,A = B[x], and the action σ is given by t b = b for every× element b B and · ∈ t x = x + c0t with c0 B 0 .Theargumentintheaboveproof implies· the following result.∈ \{ } 22 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

Corollary 3.2. Let X be an affine variety defined over k equipped with a fixed point free G -action σ : G X X.LetK/k be a finite a a × → algebraic extension such that σK : Ga XK XK is a translation. Then σ is a translation. × → Proof. Let X =SpecA and let δ be the lnd associated with the action σ.LetB =Kerδ.ThenwehaveB := B K =Kerδ with K ⊗k K δK = δ k K. By the assumption, we have AK = BK[x]foravariablex. Let Y =Spec⊗ B and let q : X Y be the quotient morphism. Then 1 → 1 qK : XK YK is an A -bundle, hence q : X Y is an A -bundle → → as in the proof of Lemma 3.1. Let = Ui i I be an affine open U { 1 } ∈ covering of Y such that Ui =SpecBi and q− (Ui)=SpecBi[xi]. For every pair (i, j), we have xj = bjixi + cji,wherebji Γ(Ui Uj, Y∗ ) and c Γ(U U , ). The lnd δ extends to an∈ lnd (denoted∩ O by ji ∈ i ∩ j OY the same letter) of Bi[xi]. Let si = δ(xi), which is an element of Bi 2 since δ (xi)=0.Thenwehavesj = bjisi.Hence si i I determines a 1 { } ∈ section in Γ(Y, − ), where is an invertible sheaf on Y determined by transition functionsL b withL respect to .NotethatG acts on each { ji} U a fiber of q without fixed points. This implies that s B∗ and hence i ∈ i = Y .Replacingxi by sixi,wemayassumethatxj = xi + cji. L ∼ O 1 Hence the obstruction cji for q to be trivial lies in H (Y, Y ). Since { } 1 O Y is affine, it follows that X ∼= Y A . This implies that σ is a translation. × 2 We are interested in generalizing Kaliman’s theorem on a fixpoint- 4 free Ga-action on A with one coordinate invariant. Kaliman’s theorem 3 states that a fixpoint-free Ga-action on A =Speck[x, y, z] is a translation. Namely, after a suitable change of coordinates, the Ga-action is given by t (x, y, z)=(x, y, z + t). Recently, Kaliman [27] proved the following result.·

Kaliman’s Theorem. Let X := A4 =Speck[u, x, y, z] and let σ : Ga X X be a proper, ﬁxpoint-free action preserving one of the coordinates,× → say u,invariant.Thenσ is a translation.

The Ga-action σ is called proper if the morphism Φ: Ga X X X deﬁned by (t, Q) (σ(t, Q),Q) is a proper (hence ﬁnite)× immersion.→ × The example of Winkelmann-→ (see Example 2.14) is given by a triangular lnd ∂ ∂ ∂ δ = u + x +(x2 2uy 1) . ∂x ∂y − − ∂z But the action is not a translation. Otherwise, the quotient threefold 3 Y := X//Ga exists and is isomorphic to A .However,asshownin AFFINE THREEFOLDS AND Ga-ACTIONS 23

Example 2.14, Y has a unique singular point. So, Winkelmann’s Ga- action is not proper. If the action σ is furthermore triangularizable, the above theorem of Kaliman was proved by Dubouloz-Finston-Jaradat [15]. A crucial point of their arguments is stated as follows. Let be 4 Y the geometric quotient of A by Ga in the category of algebraic spaces (or Deligne-Mumford stacks). In fact, exists by a general theory. 4 Y The algebraic quotient Y := A // G a exists as Spec Ker δ since the ring Ga Ker δ = k[u, x, y, z] is finitely generated over k provided u is Ga- invariant [3]. If the Ga-action is fixpoint-free, triangular and proper, it is shown that the canonical morphism π : A1 =Speck[u]has Y→ all fibers isomorphic to A2 and the quotient morphism ρ : A4 is →Y an A1-bundle in the étale topology of . A key result is to show that 4 2 Y 1 ∼= A // G a and it is an A -bundle over A in the Zariski topology. HenceY Ker δ = k[u, x, y]afterasuitablechangeofcoordinates. Mostly for a technical reason, we need to assume that the ground field k has infinite transcendence degree over the algebraic closure of the prime field Q.So,weassumethatk is the complex field C. 4 Lemma 3.3. Consider a fixpoint-free Ga-action on X := A .Assume that the G -action fixes invariant a variable u of A := Γ(X, ).Then a OX there exists w k[u] such that Aw := A[1/w] is Ga-isomorphic to k[u] [x, y, z] with∈ a G -action t (u, x, y, z)=(u, x, y, z + t). w a · 4 1 Proof. Consider the given Ga-morphism f : X = A A = 1 → Spec k[u]. Let g :Speck[u, x, y, z] A =Speck[u]betheGa- morphism with the action as a translation→ t (u, x, y, z)=(u, x, y, z+t). · We will compare these two Ga-actions. By Kaliman’s theorem [26], the fibers of f and g over any closed point of Spec k[u]areGa-isomorphic. By the Generic Equivalence Theorem [37] (where we use the assumption that k has infinite transcendence degree over Q), there exists afiniteGaloisextensionL K = k(u)andaG -isomorphism over L ⊃ a L A ∼ L[x, y, z]. ⊗k[u] −→ Let Γbe the Galois group of L/K.Wehave (K A)Ga = K AGa ⊗k[u] ⊗k[u] and (L A)Ga = L AGa = L (K AGa ). ⊗k[u] ⊗k[u] ⊗K ⊗k[u] Ga Since L k[u] A ∼= L[x, y]andformsofK[x, y] are trivial [31], there ⊗ Ga Ga [2] exist x1,y1 (K k[u] A) such that (K k[u] A) = K[x1,y1] ∼= K . Let z be a∈ variable⊗ for L A over L⊗[x ,y ]. For γ Γ, γ(z )= 1 ⊗k[u] 1 1 ∈ 1 a z +b with a L∗ and b L[x ,y ]. By Hilbert’s Theorem 90, the γ 1 γ γ ∈ γ ∈ 1 1 24 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

γ 1 multiplicative version, we have α = a(a )− for some a L∗ and all γ ∈ γ Γ. Free to replace z1 by az1,wecanassumeaγ = 1 for all γ.Then ∈ γ bγ = b b for some b L[x1,y1] by Hilbert’s Theorem 90, the additive version.− (Note that Γ∈acts on the coeﬃcients of b only.) Free to replace z1 by z1 +b,wecanassumez1 K k[u] A,i.e.,K k[u] A = K[x1,y1,z1]. ∈ ⊗ ⊗ 1 We have t z = z + bt with b K∗.Sincewecanreplacez by z b− , · 1 1 ∈ 1 1 we can assume t z1 = z1 + t. By the above· argument, there exists w k[u]suchthatA := ∈ w A[1/w] is Ga-isomorphic to k[u]w[x, y, z]. After multiplying variables by powers of w if necessary, we have A = k[u] [x ,y ,z ]withx ,y ,z w w 1 1 1 1 1 1 ∈ A, x ,y ﬁxed by G and t z = z + ct for c (k[u] )∗. 2 1 1 a · 1 1 ∈ w

Remark 3.4. Let X and the Ga-action be the same as in Lemma 3.3. Let Y := X//Ga and let q : X Y be the quotient morphism. Suppose that q is a flat morphism. →Then B := Γ(Y, ) is a reg- OY ular, factorial, affine domain of dimension three with B∗ = k∗ and 1 1 B[w− ]=k[u, w− ,x ,y ]withw k[u]. Hence the partial derivatives 1 1 ∈ (∂/∂x1)and(∂/∂y1)multipliedbypowersofw extend to the lnds δ1 and δ2 which define independent Ga-actions σ1 and σ2 on Y .Bythe definition, σ1 and σ2 commute. Furthermore, it is easy to show that Y is simply connected. In fact, any topological covering Y Y factors → the morphism q as X Y Y because X = A4 is simply connected. → → Since the general fibers of q is irreducible and reduced," Y Y is birational and hence biregular." → Let Ri =Kerδi and Zi =SpecRi for i =1, 2. By Theorem" 1.1, 2 it follows that Z1 = Z2 = A . Since δi(u)=0fori =1, 2, the ∼ ∼ 1 quotient morphisms qi : Y Zi are morphisms over A =Speck[u]. → Since σ1 and σ2 commute, σ2 induces a Ga-action on Z1 such that Z1// G a =Speck[u]. This is the case with Z2.Hencethereexistsa 3 morphism ρ := (q1,q2):Y Z1 A1 Z2 = A .ByZariski’sMain → × ∼ 1 Theorem, ρ is an isomorphism if and only if two A -fibrations q1 and q2 share no fiber components. As a matter of fact, the Koras-Russell threefold, which we denote by Y for the sake of consistency of the notations, is a smooth factorial affine threefold equipped with two commuting Ga-actions (cf. Exam- ple 2.13). The quotients of Y by these Ga-actions are isomorphic to A2. Hence there is a birational morphism ρ : Y A3. But the quo- → tient morphisms q1 and q2 share fiber components. So, we can pose here a question. Does there exists a dominant morphism form A4 to the Koras-Russell threefold? In other words, under the present situ- 4 ation where Y is obtained as the quotient threefold A // G a,doesthe phenomenon of sharing fiber components of both A1-fibrations occur? AFFINE THREEFOLDS AND Ga-ACTIONS 25

