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Preprint submitted on 14 Sep 2004
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Danielewski-Fieseler surfaces
Adrien Dubouloz
To cite this version:
Adrien Dubouloz. Danielewski-Fieseler surfaces. 2004. �hal-00001044v3�
ccsd-00001044, version 3 - 14 Sep 2004
ADRIEN DUBOULOZ
Abstract. We study a class of normal affine surfaces with additive group actions which contains in particular the Danielewski surfaces inA3given by the equationsxnz=P(y), wherePis a nonconstant polynomial with simple roots.
We call them Danielewski-Fieseler Surfaces. We reinterpret a construction of Fieseler [7] to show that these surfaces appear as the total spaces of certain torsors under a line bundle over a curve with an r-fold point. We classify Danielewski-Fieseler surfaces through labelled rooted trees attached to such a surface in a canonical way. Finally, we characterize those surfaces which have a trivial Makar-Limanov invariant in terms of the associated trees.
Introduction The surfaces Sj =
xjz=y2+ 1 , j ≥1, in C3 = Spec (C[x, y, z]) admit free actions of the additive group Ga,C induced by the locally nilpotent derivations
∂j =xj∂y + 2y∂z of C[x, y, z] respectively. Danielewski [4] observed that the as- sociated quotient morphism π : Sj → Sj//Ga,C ≃ Spec (C[x]) is an A1-fibration which factors throught anA1-bundleρ:Sj→X over the affine line with a double origin δ : X → Spec (C[x]). Indeed, Sj is obtained as the gluing of two copies Spec (C[x] [Ti]) ofC2,i= 1,2, overC∗×Cby means of theC[x]x-algebras isomor- phisms C[x]x[T1] →∼ C[x]x[T2],T1 7→2x−j +T2. This interpretation was further generalized by Fieseler [7] to describe certain invariant neighbourhoods of the fibers of a quotientA1-fibrationπ:S→Z associated with a nontrivial action ofGa,Con a normal affine surfaceS. More precisely, he established that if a fiber π−1(z0) is reduced then the induced morphismp2:S×ZSpec (OZ,z0)→Spec (OZ,z0), where OZ,z0 denotes the local ring of z0 ∈Z, factors through anA1-bundle ρ:S →X over the schemeX obtained from Spec (OZ,z0) by replacingz0by as many points as there are connected components inπ−1(z0). More generally, given a fieldkof car- acteristic zero and a pair (X= Spec (A), x0= div (x)), where Ais either discrete valuation ring with uniformizing parameterxand residue fieldkor of a polynomial ringA=k[x], the same description holds for the following class of surfaces.
Definition 0.1. ADanielewski-Fieseler surface with base(A, x) (A DFS, for short) is an integral affineX-schemeπ:S→X of finite type, restricting to a trivial line bundle over X∗ = X \ {x0}, and such that π−1(x0) is nonempty and reduced, consisting of a disjoint union of curves isomorphic to affine lineA1k.
In this paper, we give a combinatorial description of DFS’s in terms of (A, x)- labelled rooted trees, that is, pairs γ = (Γ, σ) consisting of a rooted tree Γ and a cochainσ∈An, indexed by the terminal elements of Γ, satisfying certain conditions with respect to the geometry of Γ (see (1.5) below). Then, as an application, we
2000Mathematics Subject Classification. 14L30,14R05,14R20,14R25.
1
characterize the DFS’s with base (k[x], x) which have a trivial Makar-Limanov invariant [10].
The paper is divided as follows. In section 1, we collect some preliminary results on labelled rooted trees. In section 2, we reinterpret the ’cocycle construction’ of Fieseler [7] to describe DFS’s as torsors under certain line bundles on a nonseparated scheme. In section 3, we associate to every labelled tree γ, a DFSπγ :S(γ)→X which comes equipped with a canonical birationalX-morphismψγ :S(γ)→A1X. For instance, given a polynomial P ∈ k[y] with simple rootsy1, . . . , yn ∈ k, the Danielewski-Fieseler surface with base (k[x], x)
π:SP,m= Spec (k[x, y, z]/(xmz−P(y)))−→X = Spec (k[x]),
equipped with the morphismpry:SP,m→A1X= Spec (k[x] [y]) corresponds to the following labelled treeγP,m= (Γm,n, σ).
Γm,n=
m
n σ={yi}i=1,...,n∈k[x]n
In Theorem (3.2), we establish that the category−→
D(A,x)of DFS’s with base (A, x) equipped with certain birational morphisms as above and the category T(A,x) of (A, x)-labelled trees are equivalent. Then, in Theorem (3.10), we classify these DFS’s up toX-isomorphisms in terms of the corresponding trees. In section 4, we decompose birationalX-morphisms between two DFS’s into a succession of simple affine modifications [11], which we call fibered modifications (Theorem 4.5). This leads to a canonical procedure for constructing embeddings of a DFS π: S →X into a projectiveX-scheme ¯π : ¯S →X (Proposition 4.7). Section 5 is devoted to the study of DFS’s with base (k[x], x), which we call simplyDanielewski surfaces.
We recall that the Makar-Limanov invariant [10] of an affine varietyV /kis defined as the intersection in Γ (V,OV) of all the invariant rings of Ga,k-actions on V. Makar-Limanov [12] and [13] established that a surface SP,m≃S(γP,m) as above has trivial Makar-Limanov invariant ML (SP,m) = k if and only if m = 1 and deg (P)≥1. More generally, in case thatk= ¯k is algebraically closed, we give the following characterization (Theorem 5.4).
Theorem 0.2. A Danielewski surfaceS(γ)has a trivial Makar-Limanov invariant if and only ifγ is a comb,i.ea labelled treeγ= (Γ, σ)with the property that all but at most one of the direct descendants of a given e∈Γare terminal elements of Γ.
Finally, we obtain the following description of normal affine surfaces with a trivial Makar-Limanov invariant, which generalizes previous results obtained by Daigle- Russell [3] and Miyanishi-Masuda [14] for the particular case of log Q-homology planes (Theorem 5.9).
