On a class of Danielewski surfaces in affine 3-space

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On a class of Danielewski surfaces in affine 3-space

Adrien Dubouloz, Pierre-Marie Poloni

To cite this version:

Adrien Dubouloz, Pierre-Marie Poloni. On a class of Danielewski surfaces in affine 3-space. 2006.

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ccsd-00019635, version 2 - 26 Aug 2006

ON A CLASS OF DANIELEWSKI SURFACES IN AFFINE 3^-SPACE

ADRIENDUBOULOZANDPIERRE-MARIEPOLONI

Abstrat. In[16℄and[17℄,L.Makar-Limanovomputedtheautomorphismgroupsofsurfaes

inC³denedbytheequationsxⁿz−P(y) = 0^,wheren≥1^andP(y)^isanonzeropolynomial.

Similarresults havebeen obtainedbyA. Crahiola [3℄for surfaes dened by the equations

xⁿz−y²−σ(x)y= 0^,wheren≥2^andσ(0)6= 0^{,denedover anarbitrarybaseeld. Here}

weonsiderthemoregeneralsurfaesdenedbytheequationsxⁿz−Q(x, y) = 0^,wheren≥2

and Q(x, y) ^{isa polynomial with oeients in an arbitrary base eld} k^{. Weharaterise}

amongthemtheoneswhihareDanielewskisurfaesinthesenseof[8℄,andweomputetheir

automorphismgroups. We studylosed embeddingsof these surfaes inane3^{-spae. We}

showthatingeneraltheirautomorphismsdonotextendtotheambientspae. Finally,wegive

expliitexamplesofC^∗-ationsonasurfaeinA³

C^{whihanbeextended}holomorphiallybut notalgebraiallytoC^∗-ationsonA³_C.

Introdution

Sine they appeared in aelebrated ounterexampleto the CanellationProblem due to W.

Danielewski[5℄,thesurfaesdened bytheequationsxz−y(y−1) = 0 ^andx²z−y(y−1) = 0

in C³ andtheir naturalgeneralisations, suh assurfaesdened bythe equationsxⁿz−P(y) = 0^{, where} P(y) ^{is a nononstant} polynomial, have been studied in many dierent ontexts. Of partiular interest is the fat that theyan be equipped with nontrivialations of the additive

group C₊. The general orbitsof these ations oinide with the general bers of A¹-brations

π:S→A¹,that is,surjetivemorphismswithgeneriberisomorphito ananeline. Normal ane surfaes S ^{equipped with an} A¹-bration π : S → A¹ anbe roughly lassied into two lassesaordingthefollowingalternative: eitherπ:S→A¹ isauniqueA¹-brationonS ^upto

automorphismsofthebase,orthereexists aseondA¹-brationπ^′ :S →A¹ withgeneralbers distintfromtheonesofπ^.

Due to the symmetrybetween thevariables x ^andz^{, a surfaedened bythe equation} xz− P(y) = 0 ^{admits twodistint} A¹-brations over theane line. In ontrast, it wasestablished byL.Makar-Limanov[17℄that onasurfaeSP,n ^{denedbytheequation}xⁿz−P(y) = 0ⁱⁿ C³, where n ≥ 2 ^{and where} P(y) ^{is a polynomialof degree} r ≥ 2^{, the projetion}pr_x : SP,n → C

isauniqueA¹-brationupto automorphismsof thebase. Inhis proof, L.Makar-Limanovused the orrespondene between algebrai C₊-ations on an ane surfae S ^{and loally nilpotent}

derivations of the algebra of regular funtions on S^{. It turns out that his proof is} essentially independentofthebaseeld k ^{providedthat wereplaeloallynilpotent}derivationsbysuitable systemsof Hasse-Shmidtderivationswhentheharateristiofk^{ispositive(seee.g.,[3℄).}

The fat that an ane surfae S ^{admits a unique} A¹-bration π : S → A¹ makesits study simpler. Forinstane, everyautomorphism of S ^{must preservethis bration. Inthis ontext, a}

resultduetoJ.Bertin[2℄assertsthattheidentityomponentoftheautomorphismsgroupofsuh

asurfaeisanalgebraipro-groupobtainedasaninreasingunionofsolvablealgebraisubgroups

of rank≤1^{. For surfaes dened by theequations}xⁿz−P(y) = 0 ⁱⁿ C³, the piturehasbeen ompletedbyL.Makar-Limanov[17℄whogaveexpliitgeneratorsoftheirautomorphismsgroups.