The quotient morphism q : X Y is equi-dimensional by the flat- ness condition. By Theorem 2.8,→q has no multiple components. It seems that all reducible fibers of q are disjoint unions of irreducible 1 3 components isomorphic to A .Suppose,forexample,thatY ∼= A .For any point P of Y ,wetakeagenerallinearplaneH passing through P . 1 Then XH := q− (H)isasmoothaffine threefold admitting a fixed point free Ga-action and having H as the quotient by the induced Ga-action. 1 1 By Theorem 2.5, the fiber q− (P ) is a disjoint union of the A .Suppose that Sing(q) = . Then Sing(q) has pure codimension one by Lemma 2.7, (1). Let S' be∅ an irreducible component of Sing(q) passing through P . Since Y is factorial, S is defined by s = 0 with s B := Γ(Y, Y ). Since s is a prime element of A as well, all irreducible∈ componentsO of 1 1 q− (P ) are contained in an irreducible subvariety q− (S)ofcodimen- 1 sion one in X.Namely,eachirreduciblecomponentofq− (P )should have a moduli in X = A4. 2

The following example shows that Rentschler’s theorem does not hold in the relative case. The theorem states that if δ is a locally nilpotent derivation on a polynomial ring k[x, y], then, after a suitable change of variables, we have Ker δ = k[x]andtheassociatedGa-action is given by t (x, y)=(x, y + f(x)t)withf(x) k[x]. · ∈ Example 3.5. Let δ be a locally nilpotent derivation on A = k[x, y, z] defined by ∂ ∂ δ = 2z + x . − ∂y ∂z 3 Then the associated Ga-action on A =SpecA is given by t (x, y, z)= · (x, y 2zt xt2,z+xt), and the G -invariant subring is AGa = k[x, xy+ − − a z2]. Let q : A3 A2 =SpecAGa .Thenq has a fiber of multiplicity 2 over the point of→ origin x = xy + z2 =0.IfRentschler’stheoremholds with this example, one can choose variables x, y, z so that AGa = k[x, y] and the Ga-action is given by t z = z + f(x, y)t with f(x, y) k[x, y]. Then the quotient morphism has· no multiple fibers. ∈ 2

Remark 3.6. In Example 3.5, the Ga-ﬁxed point locus coincides with 3 the multiple component x = z =0 .IftheGa-action on A is ﬁxed- point free, then one can take{ Kerδ =}k[x, y]andt Z = z+t by Kaliman [26] after a suitable change of coordinates x, y,· z with x unchanged. { } So, the relative Rentschler’s Theorem holds. Does it hold if the Ga- action is proper? 2 26 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

4 1 4. Relative effective Gm-actions on A over A It is not clear to us how to generalize the results of the previous section to more general Ga-actions since not much appears to be known about their deformations. Actions of Gm,however,areveryrigidanda very complete answer can be given. We will tacitly assuume k = C in some places. It is not difficult to remove this assumption aposteriori, see [36]. Theorem 4.1. Let X =SpecA be a smooth affine variety and q : X Y := A1 =Speck[u] → amorphismwitheveryclosedfiberisomorphictoA3.SupposethatX admits an effective relative Gm-action with trivial action on Y .Then 3 X is equivariantly isomorphic to A Y with diagonal Gm-action on × A3.

Proof. The Gm-action on X has a fixpoint, in fact at least one in each fiber of q. As is well known, the induced action on the tangent space at a fixpoint is diagonalizable with weights independent of the fixpoint chosen, see [32]. The induced actions on the fibers of q are diagonalizable, see [30], and it follows that the weights are the same for all fibers. We denote them by a, b, c.Weassumea b c.We will have to distinguish three cases. ≥ ≥ (1) a, b, c 0. (2) a>0,b≥ =0,c<0. (3) a>0,b>0,c<0. 3 Let Z =Speck[ξ,η,ζ] ∼= A be the Gm-variety with ξ,η,ζ given weights a, b, c.Wewillcompareq to the Gm-morphism q# : Z Y Y . By the Generic Equivalence Theorem [37] (where we again× use→ the assumption that k has infinite transcendence degree over Q)andthe 3 theorem on the absence of non-trivial forms of Gm-actions on A [36], we obtain: 1 there exists a dense open set U Y such that q− (U) is G - ( ) ⊂ m ∗ isomorphic with Z U. × 1 Since q− (U) is factorial and closed fibers of q are irreducible, we have by Nagata’s lemma that ( ) A is factorial. ∗∗ (I) Assume that we have case (1) above. Then the Gm-action on X is fixpointed and X is a vector bundle over the fixpoint set T = XGm , see [4] and [32]. We consider the case a>0,b= c =0,theothercases are similar, and easier. Consider π = q : T Y .By()wehave |T → ∗ AFFINE THREEFOLDS AND Ga-ACTIONS 27

1 1 π− (U) ∼= Speck[η,ζ] U.Moreover,foreachP Y we have π− (P ) ∼= Spec k[η,ζ]. By Sathaye’s× theorem [46] we obtain∈ T = Spec k[η,ζ] Y 1 ∼ × and X = Spec k[η,ζ] Y A ,withactionofGm on the last factor. ∼ × × + (II) Assume that we have case (2) above. We consider X (resp. X−), the locus of Q X such that limt 0 t Q, t Gm (resp. lim1/t 0 t Q, t 5 ∈ → · ∈ → · ∈ Gm)exists. Note that in Z Y the (+)- and ( )-locus are defined by ζ =0andξ =0respectively.By(× )wecanfind− x, y, z A so 1 ∗ 1 ∈+ that q− (U)=SpecS[x, y, z], S a localization of k[u], and q− (U) and 1 + q− (U)− defined by z =0andx = 0. The zero locus of z on X has X as an irreducible component. Any other irreducible component will be a fiber u = λ of q,i.e.,wehavez (u λ)A and can replace z by z/(u λ). ∈ +− − We may assume therefore that X is the zero-locus of z and X− the zero-locus of x.Finally,notethatη k[u, ξ,η,ζ]Gm k[u, ξ,η,ζ] is the defining equation for the fixpoint set∈ inside the quotient.⊂ By ( )wecan 1 Gm 1 ∗ assume that y defines q− (U) inside q− (U)//Gm.Appealingonce 1 3 more to ( )andSathaye’stheoremwefindthatq− (U)//Gm ∼= A . ∗ Gm If the zero-locus of y in X//Gm has a a component other than X , it will be a fiber of X//G Y and y is divisible by some u λ in m → − AGm ,henceinA.Againwemayassumethisdoesnothappen.Now in each A/(u λ)A the images x, y, z of x, y, z define, respectively, the ( )-locus, the− fixpoint set inside the quotient and the (+)-locus. They then− generate A/(u λ)A. It follows that A = k[u, x, y, z]. − (III) Assume that we have case (3) above. + 2 1 We have X ∼= Spec k[u] A and X− ∼= Spec k[u] A with Gm × × X ∼= Spec (k[u]). We have: X+ =Speck[A/zA], where z A is an irreducible semi- invariant of negative weight, unique∈ up to a constant.

As in ( )wecanﬁndhomogeneousξ,η,ζ A of weights a b>0 and c<0and∗ h k[u],h=0sothat ∈ ≥ ∈ ' 1 Ah = k[u, h(u)− ,ξ,η,ζ]. After factoring out factors from k[u]wecanassume ζ = z. We now follow the line of argument in [35, Proposition 1.8] to recon- struct X from X =SpecA/(z 1)A and the G -action. Note that 1 − m X1 is irreducible and has a Ω:= Z/cZ-action relative to Spec (k[u]).