Theorem 0.3. Every normal affine surface S/¯k with a trivial Makar-Limanov invariant is isomorphic to a cyclic quotient of a Danielewski surface.
1. Preliminaries on rooted trees
Here we give some results on rooted trees and labelled rooted trees which will be used in the following sections.
Basic facts on rooted trees. Let (G,≤) be a nonempty, finite, partially ordered set (aposet, for short). A totally ordered subsetC⊂Gis called asubchain of length l(C) = Card (C)−1. Asubchain of maximal length is called amaximal subchain.
For every e ∈ G, we let (↑e)G = {e′∈G, e′ ≥e} and (↓e)G = {e′∈G, e′ ≤e}.
Theedges E(G) of Gare the subsets −→
ee′={e′ < e} of Gwith two elements such that (↑e′)G∩(↓e)G=−→
ee′.
Definition 1.1. A (rooted)treeΓ is poset with a unique minimal elemente0called theroot, and such that (↓e)Γ is a chain for everye∈Γ. A (rooted)subtree of Γ is a sub-poset Γ′⊂Γ which is a rooted tree for the induced order. We say that Γ′ is maximal if there existse′∈Γ such that Γ′= (↑e′)Γ.
1.2. An elemente∈Γ such thatl(↓e)Γ=mis said to beat levelm. The maximal elementsei=ei,mi, wheremi=l(↓ei)Γ, of Γ are called theleaves of Γ. We denote the set of those elements byL(Γ). The corresponding subchains
(1) (↓ei,mi)Γ ={ei,0=e0< ei,1<· · ·< ei,mi−1< ei,mi =ei}, i= 1, . . . , n.
are the maximal subchains of Γ. We say that Γ hasheight h(Γ) = max (mi). The first common ancestor of two elemente, e′∈Γ\ {e0}is the maximal element of the chain ((↓e)Γ\ {e})∩((↓e′)Γ\ {e′}). Thechildren ofe∈Γ\L(Γ) are the minimal elements of (↑e)Γ\ {e}. We denote the set of those elements by ChΓ(e).
e0 e
e2,4
e3,4
e4,4
e1,3
(↑e)Γ (↓e)Γ
Figure 1. A tree Γ rooted ine0.
Definition 1.3. Amorphism of (rooted)treesis an order-preserving mapτ: Γ′→ Γ satisfying the following properties:
a) The image of a maximal chain of Γ′ byτ is a maximal chain of Γ.
b) For everye′∈Γ′,τ−1(τ(e′)) is eithere′ itself or a maximal subtree of Γ′. Injective and bijective morphisms are referred to as embeddings and isomorphisms respectively.
1.4. This definition implies that for every leafe′i,m′
i of Γ′ at levelm′i,τ e′i,m′
i
is a leafej(i),mj(i) of Γ at level mj(i) ≤m′i, such thatτ
e′i,k
=ej(i),min(k,mj(i)) for everyk= 0, . . . , m′i(see (1) for the notation).
Labelled rooted trees. In this subsection, we fix a pair (A, x) consisting of a domainAand an elementx∈A\ {0}.
Definition 1.5. An (A, x)-labelled tree is a pairγ= (Γ, σ) consisting of a a tree Γ with the leaves e1,m1, . . . , en,mn and a cochainσ={σi}i=1,...,n ∈An such that σj−σi ∈xdijA\xdij+1A whenever the first common ancestor ofei,mi and ej,mj
is at level dij < min (mi, mj). A cochain σ with this this property is said to be Γ-compatible.
Amorphism of (A, x)-labelled trees τ : (Γ′, σ′)→(Γ, σ) is a morphism of trees τ : Γ′→Γ such that ifej(i),mj(i) ∈L(Γ) is the image of a leafe′i,m′
i ∈L(Γ′) byτ, thenσi−σj(i)∈xmj(i)A.
The category of (A, x)-labelled just defined is denoted byT(A,x).
Example 1.6. Every isomorphism class of (k[x], x)-labelled trees contains a tree γ = (Γ, σ) with the leavese1,m1, . . . , en,mn, such thatσi =Pmi−1
j=0 wi,jxj ∈k[x]
is a polynomial of degree < mi for every i = 1, . . . , n. In turn, this cochain σ is uniquely determined by the choice of aweight function
wσ:E(Γ)−→k, −−−−−−→ei,j+1ei,j7→wσ(−−−−−−→ei,j+1ei,j) =wi,j
with the property thatwσ
−→ e′e
6=wσ
−→
e′′e
whenevere′ande′′are children of the samee∈Γ. A tree Γ equipped with such a function w is referred to as afine k- weighted tree. A morphism of treesτ: Γ′→Γ gives rise to a morphism of (k[x], x)- labelled τ : (Γ′, σ′)→ (Γ, σ) if and only if wσ
−−−−−−−→
τ(e)τ(e′)
=wσ′
−→ ee′
whenever τ(e)6=τ(e′). If it is the case, then we say thatτ is amorphism of fine k-weighted trees. In this way, we obtain a bijection between morphisms of (A, x)-labelled trees and morphisms of fine k-weighted trees. We conclude that the categoriesT(k[x],x)
of (k[x], x)-labelled trees andTwk of finek-weighted trees are equivalent.
Gluing trees. Intuitively, (A, x)-labelled trees are constructed by gluing (A, x)- labelled chains. Similarly, a morphismτ:γ′ →γof (A, x)-labelled trees is uniquely determined by its values on maximal subchains ofγ′. More precisely, we have the following results.
Proposition 1.7. Suppose that(n, m, d, σ)is a data consisting of an integern≥1, a multi-indexm= (m1, . . . , mn)∈Zn>0, a matrix d= (dij)i,j=1,...,n ∈Matn(Z≥0) and a cochainσ={σi}i=1,...,n∈An with the following properties.
1) For everyi6=j ,dij=dji<min (mi, mj).