SimilarresultshavebeenobtainedoverarbitrarybaseeldsbyA.Crahiola[3℄forsurfaesdened

bytheequationsxⁿz−y²−σ(x)y= 0^,whereσ(x)^{isapolynomialsuh that}σ(0)6= 0^.

MathematisSubjetClassiation(2000):14R10,14R05.

Keywords:A¹-brations,Danielewskisurfaes,automorphismgroups,extensionofautomorphisms.

The latter surfaes are partiular examples of a general lass of A¹-bred surfaes alled Danielewski surfaes [8℄, that is, normal integralanesurfaeS ^{equipped with an}A¹-bration

π:S →A¹

k ^{overananelinewithaxed}k^{-rationalpoint}o^{,suhthateveryber}π⁻¹(x)^,where x∈ A¹

k\ {o}^{, is} geometriallyintegral, and suh that everyirreduible omponentof π⁻¹(o)^is

geometriallyintegral. Inthis artile,weonsiderDanielewskisurfaesSQ,n ⁱⁿ A³

k ^{dened byan}

equationoftheformxⁿz−Q(x, y) = 0^,wheren≥2^andwhereQ(x, y)∈k[x, y]^{isapolynomial}

suh that Q(0, y)^{splits with} r ≥2 ^{simple roots in} k^{. This lass ontains mostof the surfaes}

onsideredbyL.Makar-Limanovand A.Crahiola.

The paperis organised as follows. First, we briey reall denitions about weighted rooted

treesandthenotionofequivaleneofalgebraisurfaesinanane3^{-spae. Insetion}2^,wereall

from [8℄ the main fats about Danielewski surfaes and we review the orrespondene between

these surfaes andertain lassesof weightedtrees in aform appropriateto ourneeds. Wealso

generaliseto arbitrarybase eldsk ^{someresultswhihareonlystated foreldsof} harateristi zeroin [7℄and[8℄. Inpartiular, theaseof Danielewskisurfaeswhihadmit twoA¹-brations with distintgeneralbersis studied in Theorem2.11. Weshowthat these surfaesorrespond

to Danielewskisurfaes S(γ)^{dened by thene} k^{-weightedtrees}γ ^{whih are alled ombs and}

wegiveexpliitembeddingsofthem. ThisresultgeneralisesTheorem4.2in[9℄.

In setion 3^{, we lassify} Danielewski surfaes SQ,h ⁱⁿ A³

k ^{dened by equations of the form}

x^hz−Q(x, y) = 0 ^{and determine their} automorphism groups. We remark that suh a surfae

admits many embeddings as a surfae SQ,h^{. In partiular, we establish in Theorem 3.2 that}

thesesurfaes analwaysbeembeddedassurfaeSσ,h ^{denedbyanequationoftheform} x^hz− Qr

i=1(y−σi(x)) = 0 ^{for asuitable olletion of} polynomials σ ={σi(x)}_i=1,...,r^{. We say that}

thesesurfaesSσ,h ^{arestandardform of}DanielewskisurfaesSQ,h^{. Next,weompute(Theorem}

3.10)theautomorphismgroupsofDanielewskisurfaes instandardform. Weshowinpartiular

thatanyofthemomesastherestritionof analgebraiautomorphismoftheambientspae.