5 + It is straightforward to give a purely algebraic deﬁnition of X and X−. 28 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

3 CLAIM 1: X1 ∼= A . [2] In fact, consider k[u] B = A/(z 1)A.WehaveBh ∼= k[u]h and [2] ⊂ − B/(u λ)B ∼= k for all λ k.TheresultfollowsfromSathaye’s theorem− [46]. ∈ CLAIM 2: We have an Ω-isomorphism

ϕ : X =SpecB Spec k[u, x, y], 1 → where Ωacts trivially on u and with weights a, b modulo c on x, y. Any such ϕ induces a Gm-isomorphism Φ: X X1 =SpecAz 1 \ → Spec k[u, v, w, z, z− ], where u, v, w, z have weights 0,a,b,c.

In fact, ϕ exists by CLAIM 1 and since finite group actions on A3 that fix a variable are linearizable [37]. Φis defined by

Φ(t p)=t ϕ(p)forp X . · · ∈ 1 See [35, 36] for details. To simplify notation, we identify k[u, x, y]withasubringofB and k[u, v, w, z]withasubringofAz.Wecan,andwill,arrangex and y so that they vanish on X− X1. This is automatic if a =0modc and b =0modc. ∩ ' ' CLAIM 3: For each point p0 Spec k[u]thereexistsaZariskiopen 1 ∈ 3 neighborhood U so that q− (U) is equivariantly isomorphic to U A . ×

By an equivariant version of the theorem of Bass-Connell-Wright [2], proving this claim will prove the result.

We assume for simplicity that u(p0) = 0intheproofofCLAIM3. We Gm + have p X = X X−. By [34] there exists a G -invariant analytic 0 ∈ ∩ m open neighborhood of p0 in X analytically and equivariantly isomorphic 4 to an analytic C∗-neighborhood of 0 C with linear action of C∗. We may assume that the isomorphism∈ is given by u, F, G, z,where F, G are homogeneous analytic functions on X of weights a, b.Wecan + assume that x0 = F X+ ,y0 = G X+ are algebraic on X ,whereu, x0,y0 are homogeneous coordinates| on| X+ =Spec(A/z). Moreover u, f = 1 F ,g = G are local analytic coordinates at p = q− (p ) X−. |X1 |X1 1 0 ∩ Let the notation be as in CLAIM 2. Let us consider x = x(u, f, g) and y = y(u, f, g)aspowerseriesinu, f, g, homogeneous of weights a and b mod c.Lett be a parameter for Gm.Thenv and w are AFFINE THREEFOLDS AND Ga-ACTIONS 29 characterized by tav = x(u, taf,tbg)andtbw = y(u, taf,tbg). Consider a term u&f mgn appearing in x (resp. y). We have am + bn = a mod c (resp. am + bn = b mod c). We obtain a contribution u&F mGnzs to v (resp. w) as series in u, F, G, z,wheream + bn a = cs (resp. am + bn b = cs). Negative powers of z occur only− if am− + bn < a (resp. −am + bn− < b). For v this occurs for terms u&gn with bn < a,andsinceb a this does not occur for w. ≤ The linear forms of u, x, y are linearly independent. Since x, y vanish for u in a neighbourhood of 0, those for x, y are free of u.Writethem as

l1 = a11f + a12g,l2 = a21f + a22g. We obtain: CLAIM 4: (i) w is a power series in u, F, G, z with linear form

L2 = a21F + a22G in case a = b and a22G otherwise. (ii) L depends on F, G only and w is regular at p if and only if L =0. 2 0 2 '

If f and g have diﬀerent weights mod c,i.e.,ifa = b mod c,then a = a =0,a =0,a =0.Ontheotherhand,if' a = b mod c,we 12 21 11 ' 22 ' can change ϕ by any linear automorphism of A3 that ﬁxes u.Sowe can arrange (with a new choice of y if necessary) that a =0. 22 ' Since w is homogeneous for Gm we obtain:

CLAIM 5: w is regular and dw = 0 at each point of a Gm-invariant ' 1 Zariski-neighborhood W of p0 in X,wherewecanassumeW = q− (U) with U an open Zariski-neighborhood of 0 in Spec(k[u]). Write U =Spec(R),R = k[u, 1/h],h k[u],h(0) =0,andput D = A , K = R[w, z] D. We will show that∈ D = K[1].' h ⊂ CLAIM 6: w is a variable in D/z,i.e.,D/z = R[w][1].

In fact, let p# U. We can repeat the above discussion with new local 0 ∈ analytic coordinates u#,F#,G#,z,u#(p0# )andu#,f#,g#.Wecanconsider + 2 D/z, with Spec (D/z)=W ,asSymR(I/I ), where I is the ideal in D/z of the ﬁxpoint set W Gm .ByCLAIM5andandCLAIM4(applied to the new analytic coordinates) we ﬁnd that w induces a nowhere zero homogeneous element of the (free) R-module I/I2. 30 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

Let an overline denote images mod u λ,λ = u(p0# ). Then w is obtained via the procedure in CLAIM 2− by setting u = λ.Wehave 1 3 w D and dw =0ateachpointofq− (p0) ∼= A by CLAIM 5 and CLAIM∈ 4. As explained' in [35, 1.8, 1.10], we can, by a modiﬁcation of ϕ that keeps ly unchanged, ﬁnd a homogeneous v# D so that ∈ D = k[v#, w, z]. We therefore have: CLAIM 7: For each maximal ideal m =(u λ)R Spec (R)wehave − ∈ D/mD = k[w, z][1]. Consider K D and S = 1,z,z2,... . Using CLAIM 2 we can 1 ⊂ 1 { } write S− D =(S− K)[τ]withτ D and homogeneous. Using CLAIM 6andCLAIM7itisstraightforwar∈ dtoverifythatthereexisthomo- geneous κ K and σ S so that ∈ ∈ τ # =(τ κ)/σ D − ∈ and

D = K[τ #]. According to [45, Theorem 2.3.1], we have to check that K is S-inert in D,i.e., 1 (i) D S− K = K, ∩ (ii) Q,theﬁeldofquotientsofK = K/(zD K) D/zD, is algebraically closed in the ﬁeld of quotients∩ of D/zD⊂ , (iii) Q D/zD = K. ∩ Now (i) is readily deduced from CLAIM 6, and (ii) and (iii) are clear by CLAIM 7. 2

Remark 4.2. It is an open question whether the conclusion X ∼= A3 Y is correct without reference to any group action, as it is for × A2-ﬁbers, see [29], or [37] for an argument in the spirit of the present paper. The following results are immediate consequences of Theorem 4.1. Theorem 4.3. Suppose F A = k[4] is invariant under an eﬀective ∈ [3] Gm-action on A and for all λ k we have A/(F λ)A ∼= k .Then F is a variable in A. ∈ −

4 Theorem 4.4. A Gm-action on A that ﬁxes a variable is linearizable. Any variable that is ﬁxed by the action is part of a system of variables that diagonalizes the action. AFFINE THREEFOLDS AND Ga-ACTIONS 31