2) For every triplei, j, k,min (dij, dik) = min (dji, djk).
3) For everyi6=j,σj−σi∈xdijA\xdij+1A.
Then there exists a treeΓ, unique up to a unique isomorphism, with the roote0and the leaves ei at levels mi,i= 1, . . . , n, such that σisΓ-compatible.
Proof. Up to an isomorphism, γi = (Ci={ei,0< ei,1<· · ·< ei,mi}, σi) is the unique chain of height mi ≥ 1 such that σi is Ci-compatible. For every i 6= j, we let Cij = ↓ei,dij
Ci ⊂Ci. Condition (2) guarantees that there exist isomor- phisms of chains φij : Cij
→∼ Cij such that φij(Cij∩Cik) = Cji∩Cjk for every triplei, j, k, and such that the cocycle conditionφik =φjk◦φijholds on onCij∩Cik. Moreover, (1) implies that for every i= 1, . . . , n,ei,0 belongs toCij whereasei,mi
does not. Therefore, the quotient poset Γ =Fn
i=1Ci/(Cij ∋ei,k ∼φij(ej,k)∈Cji) is a tree rooted in the common image e0 =ei,0 of the rootsei,0 of the chainsCi,
and with the leavesei=ei,mi at levelmi,i= 1, . . . , n. Finally, (3) means exactly
thatσ is Γ-compatible. This completes the proof.
Proposition 1.8. Let γ = (Γ, σ) and γ′ = (Γ′, σ′) be two (A, x)-labelled trees.
Suppose we are given a collection of morphisms of(A, x)-labelled trees τi:γi′ = (Γ′i, σi′) = ((↓e′i)Γ′, σi′)−→γ, whereL(Γ′) ={e′1, . . . , e′n′}, restricting to a same morphism of trees τij =τji : Γ′i∩Γ′j →τi(Γ′i)∩τj Γ′j
for everyi6=j. Then there exists a unique morphism of(A, x)-labelled treesτ:γ′→γ such that τ|γi′=τi for everyi= 1, . . . , n′.
Proof. The conditions guarantee that there exists a unique order-preserving map τ : Γ′ → Γ such that τ |Γ′i= τi, and such that the preimage τ−1(e) of a given e ∈ τ(Γ′) is the maximal subtree of Γ′ rooted in the unique minimal element of τ−1(e). Thus τ is a morphism of trees compatible with the cochains σ′ and σ in the sense (1.5), whence a morphism of (A, x)-labelled trees.
Blow-downs of trees. By definition, (see (1.5)), A morphism of treesτ : Γ′→Γ factors through the retraction of a collection of maximal subtrees of Γ′, followed by an embedding. Therefore, for every element e ∈ Γ′\L(Γ′) such that ChΓ′(e)⊂ L(Γ′), the image of the subtree Γ′e = (↑e)Γ′ of Γ′ is either a subtree τ(Γ′e) of Γ isomorphic to Γ′eor the unique elementτ(e)∈L(Γ). In the second case,τ factors through the morphism of trees
τe: Γ′−→Γ′\ChΓ′(e) e′7→
e′ ife′ ∈Γ′\ChΓ′(e) e ife′ ∈ChΓ′(e)
Definition 1.9. Let γ = (Γ, σ) be an (A, x)-labelled tree, and lete ∈ Γ\L(Γ) be an element such that ChΓ(e) ⊂ L(Γ). A blow-down of the leaves at e is an (A, x)-labelled tree ˆγ(e) =
Γ (e)ˆ ,σˆ(e)
with underlying tree ˆΓ (e) = Γ\ChΓ(e) for which the morphism of trees τe above is a morphism of (A, x)-labelled trees τe:γ→γˆ(e). Since two labelled trees ˆγ(e) with this property are isomorphic, the morphismτe itself will be usually referred to asthe blow-down of the leaves ate.
As a consequence of the above discussion, we obtain the following description.
Proposition 1.10. A morphism of(A, x)-labelled trees factors into a sequence of blow-downs of leaves followed by an embedding.
Equivalence of labelled trees. An (A, x)-labelled tree γ = (Γ, σ) rooted ine0
is calledessential if Card (ChildΓ(e0))6= 1. Theessential subtree Es (Γ) of a given tree Γ is the maximal subtree of Γ rooted either in the unique leaf of Γ if Γ is a chain, or in the first common ancestor ˜e0of the leaves of Γ otherwise. For instance, the tree Γ of Figure 1 above is not essential, and its essential rooted subtree is the maximal subtree (↑e)Γof Γ rooted ine. If an (A, x)-labelled treeγ= (Γ, σ) is not essential then there existsc∈Aand an Es (Γ)-compatible cochain Es (σ) ={˜σi}i=1,...,n ∈An such that σi =c+xmσ˜i for everyi= 1, . . . , n, wheremdenotes the height of the root ˜e0 of the essential subtree Es (Γ) of Γ. A cochain Es (σ) with this property is called an essential cochain for γ, and we say that Es (γ) = (Es (Γ),Es (σ)) is an essential labelled subtree for γ.
Definition 1.11. We say that two (A, x)-labelled treesγ= (Γ, σ) andγ′= (Γ′, σ′) are equivalent if there exist essential cochains Es (σ) and Es (σ′) forγ and γ′ re- spectively, an isomorphism of trees τ: Es (Γ′)→∼ Es (Γ) and a pair (a, b)∈A∗×A such that
aEs (σ′)i−Es (σ)j(i)+b∈xmj(i)A wheneverej(i),mj(i)∈L(Es (Γ)) is the image ofe′i,m′
i∈L(Es (Γ′)) byτ.
Example 1.12. By definition, two essential cochains for a given labelled tree γ differ by the addition of a constantb∈A. Therefore, the essential labelled subtrees forγare all equivalent in the sense of definition (1.11).