FinallyweonsidertheproblemofextendingautomorphismsofagivenDanielewskisurfaeSQ,h

toautomorphismsoftheambientspaeA³

k^{. Weshowthatthisisalwayspossibleintheholomorphi}

ategorybut notin the algebraione. Wegive expliitexampleswhih ome from the studyof

multipliativegroupations on Danielewskisurfaes. For instane, weprove that everysurfae

S⊂A³_C denedbytheequationx^hz−(1−x)P(y) = 0^,whereh≥2^andwhere P(y)^hasr≥2

simpleroots,admitsanontrivialC^∗-ationwhihisalgebraiallyinextendablebutholomorphially extendabletoA³_C. Inpartiular,eventheinvolutionofthesurfaeS^{denedbytheequation}x²z− (1−x)P(y) = 0^induedbytheendomorphismJ(x, y, z) = (−x, y,(1 +x) ((1 +x)z+P(y)))^of A³

k ^{doesnotextendtoanalgebrai}automorphismofA³

k^.

1. Preliminaries

1.1. Basifats onweighted rootedtrees.

Denition1.1. Atreeisanonempty,nite,partiallyorderedsetΓ = (Γ,≤)^{withauniquemin-}

imalelemente0 ^{alledtheroot,andsuhthatforevery}e∈Γ^thesubset(↓e)_Γ={e^′∈Γ, e^′≤e}

isahainfortheinduedordering.

1.2. Aminimalsub-hain

←−

e^′e={e^′< e}^{withtwoelementsofatree}Γ^{isalledan edge of}Γ^{. We}

denote the set of alledges in Γ ^by E(Γ)^{. An element} e ∈Γ ^{suh that Card}(↓e)_Γ = m ^{is said}

to beat level m^{. The maximalelements} ei=ei,mi^{, where}mi =^Card(↓ei)_Γ ^of Γ ^{arealled the}

leaves ofΓ^{. Wedenotethesetofthoseelementsby}L(Γ)^{. Themaximalhainsof}Γ^arethehains

(1.1) Γe_i,mi = (↓ei,mi)_Γ={ei,0=e0< ei,1<· · ·< ei,mi}, ei,mi∈L(Γ).

Wesaythat Γ^hasheight h= max (mi)^{. Thehildren ofanelement}e∈Γ^{are theelementsof}Γ

atrelativelevel1^withrespettoe^{,i.e.,themaximalelementsofthesubset}{e^′ ∈Γ, e^′> e}^ofΓ^.

Denition 1.3. A ne k^{-weighted tree} γ = (Γ, w) ^{isa tree}Γ ^{equipped witha weightfuntion} w :E(Γ) →k ^{with valuesin aeld} k^{, whih assigns anelement}w←−

e^′e

of k^{to everyedge}←− e^′e

ofΓ^{,in suh awaythat} w←−−

e^′e1

6=w←−−

e^′e2

whenevere1 ^ande2 ^{are distinthildrenofasame}

elemente^′.

Inwhatfollows,wefrequentlyonsiderthefollowinglassesoftrees.

Denition1.4. LetΓ ^{bearootedtree.}

a)IfalltheleavesofΓ^{areatthesamelevel}h≥1^{andifthereexists auniqueelement}e¯0∈Γ

forwhih Γ\ {¯e0} ^{isanonemptydisjointunionofhainsthenwesaythat}Γ^isarake.

b)IfΓ\L(Γ)^{isahain thenwesaythat} Γ^isaomb. Equivalently,Γ ^{isaombifandonlyif}

everye∈Γ\L(Γ)^{hasat mostonehildwhihis notaleafof}Γ^. e0 e¯0

Arakerootedin e0^.

e0

Aombrootedin e0^.

1.2. Algebraiand analyti equivaleneof losed embeddings.

Herewebrieydisussthenotionsofalgebraiandanalytiequivalenesoflosedembeddingsof

agivenanealgebraisurfaeinanane3^-spae.

LetS ^{be anirreduibleane surfaeandlet}iP1 :S ֒→A³

k ^and iP2 :S ֒→A³

k ^beembeddings

of S ^{in asame ane}3^{-spaeaslosed subshemesdened bypolynomialequations} P1 = 0^and P2= 0respetively.

Denition 1.5. In the above setting, wesay that the losed embeddings iP1 ^and iP2 ^{are alge-}

braiallyequivalent ifoneofthefollowingequivalentonditionsissatised:

1)Thereexists anautomorphismΦ^ofA³

k ^suhthatiP2 =iP1◦Φ^.