5. Danielewski surfaces of non-hypersurface type The Danielewski surfaces are the hypersurfaces X(P ) in A3 =Spec k[x, y, z]definedbyxmy = P (z), where m 1andP (z) k[z]with ≥ ∈ gcd(P (z),P#(z)) = 1. We consider here the simplest case X with m =1 and P (z)=z2 1. The following properties are well known about X. − Lemma 5.1. Let X = xy = z2 1 in A3.Thenwehave: 1 { − } 1 1 (1) X has two A -fibrations ρ1 : X A and ρ2 : X A defined → →1 by ρ1(x, y, z)=x and ρ2(x, y, z)=y.ThesetwoA -fibrations are obtained as orbits of the Ga-actions σ1 and σ2 which are defined respectively by the locally nilpotent derivations δ1 and δ2 such that δ1(x)=0,δ1(y)=2p1(x)z, δ1(z)=p1(x)x with p1(x) k[x] and δ2(x)=2p2(y)z, δ2(y)=0,δ2(z)=p2(y)y. ∈ 1 1 (2) X = F0 D,whereD is the diagonal of F0 = P P . \ × (3) X is simply connected, H1(X; Z)=0and H2(X; Z)=Z. (4) An algebraic subgroup of Aut(X) is either Ga = exp(δit) t k for i =1, 2 or PGL(2) which is generated by{exp(δ t)|and∈ } 1 exp(δ2s),wherep1(x)=p2(y)=1. The following results have been known to the experts (see [24]), and exhibit possible directions for the Danielewski surfaces to be generalized in the non-hypersurface case and the higher-dimesnional case. Lemma 5.2. Let X be as in Lemma 5.1. Then we have: (1) X = T PGL(2),whereT is a maximal torus of PGL(2). ∼ \ (2) The embedding X. F0 is PGL(2)-equivariant, where PGL(2) 1 →1 acts on F0 = P P diagonally. × (3) Write T PGL(2) = T # SL(2),whereT # is the maximal torus t \0 \ ab 1 t k∗ .Let be an element of SL(2). 0 t− ∈ cd #$ % & ' $ % Then the left action& of T # on SL(2) is given by (a, b, c, d) 1 1 -→ (ta, tb, t− c, t− &d),andhencetheT #-invariant subring A of the coordinate ring B := k[a, b, c, d]/(ad bc =1)of SL(2) is generated over k by x = ac, y = bd, w = ad− and z = bc.Therelation among x, y, w and z are w = z +1 and xy = wz.Eliminating w,wehavexy = z(z +1),whichisadefiningequationofthe Danielewski surface after a suitable change of coordinates. We employ this equation as the defining equation of X. (4) The Ga-action σ1 on X corresponding to the locally nilpotent derivation δ1 with p1(x)=1in Lemma 5.1 is given by the 1 u right multiplication of u k .TheA1-fibration 01 ∈ #$ % & ' & & 32 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

1 ρ1 : X A which is the quotient morphism by σ1 is given by (x, y,→ z) x and has the following splitting data with the -→10 section T for x A1 x 1 ∈ $ % 10 1 u 1 u T = T ( ) x 1 01 xux+1 ∗ $ %$ % $ % where y = u(ux +1),w= ux +1,z= ux.Ifa =0,thenthe ab ' T -residue class T is written in the form ( ) as shown cd ∗ in the following computation:$ %

ab a 1 0 ab T = T − cd 0 a cd $ % $ %$ % 1 a 1b 1 u = T − = T , ac ad xw $ % $ % 1 with x = ac, u = a− b, w = ad and z = ad 1.Ifa =0,then bc = 1 and − − 0 b b 0 01 T = T 1 cd 0 b− 1 bd $ % $ %$ − % 01 01 1 y = T = T , 1 y 10 01− $ − % $ − %$ % where y = bd.Thustheﬁberofρ1 over the point x =0consists 10 01 of two G -orbits passing through T and T , a 01 1 y respectively. $ % $ − % (5) The Ga-action σ2 on X corresponding to the locally nilpotent derivation δ2 with p2(y)=1is given by the right multiplication 10 1 1 of v k .TheA -ﬁbration ρ2 : X A which v 1 ∈ → #$ % & ' is the quotient& morphism by σ2 is given by (x, y, z) y and & 1-→y has the following splitting data with the section T for 01 $ % y A1 ∈ 1 y 10 1+vy y T = T ( ) 01 v 1 v 1 ∗∗ $ %$ % $ % where x = v(1 + yv),w=1+vy and z = vy.Ifd =0,then ab ' the T -residue class T is written in the form ( ) cd ∗∗ $ % AFFINE THREEFOLDS AND Ga-ACTIONS 33

ab d 0 ab T = T 1 cd 0 d− cd $ % $ %$ % ad bd wy = T 1 = T , cd− 1 v 1 $ % $ % 1 where y = bd, v = cd− and z = ad 1.Ifd =0,thenbc = 1 and − − ab b 0 ac 1 T = T 1 − c 0 0 b− 10 $ % $ %$ − % x 1 01 10 = T = T , −10 10 x 1 $ − % $ − %$ − % where x = ac.Thustheﬁberofρ2 over the point y =0has 10 01 two G -orbits passing through T and T , a 01 1 y respectively. $ % $ − % 1 (6) Two A -ﬁbrations ρ1 and ρ2 are transformed to each other by the action of the Weyl group: 0 1 1 u 01 T 10− xux+1 10 $ % $ %$ − % 0 1 01 0 1 1 u 01 = T 10− 10 10− xux+1 10 $ % $ − %$ %$ %$ − % ux +1 x = T . u −1 $ − % 2

The group SL(2) is a quadric hypersurface xy zu = 1 in A4.A slight modiﬁcation of the deﬁning equation of SL(2)− will give similar but interesting results.

Theorem 5.3. Let X(m, 1) be the hypersurface xy zmu =1in A4 = Spec k[x, y, z, u],wherem 2.Thenthefollowingassertionshold.− ≥ (1) X(m, 1) is factorial and simply-connected. (2) X(m, 1) has a locally nilpotent derivation (simply lnd) δ deﬁned by

m 1 δ(x)=δ(u)=0,δ(y)=mz − u and δ(z)=x.

t 1 (3) X(m, 1) has a Gm-action deﬁned by (x, y, z, u)=(tx, t− y,tz, m m t− u).LetR = k[x, y, z, u]/(xy z u =1)be the coordinate − 34 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

Gm ring of X(m, 1).ThentheGm-invariant subring A = R is given by m i i A = k[xy, yz, x − z u (0 i m)]. ≤ ≤ Let B be an aﬃne k-domain deﬁned by

XYi 1 = Yi(1 + Ym), 1 i m B = k[X, Y0,Y1,...,Ym] − ≤ ≤ YiYj = Yi" Yj" ,i+ j = i# + j# / $ % Then there exists an isomorphism θ : B ∼ A defined by m i i −→ θ(X)=yz, θ(Y )=x − z u for 0 i m. i ≤ ≤ (4) Let V (m, 1) = Spec A = X(m, 1)// G m.ThenV (m, 1) is a smooth affine surface. The lnd δ on R induces an lnd δ on A such that δ(X)=(m +1)Ym +1 and δ(Yi)=iYi 1 for 0 − 1 ≤ i m.HenceKer δ = k[Y0].Letρ2 : V (m, 1) A be ≤ → the quotient morphism by the Ga-action associated to δ,where 1 A =Speck[Y0].IfY0 =0then ' i 1 m 1 m Y1 X = m Y1(Y0 − + Y1 ) and Yi = i 1 (2 i m). Y0 Y0 − ≤ ≤ 1 1 1 1 1 Hence ρ2− (A ) = A A ,andρ2− (0) is a disjoint union of ∗ ∼ ∗ × two reduced irreducible components isomorphic to A1 defined by (Y0 = = Ym 1 =0,Ym = 1) and (Y0 = = Ym 1 = Ym = 0) respectively.··· − − ··· − 1 (5) Let ρ1 : V (m, 1) A =Speck[X] be defined by the inclusion k[X] . A.IfX →=0then → ' Ym m i Yi = m i (1 + Ym) − (0 i m)() X − ≤ ≤ ∗ 1 1 1 1 1 whence ρ1− (A ) = A A ,andρ1− (0) is a disjoint union of ∗ ∼ ∗ × two reduced irreducible components isomorphic to A1,forone of which Y1 = = Ym =0and Y0 is a variable, and another of which Y = ···1 and m − i+1 i Ym i =( 1) Ym 1 (0 i m). ( ) − − − ≤ ≤ ∗∗ (6) The minimal closed embedding of V (m, 1) into the affine space is V (m, 1) . Am+2,whichisgivenbytheexpressionofB → in terms of X, Y ,...,Y .Henceifm = m#, V (m, 1) is not 0 m ' isomorphic to V (m#, 1).Inparticular,V (m, 1) ∼= V (1, 1) if m 2,whereV (1, 1) is the hypersurface Danielewski' surface xy ≥= z2 1. − 1 1 (7) For m = m#,wehaveV (m, 1) A = V (m#, 1) A . ' × ∼ × AFFINE THREEFOLDS AND Ga-ACTIONS 35