2. Danielewski-Fieseler surfaces as fiber bundles
To fix the notation, we letk be a field of caracteristic zero, and we let (A, x) be a pair consisting either of a discrete valuation ringAwith uniformizing parameter x and residue field k, or a polynomial ring k[x] in one variable x. We letX = Spec (A), x0 = div (x)∈ X , and we denote byX∗ = X\ {x0} ≃ Spec (Ax) the open complement ofx0 inX.
Definition 2.1. ADanielewski-Fieseler surface(a DFS for short) with base (X, x0) (or, equivalently, with base (A, x)) is an integral affine X-scheme π : S → X of finite type such thatS |X∗ is isomorphic to the trivial line bundleA1X
∗ overX∗and
such that the scheme-theoretic fiberπ−1(x0) is nonempty and reduced, consisting of a disjoint union of curves isomorphic to the affine line A1k. A DFS with base (k[x], x) is simply referred to as aDanielewski surface.
A morphism of Danielewski-Fieseler surfaces with base (X, x0) is a birational X-morphismβ :S′→S restricting to anX∗-isomorphismβ∗:S′|X∗
−→∼ S|X∗. Danielewski-Fieseler surfaces together with these morphisms form a sub-category D(A,x)(orD(X,x0)) of the category Sch/X
ofX-schemes.
2.2. The total space S of DFS π : S → X is a smooth affine surface over k.
Indeed, X/k is itself affine and smooth, and the local criterion for flatness ([2, III,§5]) guarantees thatπis a faithfully flat morphism, whence a smooth morphism as its geometric fibers are regular. Note that in contrast with general DFS’s, a Danielewski surface is an affine k-scheme of finite type since in this caseX ≃A1k by definition. In general, an arbitraryA1-fibrationπ:S →A1k on a smooth affine surfaceSdoes not give rise to a structure of Danielewski surface onS, but this does not preventS from being a Danielewski surface for another suitable A1-fibration.
For instance, the Bandman and Makar-Limanov surface [1]S⊂Spec (k[x] [y, z, u]) with equations
xz−y(y−1) = 0, yu−z2= 0, xu−(y−1)z= 0,
equipped with the A1-fibrationpru :S →Spec (k[u]) is not a Danielewski surface with base (k[u], u). Indeed the fiber pr−1u (0) is not reduced. However, it is a Danielewski surfaceprx:S→Spec (k[x]) with base (k[x], x).
Following Fieseler [7], DFS’s with base (X, x0) can be described as certain A1- bundles ρ: S →Y over a curve δ:Y →X with an n-fold point overx0. In this section, we give a new interpretation of Fieseler’s ’cocycle construction’.
Torsors under a line bundle. Given an invertible sheafL on a scheme X, the line bundle
V(L) =Spec(S(L)) =Spec
M
n≥0
Ln
−→X
is equipped with the structure of a commutative group scheme for the group law m:V(L)×XV(L)→V(L) defined by means of theOX-algebras homomorphism S(L)−→S(L)⊗S(L) induced by the diagonal homomorphism ∆ :L → L ⊕ L.
For instance, if L =OX thenV(L) is simply the additive group scheme Ga,X = Spec(OX[T]).
2.3. A scheme π : S → X locally isomorphic to the trivial line bundle A1X is called anA1-bundle. IfS comes further equipped with an action of a line bundle V(L)/X for which there exists an open coveringU = (Xi)i∈I ofXsuch thatS|Xi
is equivariantly isomorphic to V(L) |Xi acting on itself by translations for every i ∈I, then we say that S is a V(L)-torsor. We recall [9, XI.4.7] that the set of isomorphism classes ofV(L)-torsors is a group isomorphic toH1(X,L∨). Indeed, an automorphism of the symmetricOX-algebraS(L) is equivariant for the co-action ofS(L) on itself by translations if and only if it is induced by an homomorphism ofOX-modules (g,Id) :L → OX⊕ L, whereg∈HomX(L,OX). Therefore, given a V(L)-torsorπ:S→X which becomes trivial onU, there exists a ˇCech cocycle g ={gij}i,j∈I ∈C1(U,L∨) such that S is isomorphic to the scheme W(U,L, g) defined as the spectrum of the quasi-coherentOX-algebra obtained by gluing the symmetric algebras S(L |Xi) overXij =Xi∩Xj by means of the OXij-algebras isomorphismsS(gij,Id) induced by the OXij-modules homomorphisms (gij,Id) ∈ HomXij L |Xij,OXij⊕ L |Xij
, i 6= j. The isomorphism class of W(U,L, g) is simply the image in H1(X,L∨) ≃ Hˇ1(X,L∨) of the class [g] ∈ Hˇ1(U,L∨) of g∈C1(U,L∨).
2.4. If the base schemeX is integral, then every A1-bundle π: S →X admits a structure of aV(L)-torsor for a suitable invertible sheafLonX. Indeed, sinceX is integral, the transition isomorphismsτij =τi◦τj−1associated with a given collection of trivialisations τi :S|Xi
→∼ Spec (OXi[T]),i ∈I, are induced by automorphisms T 7→g˜ij+fijT ofOXij[T], where ˜gij ∈Γ (Xij,OX) and fij ∈Γ (Xij,OX∗). For a triple of indicesi6=j6=k, the identityτik =τjk◦τij overXi∩Xj∩Xk guarantees that (fij)i,j∈I ∈ C1(U,OX∗) is a ˇCech cocycle defining an invertible sheaf L on X, trivial on U, with isomorphisms φi : L |Xi
→ O∼ Xi, i ∈ I. By construction, g=
˜
gij·φi|Xij i,j∈I ∈C1(U,L∨) is a ˇCech cocycle for which S isX-isomorphic to theV(L)-torsorW(U,L, g).
Schemes with an n-fold divisor. Given a principal divisorx0 = div (x) on an integral scheme X and an integer n ≥ 1, we letδn : X(n) = X(x0, n) → X be the scheme obtained by gluingncopiesdi :Xi(n)→∼ X ofX by the identity over the open subsetsXi(n)∗=Xi(n)\xi(n), wherexi(n) =d−1i (x0). We denote by U(n)= (Xi(n))i=1,...,n the canonical open covering of X(n). For everyi 6=j, we letXij(n) =Xi(n)∩Xj(n)≃X\ {x0}.