2)ThereexistsanautomorphismΦ^ofA³

k^{andanonzeroonstant}λ∈k^∗suhthatΦ^∗P1=λP2^.

3)Thereexists automorphismsΦ^andφ^ofA³

k ^andA¹

k respetivelysuh thatP2◦Φ =φ◦P1^.

1.6. Overtheeldk=Cofomplexnumbers,oneanalsoonsiderholomorphiautomorphisms.

With thenotationofdenition 1.5,twolosed algebraiembeddingsiP1 ^andiP2 ^ofagivenane

surfaeSⁱⁿA³_CarealledholomorphiallyequivalentifthereexistsabiholomorphismΦ :A³_C→A³_C

suhthatiP2 =iP1◦Φ^{. Clearly,theembeddings}iP2 ^andiP1 ^areholomorphiallyequivalentifand onlyifthereexistsabiholomorphismΦ :A³_C→A³_CsuhthatΦ^∗(P1) =λP2^{foraertainnowhere}

vanishingholomorphifuntion λ^{. Sinetherearemanynononstantholomorphifuntions with}

thispropertyonA³_C,Φ^{neednotpreservethealgebraifamiliesoflevelsurfaes}P1:A³_C→A¹_Cand

P2:Aⁿ_C→A¹_C. Soholomorphiequivaleneisaweakerrequirementthanalgebraiequivalene.

2. Danielewski surfaes

Forertainauthors,aDanielewskisurfaeisananesurfaeS^whihisalgebraiallyisomorphi toasurfaeinC³denedbyanequationoftheformxⁿz−P(y) = 0^,wheren≥1^andP(y)∈C[y]^.

These surfaes ome equipped witha surjetive morphism π = pr_x |S: S → A¹ restriting to a trivialA¹-bundleovertheomplementoftheorigin. Moreover,iftherootsy1, . . . , yr∈CofP(y)

aresimple,thenthebrationπ= pr_x|S:S→A¹fatorsthroughaloallytrivialberbundleover theaneline withanr ^{-foldorigin(seee.g.,[5℄and [11℄). In[8℄, therstauthorused theterm}

Danielewskisurfaetoreferto anane surfaeS ^{equippedwith amorphism}π:S →A¹ whih fatorsthroughaloally trivialberbundlein asimilar wayasabove. Inwhat follows,wekeep

this pointof view, whih leadsto anatural geometrigeneralisation of thesurfaes onstruted

by W. Danielewski [5℄. Wereall that an A¹-bration overan integralsheme Y ^{is afaithfully}

at (i.e., atand surjetive) ane morphism π : X → Y ^{with generi ber isomorphi to the}

aneline A¹

K(Y) ^{overthe funtion eld} K(Y)^of Y^{. The followingdenition is a}generalisation toarbitrarybaseeldsk ^{oftheoneintroduedin[8℄.}

Denition 2.1. A Danielewski surfae is an integral ane surfae S ^{dened over a eld} k^,

equippedwithanA¹-brationπ:S→A¹

k ^{restritingtoatrivial}A¹-bundleovertheomplement oftheak^{-rationalpoint}o^ofA¹

k ^{andsuhthattheber}π⁻¹(o)^{isredued,onsistingofadisjoint}

unionofanelinesA¹

k ^overk^.

Notation2.2. Inwhatfollows,wexanisomorphismA¹

k≃Spec (k[x])^{andweassumethatthe} k^{-rationalpoint}o^issimplythe"origin"ofA¹

k^{,that is,thelosedpoint}(x)^ofSpec (k[x])^.

2.3. In the following subsetions, we reall the orrespondene between Danielewski surfaes

and weighted rooted trees established by the rst author in [8℄ in a form appropriate to our

needs. Although the results given in lo. it. are formulated for surfaes dened over a eld

of harateristi zero, most of them remain valid withoutany hanges over a eld of arbitrary

harateristi. We provide full proofs only when additional arguments are needed. Then we

onsider Danielewski surfaes S ^{with a trivial anonial sheaf} ωS/k = Λ²Ω¹_S/k^{. We all them}

speial Danielewskisurfaes. Wegiveaomplete lassiationof thesesurfaes intermsof their

assoiatedweightedtrees.