Proof. (1) The subvariety V (x)ofX(m, 1) deﬁned by x =0isiso- morphic to C A1,whereC is the irreducible plane curve zmu +1= × 1 2 0. Further, D(x) := X(m, 1) V (x) ∼= A A ,whichisfactorial. Hence X(m, 1) is factorial by\ Nagata [42].∗ × Next apply Nori’s exact sequence (see Lemma 6.2 below) to the projection p : X(m, 1) → A1, (x, y, z, u) x.Wethenhaveanexactsequence -→ 2 1 π1(A ) π1(X(m, 1)) π1(A ) (1). → → → This shows that X(m, 1) is simply connected. (2) Straightforward. Gm m i i (3) We show that R = k[xy, yz, x − z u (0 i m)]. Under the a b c d ≤ ≤ given Gm-action, a monomial x y z u is Gm-invariant if and only if a + c = b + md,wherea, b, c, d are non-negative integers. We consider 3casesseparately. (i) If 0 c

Gm m i i Hence R is generated by xy, yz and x − z u for 0 i m. It is clear that the homomorphism θ : B A is≤ well-deﬁned≤ and surjective. We show that it is an isomorphism.→ If X =0then ' Ym m i − Yi = m i (1 + Ym) (0 i m). X − ≤ ≤ Hence dim B =2,andhenceθ is birational. Consider the case where X = 0. Since XYm 1 = Ym(1 + Ym), we have Ym =0or 1. Further, − − since XYi = Yi+1(1+Ym)for0 i 0. In fact, YmYm 2 = Ym 1 implies Ym 2 = Ym 1 with α2 =2. − αi 2 − − αi−1 − Suppose that Ym i+2 Ym −1 and Ym i+1 Ym −1 with 0 <αi 2 < − ∼ − 2 − ∼ − αi − αi 1 for i 2. Since Ym i+2Ym i = Ym i+1,wehaveYm i Ym 1 with − ≥ − − − 2 − ∼ − αi =2αi 1 αi 2 > 0. Since Ym i 1Ym i+1 = Ym i,wehaveYm i 1 αi+1 − − − − − − − − − ∼ Ym 1 with αi+1 =2αi αi 1 =(4αi 1 2αi 2) αi 1 =3αi 1 2αi 2, − − − − − − − − − − − where αi+1 αi = αi 1 αi 2 > 0. So, we are done. − − − − 36 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

Suppose that X = Ym 1 =0andYm = 1. We show that Yi =0 − − for 0 i

Remark 5.4. In [14], Dubouloz-Finston treated a hypersurface X(m, n) = xmy znu =1 in A4 =Speck[x, y, z, u]whichisquasi-homogeneous { − } t m n with respect to the grading (x, y, z, u)=(tx, t− y,tz,t− u)andshowed that V (m, n) := X(m, n)// G m is a Danilov-Gizatullin surface of degree d := m+n. Namely V (m, n) is obtained as Fn C,whereFn is a Hirze- bruch surface of degree n and C is an ample\ section of the canonical 1 P -ﬁbration of Fn.ThisgeneralisesthefactthattheDanielewskisur- face X(1, 1) gives rise to V (1, 1) which is isomorphic to F0 ∆, where ∆is the diagonal. \ AFFINE THREEFOLDS AND Ga-ACTIONS 37

It is shown in [18] that a Danilov-Gizatullin surface V has a Ga-action 1 1 such that V//Ga ∼= A and the quotient morphism q : V A has only one reducible ﬁber with two irreducible components with→ respective multiplicity one. The following result shows that the threefolds X(m, 1) cannot be distinguished from each other by means of the ordinary topological quantities.

Theorem 5.5. Let Xm := X(m, 1) be an aﬃne hypersurface xy zmu =1.Thenthefollowingassertionshold. −

(1) H1(Xm; Z)=H2(Xm; Z)=0and H3(Xm; Z)=Z. (2) Let Nm be the boundary 5-manifold at infinity, i.e., the boundary of a tubular neighborhood of the boundary divisor at infinity with respect to a suitable smooth normal compactification of Xm.ThenthehomologygroupsHi(Nm; Z) is independent of m, where 0 i 5. ≤ ≤ Proof. (1) As explained in Theorem 5.3, Xm has a fixpoint free Gm- action. Hence Xm is a C∗-fiber bundle over Vm := Xm// G m in the sense of C∞-topology. Hence we have a homotopy exact sequence

π2(C∗) π2(Xm) π2(Vm) π1(C∗) π1(Xm) π1(Vm), → → → → → where π2(Xm) ∼= H2(Xm; Z)andπ2(Vm) ∼= H2(Vm; Z)byHurewicz’s isomorphism theorem since π1(Xm) ∼= π1(Vm)=(1).NotethatH2(Vm; Z) ∼= Z because χ(Vm)=2andH2(Vm; Z)hasnotorsionbyHamm’s theorem. Hence H2(Xm; Z)=0becauseπ1(C∗) = Z and π2(C∗) = 1 ∼ ∼ π2(S ) = (1). On the other hand, let Y1 be the subvariety of X1 defined by z =0.Thenthemorphismψ : Xm X1 defined by (x, y, z, u) (x, y, zm,u) is a ramified covering which→ is totally rami- -→ 1 1 fied over Y1 and unramified over X1 Y1.HereY1 ∼= A A ,whence χ(Y ) = 0. Since χ(X )=0,wehave\χ(X Y )=0andhencewehave∗ × 1 1 \ 1 1 1 χ(X )=χ(X ψ− (Y )) + χ(ψ− (Y )) m m \ 1 1 = mχ(X Y )+χ(Y ) 1 \ 1 1 = m (χ(X Y )+χ(Y )) = 0. 1 \ 1 1 Since H3(Xm; Z)hasnotorsionagainbyHamm’stheorem,wehave H3(Xm; Z) ∼= Z. (2) By a change of coordinates, we write the defining equation of 2 2 m Xm as x + y + z u =1.LetNm be as stated above. It is also the boundary of a big closed ball taken inside Xm.ThenNm is a ramified covering of N which is totally ramified over F := N z =0 and 1 1 1 ∩{ } unramified outside F1. 38 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

2 2 Note that X1 has a natural compactification X1 = X +Y +ZU = 2 4 { V which is a quadric hypersurface in P and X1 X1 is the quadric } 2 2 3 \ surface Q = X + Y + ZU =0 in P . Since N1 is the boundary { } 1 of a tubular neighborhood of Q which is smooth, N1 is an S -bundle 1 1 1 over Q = P P .ThecomplementN1 F1 is a trivial S -bundle over 2 ∼ × \ 2 1 A = Q Z =0 ,whenceN1 F1 is isomorphic to A S . \{ } \ × Now F1 is embedded into Nm since the mapping Nm N1, (x, y, z, u) (x, y, zm,u), is totally ramified over F . Using the relative→ cohomol- -→ 1 ogy sequence with integral coefficients for the pair (Nm,F1), we have an exact sequence 0 H0(N ,F ) H0(N ) H0(F ) → m 1 → m → 1 H1(N ,F ) H1(N ) H1(F ) H2(N ,F ) → m 1 → m → 1 → m 1 H2(N ) H2(F ) H3(N ,F ) H3(N ) → m → 1 → m 1 → m H3(F ) H4(N ,F ) H5(N ) , → 1 → m 1 → m where i 2 1 H (Nm,F1) = H5 i(A S ; Z)=0(i =1, 2, 3) ∼ − 4 2 × 1 H (Nm,F1) = H1(A S ; Z) = Z ∼ × ∼ by the Lefschetz duality and by noting that the homology groups of A2 S1 are the same as those of S1 since A2 is contractible to one × i i point. Hence it follows that H (Nm) ∼ H (F1)fori =0, 1, 2. Since 5 −→ 3 Nm is simply connected, H (Nm) = H1(Nm; Z)=0.So,H (Nm) = 3 ∼ ∼ Ker (H (F1) Z). By the above→ observation and the Poincaréduality,thehomology groups H5 i(Nm; Z) is independent of m for 0 i 3. Since H1(Nm; Z) − ≤ ≤ =0andH0(Nm; Z) ∼= Z,allintegralhomologygroupsofNm are independent of m. 2

Remark 5.6. The independence of Hi(Nm; Z)ofm makes a clear m difference from the case of the Danielewski surfaces Zm := x y = 2 { z 1 ,whereH1(Zm; Z) = Z/2mZ by Fieseler [17]. − } ∼ Generalizing the results in Lemma 5.2, we ask the following problem. Problem 5.7. Let X = T PGL(n) with a maximal torus T .ThenX is a smooth affine variety of\ dimension n(n 1). − 1 n(n 1) 1 n(n 1) (1) Clarify the A 2 − -fibrations on X.ThoseA 2 − -fibrations are given by the right actions of maximal unipotent subgroups of PGL(n) and transferred to each other by the action of the Weyl group. Especially, clarify all the singular fibers. Describe the connected componets of the singular fibers in terms of the AFFINE THREEFOLDS AND Ga-ACTIONS 39

algebraic group. Show that X does not have any Ar-fibration 1 with r>2 n(n 1). − n(n 1) (2) X is simply connected as it contains A − as an open set. Use the above clarifications of singular fibers to compute the homology groups of X. (3) Compute the Picard group Pic(X) and the Picard number ρ(X). (4) If possible, prove that X does not have the cancellation property.