Definition 2.5. Given a multi-index µ= (µ1, . . . , µn)∈Zn, we let Ln,µ = OX(n)(µ1x1(n) +. . .+µnxn(n))
be sub-OX(n)-module of the constant sheaf KX(n) of rational functions on X(n) generated by (x◦δn)−µi onXi(n). The dual sheafL∨n,µ of Ln,µ is isomorphic to the sheafLn,−µ corresponding to the multi-index−µ= (−µ1, . . . ,−µn)∈Zn. We denote by sµ the canonicalrational section of Ln,µ corresponding to the constant section 1∈Γ X(n),KX(n)
.
Danielewski-Fieseler surfaces as torsors. Given a DFSπ:S→X with base (X, x0), we denote by C1, . . . , Cn the connected component of the fiber π−1(x0).
We let δn : X(n) = X(x0, n)→ X be as above, and we let ρn : S → X(n) be the uniqueX-morphism such thatπ=δn◦ρn, and such thatCi=ρ−1n (xi(n)) for everyi= 1, . . . , n.
Proposition 2.6. There exist a multi-index µ = (µ1, . . . , µn,) ∈ Zn≥0 such that µi = 0for at least one indice i, and a cocycle g ∈ C1 U(n),Ln,µ
such that S is X(n)-isomorphic to theV(Ln,−µ)-torsor ρn:W U(n),Ln,−µ, g
→X(n).
Proof. Since X is either the affine line or the spectrum of a discrete valuation ring, the Picard group Pic (X) is trivial, and so every invertible sheaf onX(n) is isomorphic toLn,−µfor a certain multi-indexµ∈Zn≥0such thatµi= 0 for at least one indicei. Moreover, every open subsetXi(n)≃X of the coveringU(n)is affine.
Therefore, a V(Ln,−µ)-torsor ρ:W →X(n) is isomorphic to W U(n),Ln,−µ, g for a certain cocycle g ∈C1(U,Ln,µ) representing its isomorphism class c(W)∈ H1(X(n),Ln,µ). So, by (2.4), it suffices to show that ρn : S → X(n) is an A1-bundle . This can be done in a similar way as in Lemma 1.2 in [7].
2.7. In view of the correspondence (2.4) betweenA1-bundles andV(L)-torsors, a DFSW U(n),Ln,−µ, g
isX-isomorphic to the surfaceS =Fn
i=1A1Xi/∼obtained by gluing ncopies ofA1X
i = Spec (A[Ti]) ofA1X by means ofAx-algebras isomor- phismsAx[Ti]→∼ Ax[Tj],Ti7→˜gij+xµi−µjTj, where ˜gij:=xµigij ∈Ax for every i6=j. In this way, we recover the ”cocycle construction” of Fieseler [7] .
2.8. A general schemeS=W U(n),Ln,−µ, g
is not affine. For instance, ifn≥2, then A1X(n) is not even separated. However, a similar argument as in Proposition 1.4 in [7] shows thatSis affine if and only if the correspondingtransition functions
˜
gij =xµigij and ˜gji=−xµjgij above have a pole inx0for everyi6=j. This means equivalently that for everyi6=j,gij ∈Γ (Xij(n),Ln,µ) is not in the image of the differential
(resi−resj) : Γ (Xj(n),Ln,µ)⊕Γ (Xi(n),Ln,µ)→Γ (Xij(n),Ln,µ). In turn, this condition is satisfied if and only if for everyi6=j,S |Xi(n)∪Xj(n)is a nontrivialV(Ln,−µ)|Xi(n)∪Xj(n)-torsor. This leads to the following criterion.
Proposition 2.9. A scheme W U(n),Ln,−µ, g
is a DFS if and only if it restricts to a nontrivialV(Ln,−µ)|U-torsor on every open subsetU ≃X(2) of X.
Morphisms of Danielewski-Fieseler surfaces. Given two DFS’sS′=W U(n′),Ln′,−µ′, g′ and S′ = W U(n′),Ln′,−µ′, g′
as in (2.6) and a morphism of DFS’s β : S′ →S, we letα:X(n′)→X(n) be the uniqueX-morphism such thatβ ρ−1n′ (xi(n′))
⊂ ρ−1n (α(xi(n))) for every i = 1, . . . , n′. We denote by ¯g ∈ C1 U(n′), α∗Ln,µ
the image of α∗g ∈ C1 α−1 U(n)
, α∗Ln,µ
by the restriction maps between ˇCech
complexes. We obtain a factorizationβ =prS◦β′
S′ β
′
//
ρn′
S˜=S×X(n)X(n′)≃W U(n′), α∗Ln,−µ,g¯ prS //
p2
S
ρn
X(n′) X(n′) α //X(n)
where β′ is anX(n′)-morphism restricting an isomorphism over δn−1′ (x0). This is the case if and only if the homomorphismsS(α∗Ln,−µ)|Xi(n′)→S(Ln′,−µ′)|Xi(n′)
corresponding to the restrictions βi′ : S′ |Xi(n′)→ S˜ |Xi(n′) of β′ are induced by OXi-modules homomorphisms (ci, ζi) :α∗Ln,−µ |Xi(n′)→ OXi(n′)⊕ Ln′,−µ′ |Xi(n′), i= 1, . . . , n′, whereζi ∈HomXi(n′)(α∗Ln,−µ,Ln′,−µ′) restricts to an isomorphism overXi(n′)\{xi(n′)}, and whereci∈Γ (Xi(n′), α∗Ln,µ). Moreover, it follows from the definition ofS′and ˜Srespectively that theβi’s coincide on the overlapsXij(n′), i6=j, if and only if theζi’s glue to a global sectionζ∈HomX(n′)(α∗Ln,−µ,Ln′,−µ′) such thatg′◦ζ= ¯g+∂(c), where ∂ denotes the differential of the ˇCech complex C• U(n′), α∗Ln,−µ
. Summing up, we obtain the following characterization.