2.1. Danielewskisurfaes and weighted trees.

Here wereview the orrespondene whih assoiatesto everyne k^{-weightedtree} γ = (Γ, w) ^a

Danielewskisurfaeπ :S(γ)→A¹

k = Spec (k[x]) ^{whih is thetotal spaeof an}A¹-bundle over thesheme δ:X(r)→A¹

k ^obtainedfromA¹

k ^{byreplaingitsorigin}o^byr≥1k^{-rationalpoints} o1, . . . , or^.

Notation2.4. InwhatfollowswedenotebyUr= (Xi(r))_i=1,...,r ^{theanonialopenoveringof} X(r)^{bymeansofthesubsets}Xi(r) =δ⁻¹ A¹

k\ {o}

∪ {oi} ≃A¹

k^.

2.5. Letγ= (Γ, w)^beanek^{-weightedtree}γ= (Γ, w)^ofheighth^,withleavesei^atlevelsni≤h^, i= 1, . . . , r^{. Toeverymaximalsub-hain}γi= (↓ei)^ofγ^{(see1.2forthenotation)weassoiatea}

polynomial

σi(x) =

ni−1

X

j=0

w(←−−−−−−ei,jei,j+1)x^j∈k[x], i= 1, . . . , r.

Weletρ:S(γ)→X(r)^betheuniqueA¹-bundleoverX(r)^{whihbeomestrivialontheanonial}

openoveringUr^{,andisdenedbypairsoftransitionfuntions}

(fij, gij) = x^nj−ni, x⁻ⁿⁱ(σj(x)−σi(x))

∈k

x, x⁻¹²

, i, j= 1, . . . , r.

This means that S(γ) ^{is obtained by gluing} n ^opies Si = ^Spe(k[x] [ui]) ^{of the ane plane} A²

k ^over A¹

k\ {o} ≃ ^Spe k

x, x⁻¹

by means of the transition isomorphismsindued by the

k x, x⁻¹

-algebrasisomorphisms

k x, x⁻¹

[ui]→^∼ k x, x⁻¹

[uj], ui7→x^nj−niuj+x⁻ⁿⁱ(σj(x)−σi(x)) i6=, i, j= 1, . . . , r.

This denition makes sense as the transition funtions gij ^{satisfy the twisted oyle relation}

gik = gij +x^nj−nigjk ⁱⁿ k x, x⁻¹

for everytriple of distint indies i^, j ^and k^{. Sine} γ ^{is a}

ne weightedtree,it followsthatfor everypairof distint indiesi ^and j^{,the rationalfuntion} gij = x⁻ⁿⁱ(σj(x)−σi(x)) ∈ k

x, x⁻¹

does not extend to a regular funtion on A¹

k^{. This}

ondition guarantees that S(γ) ^{is a separated sheme, whene an ane surfae by virtue of}

Fieseler'sriterion(seeproposition1.4in [11℄). Therefore,πγ =δ◦ρ:S(γ)→A¹

k=^Spe(k[x])

isaDanielewskisurfae,theberπ⁻¹(o)^{beingthedisjointunionofanelines} Ci=π_γ⁻¹(o)∩Si≃^Spe(k[ui]), i= 1, . . . , r.

2.6. A Danielewskisurfaeπ: S(γ)→ A¹

k ^{above omesanonially equippedwith abirational}

morphism (π, ψγ) :S →A¹

k×A¹

k =^Spe(k[x] [t]) ^{restriting to an}isomorphism overA¹

k\ {o}^.

Indeed,thismorphismorrespondsto theuniqueregularfuntion ψγ ^onS(γ)^whoserestritions totheopensubsetsSi≃^Spe(k[x] [ui])^ofS ^{aregivenbythe}polynomials

ψγ,i=xⁿⁱui+σi(x)∈k[x] [ui], i= 1, . . . , r.