6. Some results about A1-andP1-fibrations In this section, by a component of a variety W ,wewillmeanan irreducible component of W .Wewilldescribesomeelementaryresults about singular fibers of an A1-fibration (a P1-fibration) on a smooth affine (resp. a projective) 3-fold. Some of these arguments have been already used in [23]. Let f : X Y be a morphism from a smooth affine 3-fold X to a normal affine→ surface Y .Wedonotassumethatf is surjective. We can embed X as a Zariski-open subset of a smooth 3-fold V such that f extends to a proper morphism f : V Y . The set of points y Y 1 → ∈ 1 such that the fiber f − (y) is scheme-theoretically not isomorphic to P is called a singular value of f. For" simplicity we call the divisorial part of the set of singular" values the bad curve of f,denotedbyB(f). The 1 1 fibers f − (y)orf − (y)aredenotedby" f ∗(y)orf ∗(y)toemphasizethe scheme-theoretic fibers. " " We will use the" following general result of Nori" [43]. Lemma 6.1. Let f : X Y be a surjective morphism between smooth algebraic varieties such→ that a general fiber F of f is irreducible. As- sume that there is a closed subvariety S of Y such that S has codimen- 1 sion > 1 in Y and for any y Y S the fiber f − (y) has a reduced ∈ \ component. Then the natural sequence of homomorphisms π1(F ) π (X) π (Y ) (1) is exact. → 1 → 1 → We will need a slight extension of this result as follows [48, 1]. § Lemma 6.2. Let f : X C be a surjective morphism with X asmooth surface, C asmoothcurveandageneralfiber→ F of f irreducible. For apointy C the greatest common divisor of the multiplicities of the ∈ components of f ∗(y) is called the multiplicity of the fiber f ∗(y).Let m1F1, .., m&F& exhaust all the fibers of f with multiplicities mi > 1. Suppose that C has r places at infinity and let g be the geometric genus of C.Thenthereisanexactsequence π(F ) π (X) Γ (1), → 1 → → 40 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL where Γ is the group with generators a1,b1, ..., ag,bg,c1, .., cr,e1, .., e& and relations [a ,b ] [a ,b ]c c c =1=em1 = = em! . 1 1 ··· g g 1 · 2 ··· r 1 ··· & Now assume that f : X Y is as in Lemma 6.1, where dim Y>1. → 1 For a general hyperplane section Z of Y ,bothZ and f − (Z)aresmooth 1 and the restricted morphism f − (Z) Z has an irreducible general fiber. By a relative version of Lefschetz→ hyperplane section theorem 1 there is an isomorphism π1(f − (Z)) ∼= π1(X). We apply this to the proper morphism f : V Y , where the hyperplane section is a smooth curve C on Y .NowwecanuseLemma6.2.→ " " Lemma 6.3. Let C0 be a component of C.Thenthereisnointeger m>1 with the following property. Let y C be a general (smooth) point of both C and Y .Letz ,z be ∈ 0 1 2 local analyic coordinates at y on Y such that C0 = z1 =0 .Forany { } m point x V such that f(x)=y the function z1 is expressed as z1 = h for a holomorphic∈ function h on a neighborhood of x in V . Namely, the multiplicity of a fiber of f is 1 except for a finite set of singular values of f. " Proof. Suppose the result is false. Then there exists some integer m>1 which has the" property described in the statement. We choose 2 another small disc D in C with local holomorphic coordinates w1,w2 m on D.Letτ : D Y be the holomorphic map τ(w1,w2)=(w1 ,w2) such that the image→ of the origin in D is the point y Y and wm = z ∈ 1 1 and w = z .ConsiderthenormalizationV of the fiber product D V . 2 2 ×Y (We call V the normalized fiber product.) Using purity of branch loci, we check easily that V is a complex (smooth)" manifold and the analytic map V " V is finite and unramified. Further, every fiber of the → induced P1-fibration"V D except possibly the fiber over the origin has multiplicity" 1. Since→ the punctured disc D (0, 0) is simply- −{ } connected, by Lemma" 6.1, we have an exact sequence 1 π1(P ) π1(V F0) (1). → − → Here F is the fiber of V D over the origin. This shows that V F , 0 → " " − 0 and hence also V ,aresimply-connected.SinceF0 is a strong deformation retract" of V , we infer" that F0 is simply-connected. Denoting" the" fiber of V Y "over y by F ,wenowknowthat"F is a quotient of F → 0 0 0 modulo the finite" cyclic group Z/m" Z.ByLemma6.4below,F0 is also simply-connected. The morphism F0 F0 is finite and unramified" of degree m>1. This contradiction proves→ the lemma. 2 " AFFINE THREEFOLDS AND Ga-ACTIONS 41

In the last step of the above proof, we made use of the following result. Lemma 6.4. Let a finite group G act algebraically on a simply-connected complete reduced curve Γ.ThenΓ/G is simply-connected. Proof. Using Van kampen’s theorem we see easily that each component of Γis a cuspidal rational curve and the dual graph of Γis a tree. We will prove the result by induction of G and the number of | | components of Γ. We call a component C1 of Γa tip of Γif C1 meets the union of the remaining components of Γin a single point. First assume that all the components of Γmeet in a single point p.TheneverycomponentofΓisatipofΓ.ForanycomponentC1 of Γthe union of its translates by G,sayΓ1 is stable under G. It is easy to see that Γ1/G is an irreducible rational curve. Since any two components of Γmeet only at p,weseethatthevariouscomponentsof Γ/G meet only at the image of p in Γ/G. This implies, again by Van kampen’s Theorem, that Γ/G is simply-connected. Now we consider the case when not every component of Γis a tip of Γ. Since the dual graph of Γis a tree, there is an irreducible component C1 of Γwhich is a tip of Γ. Then each translate of C1 by G is also atipofΓ. LetΓ1 be the union of translates of C1.Theunionof the components of Γwhich are not translates of C1,sayΓ2, is also G-stable and easily seen to be connected. It is also simply-connected, being a sub-curve of Γ. By induction, both Γ1/G and Γ2/G are simply- connected. They meet in a single point in Γ/G, which is the image of Γ Γ .ThisshowsthatΓ/G is simply-connected. 2 1 ∩ 2 Next we will prove the following closely related result. Lemma 6.5. Let f : X Y be a proper morphism from a smooth 3-fold X onto a normal affi→ne surface such that a general fiber of f is P1.IfafiberF of f (taken with reduced structure) is simply-connected, then the exceptional divisor in any resolution of singularities of Y at p := f(F ) is simply-connected. Conversely, if the exceptional divisor in the resolution of a singular point p of Y is simply-connected, then 1 the corresponding fiber f − (p) is simply-connected. Proof. We will give a proof of only one implication. The other implication is similar. So assume that Fred is simply-connected. We can assume that Y is asmallSteinneighborhoodofp.Bytheaboveargumentsitfollows that X is simply-connected (since Fred is a strong deformation retract of X). Let Y Y be a resolution of singularities. We can find a → " 42 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL sequence of blowing ups with smooth centers of X,sayX X,such → that f extends to a proper morphism f : X Y . Since X X is a → → proper birational morphism and X is smooth, we know that" X is also simply-connected. " " " " AgeneralfiberofthemorphismX Y is P1.Hencewehavea" → surjection π (X) π (Y ). This implies that Y is simply-connected. 1 → 1 Since the exceptional divisor E is a strong" deformation" retract of Y , it follows that E"is simply-connected." " 2 " Lemma 6.6. Let f : X Y be a proper morphism from a smooth 3-fold onto a normal surface→ Y .LetF be a 1-dimensional fiber of f. Then Y has at worst a quotient singularity at f(F ). Proof. Let p := f(F ). Let C be a component of F and let q C be a general point. For a general transverse hyperplane section S of∈X at q, the point q is isolated in the inverse image of p for the morphism f S.It follows that that the completion of the local ring of X at q is integral| over the completion of the local ring of Y at p. This implies (using Mumford’s result on the topology of normal surface singularities) that Y has at worst a quotient singularity at p. 2 Combining Lemmas 6.5 and 6.6, we get the following: Theorem 6.7. Let f : X Y be as in the statement of Lemma 6.5. Then the fiber F is simply-connected.→ Proof. This follows since the exceptional divisor of a resolution of singularity of Y at f(F ) is a tree of non-singular rational curves. 2 Next we use the following result from [33, p. 107, (2.8.6.3)] to deduce another property of a singular fiber of a P1-fibration. Proposition 6.8. Let f : X Y be a proper morphism with X → normal, Y smooth and a general fiber P1.LetF be a 1-dimensional fiber of f.ThenH1(F , )=(0). red O If C is an irreducible component of Fred then we have a surjection H1(F , ) H1(C, ). Hence H1(C, ) = (0). Clearly, H0(C, )= red O → O O O C.HencethearithmeticgenusofC is 0. This implies that C is a smooth rational curve. This shows that every component of Fred is a smooth rational curve, and Lemma 6.5 implies that Fred is simply- connected. 7. Twisted additive group schemes and their actions Let p : Y X be an A1-fibration such that Y is affine. If X is affine and A is→ factorially closed in B where A =Γ(X, )andB = OX AFFINE THREEFOLDS AND Ga-ACTIONS 43