Proposition 2.10. A morphism of DFS’s β : S′ → S exists if and only if there exists a data(α, θ, c)consisting of:
1) anX-morphism α:X(n′)→X(n),
2) a sectionθ=tζ∈HomX(n′)(Ln′,µ′, α∗Ln,µ)such that Supp(θ)⊂δn−1′ (x0), 3) a cochainc∈C0 U(n′), α∗Ln,µ
such thatθ(g′) = ¯g+∂(c).
Furthermore,β is an isomorphism if and only ifαandθ are.
Example 2.11. Given a DFS S = W U(n),Ln,−µ, g
, there exists an integer h0 ≥ max (µi), depending on g and µ, such that for every h ≥ h0, xhs−µ is a regular section of Ln,−µ defining an homomorphism of OX(n)-modules θµ,h : Ln,µ → OX(n)with the property thatθµ,h(g1j)∈Γ X1j(n),OX(n)
extends to a sectionσj ofOXj(n)for everyj= 2, . . . , n. By (2.10), the data (δn, θµ,h, σ), where σ={σ1= 0, σj}j=2,...,n∈An, corresponds to a morphism of DFS’s ψ:S→A1X. Additive group actions on Danielewski-Fieseler surfaces. As a torsor, a DFS π : S =W U(n),Ln,−µ, g ρ
−→ X(n)→δn X comes equipped with an action mn,µ:V(Ln,−µ)×X(n)S→S of the line bundleV(Ln,−µ). Every nonzero section s ∈ Γ (X(n),Ln,µ) gives rise to a morphism of group schemes φs : Ga,X(n) → V(Ln,−µ), whence to a nontrivial action
ms=mn,µ◦(φs×Id) :Ga,X(n)×X(n)Sφ−→s×IdV(Ln,−µ)×X(n)S m−→n,µS of the additive groupGa,X by means ofX(n)-automorphisms ofS.
Proposition 2.12. Every nontrivialGa,X-action onS appear in this way.
Proof. A connected component ofπ−1(x0) is invariant under aGa,X-action. There- fore, aGa,X-action onSlifts to aGa,X(n)-actionm:Ga,X(n)×X(n)S→S. In turn, this action restricts onS |Xi(n)≃V(Ln,−µ)|Xi(n) to a Ga,Xi(n)-action mi. Thus there exists a nonzero section si ∈ Γ (Xi(n),Ln,µ) ≃ HomXi(n) Ln,−µ,OXi(n) such that the corresponding group co-action is induced by theOXi(n)-modules ho- momorphism (Id⊗1 +si⊗T) :Ln,−µ|Xi(n)→S(Ln,−µ)|Xi(n)⊗OXi(n)OXi(n)[T].
Clearly, these actions coincide on the overlapsXij(n) if and only if thesi’s glue to
a sections∈Γ (X,Ln,µ) such thatm=ms.
Corollary 2.13. A DFSπ :S →X admits a free Ga,X-action if and only if its canonical sheaf is trivial.
Proof. By construction, the canonical sheaf ofS=W U(n),Ln,−µ, g
is isomorphic toρ∗nLn,−µ. It is trivial if and only ifLn,−µ is. On the other hand, the discussion above implies that S admits a free Ga,X-action if and only if Ln,µ ≃ L∨n,−µ is
trivial.
3. Danielewski-Fieseler surfaces and labelled rooted trees Here we give a combinatorial description of DFS’s by means of labelled trees.
To state the main result of this section, we need the following definition.
Definition 3.1. A relative Danielewski-Fieseler surface with base(A, x) is a mor- phismS= ψ:S→A1X
of DFS’s with base (A, x). Amorphism of relative DFS’s is a morphism of DFS’sβ:S′→S such thatψ′ =ψ◦β.
This section is devoted to the proof of the following result.
Theorem 3.2. The category−→
D(A,x)of relative DFS’s with base(A, x)is equivalent to the category T(A,x) of (A, x)-labelled rooted trees.
In the following subsections, we construct an equivalence of categories in the form of a covariant functorS:T(A,x)→−→
D/(A,x).
DFS’s defined by labelled rooted trees. Given an (A, x)-labelled tree γ = (Γ, σ) with the leavese1,m1, . . . , en,mn, we letµ=µ(γ) = (µi=h−mi)i=1,...,n ∈ Zn≥0, whereh=h(Γ) denotes the height of Γ. Since µi≤hfor everyi= 1, . . . , n, the multiplication by the regular sectionxhs−µ ofLn,−µdefines an homomorphism θµ,h : Ln,µ = L∨n,−µ → OX(n) restricting to an isomorphism over δn−1(X∗). We let g(γ) ∈ C1 U(n),Ln,µ
be the unique cocycle such that θµ,h(g(γ)) = ∂(σ) ∈ C1 U(n),OX(n)
. If n= 1, then the scheme πγ :S(γ) =W U(n),Ln,−µ, g(γ)
→ X corresponding to this data is isomorphic to A1X. Otherwise, if n≥2, then, by (1.5), the transition functions ˜gij = xµigij = x−mi(σj−σi) ∈ Ax and ˜gji have a pole at x0. Thus S(γ) is a DFS by virtue of (2.8). The morphism of DFS’s S(γ) = ψγ :S(γ)→A1X
defined by the data (δn, θµ,h, σ) is called the canonical morphism associated with γ. The following result completes the first part of the proof of theorem (3.2).
Proposition 3.3. Every relative DFS is isomorphic toS(γ)for a suitable treeγ.
Proof. A morphism of DFS’sS= ψ:S=W U(n),Ln,−µ, g
→A1X
is given by a data (δn, θ=aθµ,h, σ), wherea∈A∗ andh≥max (µi), such thatθ(g) =∂(σ).