Γ(Y, Y ), then there exists a Ga-action on Y such that the morphism p coincidesO with the quotient morphism q : Y Y//G (see [23, Lemma → a 1.2]). However, if X is not affine, such a Ga-action does not exist for otherwise X must be isomorphic to Spec Ker δ,whereδ is the locally nilpotent derivation (lnd for short) associated to the Ga-action. In order to treat X as the quotient space of a certain group scheme acting on the variety Y ,wehavetoconsiderthelinebundleoverX equipped with the group structure which is locally isomorphic over X to the additive group scheme Ga. This idea was already exploited in Dubouloz [11] and applied to the case where p : Y X is an A1- bundle. We are interested in extending the result to→ the case where p is an A1-fibration. We begin with redefining the line bundle L with additive group structure over X. As in the previous sections, we consider all varieties and schemes defined over k.Butthemostofthetheorycanbedevelopedoveran affine scheme S =SpecR over Q.LetX be a scheme over k and let L be an invertible sheaf. Let = Ui i I be an affine open covering of U { } ∈ X such that = e ,wheree = s e for s Γ(U , ∗ ), where L|Ui OUi i j ji i ji ∈ ij OX Uij = Ui Uj.ThelinebundleL over X associated to is the affine scheme Spec∩ [ ], where [ ] = [e ] is a polynomialL ring over OX L OX L |Ui OUi i Ui in the variable ei.Definethecomultiplication∆,thecoinverseι andO the counit ε locally by ∆ (e )=e 1+1 e ,ι(e )= e ,ε(e )=0, |Ui i i ⊗ ⊗ i |Ui i − i |Ui i where the tensor products are taken over Ui .ThenL is a group O scheme over X,whichislocallyoverUi isomorphic to the additive group scheme Ga,Ui .HencewedenoteL also by Ga, and call it the -twisted additive group scheme over X. L L Lemma 7.1. (1) Let and be invertible sheaves on X.Then L M = if and only if L and M are isomorphic group schemes over X. L ∼ M (2) Let g : X# X be a morphism of schemes. Then L X X# = → × ∼ Ga,g∗ as groups schemes over X#. (3)L Let f : Y X be a morphism of schemes with Y anormal → affine scheme and let σ : Ga, Y Y be an action of X-group schemes on Y .ThentheactionL ×σ corresponds→ bijectively to a section 1 1 δ of Γ(Y, f ∗( )− ) such that δ is an lnd on Γ(f − (U ), ), TY/X ⊗ L |Ui i OY where = Ui i I is an affine open covering of X. U { } ∈ Proof. (1) Suppose that = .Thenthereexistsanaffineopen L ∼ M covering = Ui i I such that U = U ei and U = U zi for U { } ∈ L| i O i M| i O| i every i I,whereej = sjiei and zj = tjizi on Uij with sji,tji ∈ 1 ∈ Γ(U , ∗ )andsuchthatt = u s u− for u Γ(U , ∗ ). Then ji OX ji j ji i i ∈ i OX 44 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL the mapping z e := u e induces an X-group scheme isomorphism i -→ i i i between M and L. Conversely, supposethatthereexistsanX-group a scheme isomorphism" ϕ : M L,whereϕ : X [ ] X [ ] is an -algebra isomorphism such→ that ∆ϕ =(ϕO ϕL)∆→, whereO M ∆is OX ⊗ the comultiplication. We take an affine open covering of X, sji and t as above. Then ϕ := ϕ : [e ] [zU] is an { }- { ji} i |Ui OUi i →Oi i OUi isomorphism induced by ϕ(ei)=aizi + bi,whereai Γ(Ui, X∗ )and ∈ 1O b Γ(U , ). Since ϕ = ϕ on U , it follows that t = a− s a and i ∈ i OX i j ji ji j ji i bj = sjibi.Sincewehave∆ϕi =(ϕi ϕi)∆, it follows that bi =0. Hence ϕ(e )=a z for every i I.Thus,bythechangeofbase⊗ i i i ∈ zi aizi, ϕ induces an isomorphism ∼= . (2)-→ Straightforward by the definition.L M

(3) Let = Ui i I be an aﬃne open covering of X such that Ui = U { } ∈ 1 L| Ui ei for all i and ej = sjiei on Uji,andlet = f − (Ui) i I .Write O 1 V { } ∈ Ui =SpecAi and f − (Ui)=SpecBi.Thenthecoactionσ∗ restricted 1 onto f − (Ui) is given by

1 n n σ∗ : B B A [e ],b (δ ) (b)e |Ui i −→ i ⊗ i i -→ n! i i n 0 0≥ 1 such that we have on f − (Uji) 1 1 (δ )n(b)en = (δ )n(b)en n! i|Uji i n! j|Uji j n 0 n 0 0≥ 0≥ 1 for b Γ(f − (Uji, Y ). Here δi is an Ai-trivial lnd on Bi.Weneed some∈ extra explanationsO on the notations δ and δ .Thelnd i|Uji j|Uji δi as viewed an element of DerAi (Bi) corresponds to the vector field 1 associated to the Ga-action on f − (Ui)definedbyδi.Thenδi Uji is 1 | the restriction of the vector field to Vji := f − (Uji). Since this vector

field corresponds to the Ga-orbit directions of the points in Vji, δi Vji is an lnd of Γ(V , ). Similarly, δ is the lnd of Γ(V , )| in- ji OY j|Vji ji OY duced by δj.Thenδi = sjiδj. In fact, it suffices that the coincidence occurs on the open set of Vji with a closed set of codimension 2 removed. Hence we can restrict δ and δ on the smooth locus≥ i|Uji j|Uji of Vji by the normality hypothesis of Y .Let Y/X be the relative tan- 1 T 1 gent sheaf Y/X := om X (Ω , X ). Since δj = sji− δi for each pair O Y/X T 1H O (i, j)andδi Γ(f − (Ui), Y/X), we know that δi defines a section of ∈ 1 T { } 1 Γ(Y, Y/X X f ∗( )− )suchthatδi is locally nilpotent over f − (Ui). T ⊗O L 1 Conversely, given such a section δ of Γ(Y, Y/X X f ∗( )− ), it is clear 1 T ⊗O L that δi defines a Ga-action on f − (Ui) and that these locally-defined Ga-actions patch together to define an action of an X-group scheme L = Ga, on Y so that Y//L = X. 2 L AFFINE THREEFOLDS AND Ga-ACTIONS 45

Corollary 7.2. If = X ,thenGa, = Ga and Ga, -action on X is L ∼ O L ∼ L equivalent to a Ga-action on X. The following result is also known in [11].