If n ≥ 2 then mi = h−µi ≥ 1 for every i = 1, . . . , n. Indeed, otherwise there exists an indice i for which θ induces an isomorphism θi : Ln,µ |Xi(n)
→ O∼ Xi(n). Thus, for everyj6=i,gij ∈Γ (Xij(n),Ln,µ) extends to a section ofLn,µ|Xi(n) as θ(gij) = σj |Xi(n) −σi |Xi(n)∈ Γ Xij(n),OX(n)
extends to a section of OXi(n). This implies that S |Xi(n)∪Xj(n) is a trivial torsor, in contradiction with (2.9).
For the same reason, dij = ordx0(σj−σi)<min (mi, mj) for everyi 6=j. Since min (dij, dik) = min (dij, djk) for every triple of indicesi, j andk, we deduce from
(1.7) that the data
n,(mi)i=1,...,n,(dij)i,j=1,...,n, σ
corresponds to an (A, x)- labelled tree γ = (Γ, σ). Finally, the data IdX(n), aIdLn,−µ,0
defines an iso- morphism of DFS’sφ:S→∼ S(γ) such thatψγ=ψ◦φ−1:S(γ)→A1X. Corollary 3.4. Every DFS is X-isomorphic to S(γ)for a suitable treeγ.
Proof. Indeed, by (2.11), every DFSSadmits a morphism of DFS’sS= ψ:S→A1X . Example 3.5. Given three integers r ≥ 0, m ≥ 1 and n ≥ 1, we consider an (A, x)-labelled treeγ= (Γr,m,n, σ) with the following underlying tree
Γr,m,n=
r
m
n
Since there exists b∈ Aand a cochain ˜σ∈ An such thatσi =b+xrσ˜i for every i= 1, . . . , n, the corresponding DFSπγ :S(γ)→X is obtained by gluingncopies Spec (A[Ti]) ofA1X by means of the Ax-algebras isomorphisms
Ax[Ti]−→∼ Ax[Tj], Ti7→x−m(˜σj−σ˜i) +Ti, i6=j.
Since ˜σj −σ˜i ∈ A\xA for every i 6= j, the local sections ˜σi +xmTi ∈ A[Ti], i = 1, . . . , n, glue to a global one s1 ∈ B = Γ X, πγ∗OS(γ)
which distinguishes the irreducible components of πγ−1(x0). Letting P = Qn
i=1(y−σ˜i) ∈ A[y], the rational sectionx−mP(s1)∈B⊗AAxextends to a regular sections2∈B, inducing a coordinate function on every irreducible component ofπγ−1(x0). By construction, the A-algebras homomorphism A[y, z] → B, y 7→ s1, z 7→ s2 defines a closed embedding φ : S(γ) ֒→ A2X = Spec (A[y, z]) which induces an X-isomorphism betweenS(γ) and the surface
π:SP,m= Spec (A[y, z]/(xmz−P(y)))−→X.
In this coordinates, the canonical morphism ψγ : S(γ) → A1X is given as the restriction onSP,m of theX-morphismA2X →A1X, (y, z)7→xry+b.
Morphism of DFS’s defined by a morphism labelled rooted trees. Given two (A, x)-labelled treesγ′ = (Γ′, σ′) and γ= (Γ, σ), we equip the corresponding DFS’s S(γ′) = W U(n′),Ln′,−µ′, g′
and S(γ) = W U(n),Ln,−µ, g
with their canonical morphisms ψγ′ and ψγ respectively. By (1.4), the image of a leaf e′i,m′ i
of Γ′ by a morphism of (A, x)-labelled tree τ : γ′ → γ is a leaf ej(i),m(j) of Γ such that m′i ≥ mj(i), and σ′i−σj(i) ∈ xmj(i)A for every i = 1, . . . , n′. Letting α : X(n′) → X(n) be the unique X-morphism such that α(xi(n′)) = xj(i)(n) for every i = 1, . . . , n, we conclude that the invertible sheaf α∗Ln,−µ is isomor- phic to Ln′,−ν, where ν = µj(i)
i=1,...,n′ ∈ Zn≥0′ . The multiplication by the regular section xh(Γ′)−h(Γ)sν−µ′ ∈ Γ (X(n′),Ln′,ν−µ′) defines an homomorphism of OX(n′)-modules θ : Ln′,µ′ → α∗Ln,µ ≃ Ln′,ν such that θµ′,h′ = α∗θµ,h◦θ.
By construction, there exists a unique cochain σ′′ ∈ C0 U(n′),Ln′,ν
such that
α∗θµ,h(σ′′) =
σi′−σj(i) i=1,...n′ ∈C0 U(n′),OX(n′)
. Sinceα∗θµ,hrestricts to an isomorphism over δn−1′ (X∗), we conclude thatθ(gσ′(γ′)) = gσ(γ) +∂(σ′′), where gσ(γ)∈C1 U(n′),Ln′,ν
denotes the image ofα∗gσ(γ)∈C1 α−1U(n), α∗Ln,µ by the restriction maps between ˇCech complexes. By (2.10), the data (α, θ, σ′′) defines a morphism of DFS’sβτ :S(γ′)→S(γ) such thatψγ′=ψγ◦βτ, i.e. a morphism βτ :S(γ′)→S(γ) in−→
D(A,x). We say thatβτ is the morphism of relative DFS’s defined byτ :γ′→γ.
Example 3.6. We consider the blow-downτe:γ′ →γ of the leaves atebetween the following (k[x], x)-labelled treesγ′ = (Γ′, σ′) andγ= (Γ, σ).
1 e
x −x
−1 1
e 0
−1
γ′ τe γ
We lets′1∈B′ = Γ X,(πγ′)∗OS(γ′)
ands1∈B= Γ X,(πγ)∗OS(γ)
be the sec- tions corresponding to the canonical morphismsψγ′ :S(γ′)→A1X = Spec (k[x] [y]) and ψγ : S(γ) → A1X respectively. By (3.5), S(γ) is isomorphic to the surface S=
xz−y y2−1
= 0 ⊂A2X = Spec (k[x] [y, z]) via the embedding induced by thek[x]-algebra homomorphismk[x] [y, z]7→B, (y, z)7→ s1, s2=x−1s1 s21−1
. A similar argument shows that S(γ′) is isomorphic to the Bandman and Makar- Limanov surfaceS′⊂A3X = Spec (k[x] [y, z, u]) with equations
xz−y y2−1
= 0, yu−z z2−1
= 0, xu− y2−1
z2−1
= 0, via the embedding induced by thek[x]-algebras homomorphismk[x] [y, z, u]→B
(y, z, u)7→
s′1, s′2=x−1s′1
(s′1)2−1
, s′3=x−1
(s′1)2−1 (s′2)2−1 In these coordinates, the morphismβτe :S(γ′)→S(γ) defined by the blow-down τecoincide with the restriction of the projectionA3X →A2X, (x, y, z, u)7→(x, y, z).
3.7. The correspondenceγ7→S(γ), (τ :γ′→γ)7→S(τ) = (βτ:S(γ′)→S(γ)) defines a covariant functor S : T(A,x) → −→
D(A,x). It follows from (3.3) that S is essentially surjective. The following result shows that S is fully faithful, whence completes the proof of theorem (3.2).
Proposition 3.8. Every morphism β : S(γ′) → S(γ) coincides with a unique morphism βτ defined by a morphism of(A, x)-labelled treesτ :γ′→γ.
Proof. By (2.10), β is determined by a data (α, θ, σ′′) such that α(xi(n′)) = xj(i)(n),i= 1, . . . , n′,θµ′,h=α∗θµ,h◦θ,σ′=α∗σ+α∗θµ,h(σ′′), and such that
θ(gσ′(γ′)) =α∗gσ(γ) +∂(σ′′)∈C1 U(n′), α∗Ln,µ . Since α∗Ln,µ is isomorphic to Ln′,ν, whereν = µj(i)
i ∈Zn≥0′, we conclude that θ is the multiplication byxh′−hsν−µ′ ∈Γ (X(n′),Ln′,ν−µ′). Thusm′i ≥mj(i) and
˜
σ′′i = α∗θµ,h(σi′′) ∈ xh−µj(i)A = xmj(i)A for every i = 1, . . . , n′. Therefore, the formulas
Γ′i∋e′i,k7→τi e′i,k
=ej(i),min(k,mj(i))∈Γj(i), i= 1, . . . , n′k= 0, . . . , m′i, (see (1) for the notation) define a collection of morphisms of trees τi : (Γ′i, σ′i)→
Γj(i), σj(i)
inT(A,x). Moreover, we deduce that (2) ordx0(σi′′−σi′) =
ordx0 σ˜i′′′−σ˜′′i
≥mj(i) ifj(i) =j(i′) ordx0 σj(i′)−σj(i)
otherwise .
Sinceσ′ andσ are compatible with Γ′ and Γ respectively, (2) guarantees that the conditions of (1.8) are satisfied. Thus there exists a unique morphism τ :γ′ →γ inT(A,x)such thatτ |γi′=τi for everyi= 1, . . . , n′. By construction,β=βτ. Corollary 3.9. For every morphism of DFS’sβ :S′→S there exist a morphism of (A, x)-labelled treesτ:γ′→γand a commutative diagram
S′ β //
φ′ ≀
S
φ
≀
S(γ′) βτ //S(γ)
Proof. Once a morphism of DFS’sψ:S→A1Xis chosen (see (2.11)), ψ:S→A1X and ψ′=ψ◦β :S′→A1X
are objects of−→
D(A,x), whereasβ :S′→S corresponds to morphism in −→
D(A,x). So the result follows from (3.3) and (3.8).
Reading isomorphism classes of Danielewski-Fieseler surfaces from trees.
To decide when two DFS’s areX-isomorphic, we have the following criterion.
Theorem 3.10. Two (A, x)-labelled trees define X-isomorphic DFS’s if and only if they are equivalent (see (1.11) for the definition).
Proof. We first observe that if γ′ = (Γ′, σ′) is an essential labelled subtree for γ= (Γ, σ) then the DFS’s S(γ) andS(γ′) areX-isomorphic. Indeed, there exists m ∈ Z≥0 and b ∈ A such thatmi = m′i+m and σi = b+xmσi′ for every i = 1, . . . , n. Thus µ(γ′) = µ(γ) = µ ∈ Zn≥0, g(γ) = g(γ′) ∈ C1 U(n),Ln,µ
, and so, S(γ)≃ S(γ′). Therefore, it suffices to prove the assertion for DFS’s defined by essential labelled trees γ= (Γ, σ) and γ′ = (Γ′, σ′). Ifγ and γ′ are equivalent then h= h(Γ) =h(Γ′) and there exists a permutation j of{1, . . . , n} such that µ′i = µj(i) for every i = 1, . . . , n. Moreover, there exist a pair (a, b) ∈ A∗ ×A and a cochain ˜σ = {˜σi}i=1,...,n ∈ An such that σi′ = aσj(i)+b+xmj(i)σ˜j(i) for every i = 1, . . . , n. Letting σ′′ = {σ′′i =σi′−b}i=1,...,n, the same argument as above shows that S(γ′) isX-isomorphic to the DFS defined by the labelled tree γ′′ = (Γ′, σ′′). Thus, by replacing S(γ′) by S(γ′′) if necessary, we can suppose from now on thatb= 0. We letα∈AutX(X(n)) be the uniqueX-automorphism such that α(xi(n)) = xj(i)(n), and we let c ∈ C0 U(n), α∗Ln,µ
be the unique cochain such that θµ′,h(c) =
a−1xmj(i)σ˜j(i) i=1,...,n. Since Ln,µ′ ≃ α∗Ln,µ and a−1g(γ′) =α∗g(γ) +∂(c), we conclude that the data
α, aIdLn,µ′, c
defines an isomorphism of DFS’s β :S(γ′)→∼ S(γ). Conversely, anX-isomorphismβ :S′=