Lemma 7.3. Let X be a smooth variety and let f : Y X be a P1- bundle. Let S be a section of f such that the complement→ Y := Y S \ is an affine variety, and let f := f .Thenthereexistsaninvertible |Y sheaf on X such that the -twisted additive group Ga, acts on Y L L L and Y//Ga, = X.HenceY is a Ga, -torsor over X. L ∼ L Proof. We have an exact sequence 0 (S) (S) 0 . →OY →OY →OS → 1 1 Since f : Y X is a P -bundle, we have R f Y =0andhencean exact sequence→ ∗O ϕ 0 f Y f Y (S) 0, → ∗O → ∗O −→ L→ where S(S) is viewed as an invertible sheaf on X, f Y (S) is a O L ∗O rank 2 vector bundle on X and f Y = X .FurthermoreY = E ∗O O Proj S•( ), where S•( ) is the symmetric algebra of over . Y E E E OX Choose an affine open covering = Ui i I of X such that Ui = Ui ei and U { } ∈ L| O = e + e with ϕ(e )=e . E|Ui OUi OUi i i i Then we have e s" t e " j = ji ji i , e 01 e $ % $ %$ % where sji Γ(Uji, X∗")andtji Γ(Uji, X )." Since the section S is defined by∈ the surjectionO ϕ,wehave∈ O

Y = Spec Ai[ei/e] i 1 with Ui =SpecAi.Letxi = ei/e and deﬁne" an Ai-trivial lnd δi on 1 Γ(f − (U ), )byδ = ∂/∂x .Thenwehavex = s x + t and i OY i j ji i ji ∂ " ∂ sji (xj)=sji = (sjixi + tji) ∂xj ∂xi 1 on U = U U . This implies that δ = s− δ .Let be an invertible ji j ∩ i j ji i L sheaf on X with transition functions s with respect to .Thenthe { ji} U Y -group scheme Ga, acts on Y via δi i I ,andY//Ga, is identiﬁed with X. L { } ∈ L 2 46 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

1 1 Example 7.4. Let Σ0 = P P and let ∆be the diagonal of Σ0.Let × [x0 : x1](resp.[y0 : y1]) be the homogeneous coordinates of the ﬁrst factor (resp. the second factor) of Σ0.ThecomplementX := Σ0 ∆is the Danielewski surface. By the Segre embedding, we introduce regular\ functions on X as x y x y x y x y x = 0 0 ,y= 1 1 ,z= 0 1 ,u= 1 0 . x y x y x y x y x y x y x y x y 1 0 − 0 1 1 0 − 0 1 1 0 − 0 1 1 0 − 0 1 Then X is identiﬁed with a hypersurface xy = zu = z(z +1),

1 1 where u = z+1. The ﬁrst projection f :Σ0 P induces an A -bundle → morphism f : X P1 deﬁned by → f(x, y, z)=[x : u]=[z : y].

With the notations of Lemma 7.1, = P1 (2). Let = U0,U1 , L ∼ O 1 U { } where U0 = x0 =0 and U1 = x1 =0 .Thenf − (U0)=Speck[t, x] { ' } { ' } 1 with U =Speck[t],y = t(tx 1),z = tx 1,u = tx and f − (U )= 0 − − 1 Spec k[s, y]withU1 =Speck[s],x= s(sy +1),z = sy, u = sy +1.Let 2 δ0 = ∂/∂x and δ1 = ∂/∂y. Then it follows that t (∂/∂y)=∂/∂x,i.e., 2 t δ1 = δ0.Hence,with = 1 (2), L acts on X in such a way that L OP X//L = P1. AsimilarargumenttoExample7.4givesthefollowingexample.

Example 7.5. Let Σd be the Hirzebruch surface of degree d and let S be an ample section with n := (S2). Let X := Σ S which is called the d \ Danilov-Gizatullin surface (see [18]). Let f : X P1 be the restriction 1 → of the canonical -fibration of Σd.ThenGa, 1 (n) acts on X in such a P P 1 O way that X//Ga, 1 (n) = P . OP Next we consider the case where f : Y X is simply an A1-fibration. We need the following result. → Lemma 7.6. (1) Let X =SpecA, Y =SpecB and f : Y X be an → A1-fibration, where B is an affine domain and A is an affine subdomain of B.ThenthereexistsanA-trivial lnd δ of B. (2) Let δ,δ# be A-trivial lnds on B.Thenthereexistnonzeroelement a, a# A such that a#δ = aδ#. (3)∈Suppose that B is factorial and A is factorially closed in B. Further assume that both δ and δ# are reduced, i.e., there are no non- invertible elements of A which divides δ (or δ#). Then δ = uδ#,where u A∗. ∈ AFFINE THREEFOLDS AND Ga-ACTIONS 47

Proof. (1) There exists an affine open set D(a)ofX such that 1 1 1 1 N f − (D(a)) = D(a) A .HenceB[a− ]=A[a− ][x]. Let δ = a (∂/∂x), ∼ × where N is chosen in such a way that δ(bi) B with B = A[b1,...,br]. The δ is an A-trivial lnd of B. ∈ (2) Let K = Q(A). Then δ,δ# incuce K-trivial lnds of B := B K ⊗A K = K[x]. Then δ = λ(∂/∂x)andδ# = µ(∂/∂x), where λ, µ K. ∈ Then there exist a, a# A 0 such that a#δ = aδ#. ∈ \{ } (3) Note that the assumption implies that A is factorial. Since a#δ = aδ#,onecancanceloutthegcd(a, a#)froma and a#,andassumethat gcd(a, a#)=1.Letz be an arbitrary element of B.Thenaδ#(z)= a#δ(z). Since B is factorial and A is factorially closed in B,anyprime element of A remains prime in B.Hencea divides δ(b). Namely a divides δ. Since δ is reduced by the assumption, a A∗.Similarly, 1 1 ∈ a# A∗.Henceδ =(aa#− )δ# with u := aa#− A∗. 2 ∈ ∈ Theorem 7.7. Let f : Y X be an A1-fibration such that → (1) Y is a factorial affine variety of dimension less than or equal to three 6and the image of f contains all codimension one points of X. (2) Every fiber of f either is the empty set or has equi-dimension one. (3) X is normal. Then there exists an invertible sheaf on X such that the -twisted L L X-group scheme Ga, acts on the X-scheme Y and Y//Ga, = X. L L

Proof. (1) Note that f is an affine morphism. Let = Ui i I be U 1 { } ∈ an affine open covering of X.Foreveryi I, let V = f − (U ). Since ∈ i i Vi is affine, we write Vi =SpecBi and Ui =SpecAi,wherewecan assume that Ai and Bi are affine domains. Then fi := f Vi : Vi Ui 1 | → is an A -fibration. Then there exists an Ai-trivial lnd δi such that A Ker δ B ,whereKerδ is an affine domain by the assumption i ⊆ i ⊂ i i (1). Then fi splits as f : V πi Z τi U , i i −→ i −→ i where Zi =Kerδi.Thenτi is a birational morphism whose fibers are either the empty set or a finite set by the assumption (2). Since X is normal, τi is then an open immersion by Zariski’s Main Theorem. Since the image of τi contains all codimension one points of Ui, it follows that τi is biregular.

6 This assumption is used only for the ﬁnite generation of Kerδi in the subsequent proof. 48 R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL

1 (2) Let Uji = Ui Uj and Vji = Vi Vj = f − (Uji). Write Uji =SpecA and V =SpecB.∩ Since π = f ∩: V U is the quotient morphism ji i |Vi i → i by the Ga-action σi associated to the lnd δi.HenceVji is a Ga-stable open set of Vi,whencetherestrictionσi Vji corresponds to the lnd δ of B. Since B is aﬃne domain, the argument| in the step (1) shows that A =Kerδ.Similarly,σ corresponds to the lnd δ# of B such that j|Vji Ker δ# = A.NotethatB is factorial. In fact, if D is an irreducible subvariety of codimension 1 of Vji, let D be the closure of D in Y . Since Y is factorial, there exists a regular function g on Y such that D = V (g). Then D is deﬁned by g = 0. Since A is factorially closed |Vji in B, Lemma 7.6, (3) implies that δ = uδ# with u A∗. Namely we ∈ have δi Vji = sjiδj Vji.Sinceitfollowsthatski = skjsji, let be the invertible| sheaf on|X with the transition functions s with respectiveL { ji} to .ThenGa, acts on the X-scheme Y locally via δi i I so that U L { } ∈ Y//Ga, = X. 2 L

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 001, India E-mail address: [email protected]

Institute of Mathematics, Warsaw University, ul. Banacha 2, War- saw, Poland E-mail address: [email protected]

School of Science & Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan E-mail address: [email protected]

Research Center for Mathematical Sciences, Kwansei Gakuin Uni- versity, 2-1 Gakuen, Sanda 669-1337, Japan E-mail address: [email protected]

Department of Mathematics and Statistics, McGill University, 805 Sherbooke St. West, Montreal, QC, Canada E-mail address: [email protected]