Regularization of Birational Group Operations in Sense of Weil

Heldermann Verlag

Regularization of birational group op erations

in sense of Weil

Dmitri Zaitsev

Communicated byEBVinb erg

Intro duction

The present pap er deals with the classical results of A Weil on regularization

of pregroups and pretransformation spaces see Denitions and As

p ointed out in those purely algebraic results app ear to b e very useful in the

following complex analytic setting

Let D C b e a b ounded domain and AutD the group of all holomorphic

automorphisms of D By a theorem of H Cartan see also AutD is a real

Lie group In Webster gave the conditions on D such that all automorphisms

extend to the birational transformations of the ambient C Moreover as shown in

the group AutD has nitely many comp onents in this case Such prop erties

are also valid for the automorphisms of b ounded homogeneous domains if they

are realized as Siegel domains The graph of every birational transformation

denes an n dimensional compact cycle in P Thus we obtain an emb edding

of AutD in the space C of n dimensional cycles in P the Chowscheme

n n

table collection of pro jectivevarieties parameterized by The space C isacoun

the degrees of the cycles In fact it is proven in that the degree of p ossible

cycles is b ounded which means that AutD lies in nitely many comp onents of

C Theorem The group op eration of AutD extends rationally to the Zariski

closure Z of it in C and endows Z with a structure of a pregroupwhichisin

general not a group The action AutD D D extends also to a rational

n n

action Z C C Again this is a pretransformation space whichisnota

transformation space in general

The pregroups and pretransformation spaces can b e obtained by passing

from algebraic groups and their regular actions on algebraic varieties to birationally

equivalent algebraic varieties The mentioned results of Weil imply that in this

waywe obtain all p ossible pregroups and pretransformation spaces p

In the case of connected pregroups and homogeneous pretransformation spaces

a similar pro of is given in the b o ok of Merzlyakov The regularizations of pre

ISSN Heldermann Verlag

transformation spaces were generalized by Rosenlicht to the case of nonconnected

algebraic groups page Both pap ers and the b o ok utilize the

algebraic language of generic p oints which is mainly develop ed in the classical

book of Weil

In view of the ab ove applications of analytic nature we prop ose here a more

geometrical way to study pregroups and pretransformation spaces This allows to

reprove the ab ove classical results without use of generic p oints and to generalize

them to the case of several comp onents Theorems and

These results are used in to establish the following typ e of linearization

see Theorems and there

Let G beareal connected Lie group operating on a domain D C by

holomorphic automorphisms which extend to birational transformations of the

n N

ambient C Then there exists a linear representation of G on some C together

with a birational onto its closed image G equivariant mapping from C into P

Furthermore weinvestigate the p oints of pregroups and pretransformation

spaces where the ab ove regularizations are biregular Prop ositions and

These are the regular p oints of certain rational mappings This description allows

to construct the ab ove linearization of G such that the emb edding of D into P

is biholomorphic

Another application of Theorem and Prop osition is the following

typ e of a complexication

Let G beareal algebraic group operating on a complex algebraic variety

X by complex regular transformations Then X is embedded as a Zariski open

and dense subset in a complex algebraic variety Y such that the G action on X

extends to a regular action of the complexication G on Y

In fact the action of G extends uniquely to a rational action of the com

C C

plexication G which makes X to a pretransformation G space The space

Y is obtained by Theorem as a regularization of X and Prop osition im

plies that the birational mapping b etween X and Y is an emb edding In case

G is a compact Lie group op erating by holomorphic transformations on a Stein

space X such complexication was constructed by Heinzner see for X

holomorphically convex

Finally we study the uniqueness of the regularizations The regularization

of a pregroup is unique up to isomorphisms In case of pretransformation spaces

this is not true but can b e achieved b y the consideration of a restricted class of

minimal in some sense regularizations Theorem

Acknowledgments On this o ccasion I would like to express my deep grat

itude to my teacher A T Huckleb erryFurthermore I wish to thank P Heinzner

V Shevchishin and J Winkelmann for numerous useful discussions

Conventions

All our ob jects will b e dened over an algebraically closed ground eld k All op en

and closed sets are considered with resp ect to the Zariski top ologyWe follow the

b o ok of Mumford in the main terminologyFor the convenience of the reader

we recall here the basic denitions By saying that some prop erty is satised for

generic x x X X where X X are arbitrary algebraic

n n n

varieties wealways mean that it is satised for all x x in some op en dense

subset of X X

Let V W be two algebraic varieties A rational mapping f V W is

an equivalence class of the pairs U ofanopendensesubset U V and a

morphism U W Two such pairs are said to b e equivalent if the morphisms

coincide on the intersection of the corresp onding sets For every such class one

can cho ose a canonical representativeU with the maximal p ossible subset

U V such that for every other representativeU U U This denes

a onetoone corresp ondence b etw een the rational mappings and their canonical

representatives Such a representative will b e also called a rational mapping and

will b e said to b e regular on the set U and the latter will b e referred as the regular

set of f A rational mapping is said to b e regular at some p oint v V if v U

For arbitrary subsets A V and B W we dene the image of A and

the preimage of B by f Af U Aand f B f jU B

In order to have the notion of comp osition of two rational mappings we need

to intro duce dominating mappings Wesay that a rational mapping is dominating

if the preimages of all op en dense subsets are op en dense In sp ecial case of a

morphism b etween two irreducible algebraic varieties this notion coincides with

given in

Let f V V and g V V be two rational mappings If f is

dominating the comp osition g f V V is dened by the representative

f U U g f where U V and U V are the regular sets of f and g

resp ectively

More generally the representativef U U g f denes a rational

comp osition if and only if the preimage f U is dense in V A rational mapping

f is dominating if and only if the comp ositions g f exist as the rational mappings

for all p ossible rational mappings g V V

By an expression we understand an arbitrary formal comp osition

f f f f a where f V V i n are rational mappings

n n i i i

and a V is a p oint Such expression is said to b e dened if f is regular at a

and for all i n f is regular at f f a

i i

The birational mapping f V W is similarly dened as an equivalence

U W and class of the triples U U of op en dense subsets U V

isomorphisms U U Again there is a onetoone corresp ondence b etween

the birational mappings and their canonical representatives with the maximal

subsets U and U Such a representative will b e also called a birational mapping

and will b e said to b e biregular on the set U A birational mapping is said to

be biregular at some p oint v V if v U The birational mappings are always

dominating

Weshouldsay also several words ab out restrictions Let S V b e a lo cally

closed subvariety The restriction to S of a rational mapping f V W maybe

in general nowhere dened Wesay that the restriction f jS is rational if the set

is dense in S Inthiscasewe of all p oints v S where f is regular ie S U

call the rational mapping from S into W dened by the restriction f jS U the

restriction f jS Itmay b e in general regular on a larger subset of S than S U

If f V W is birational the restriction f jS is said to b e birational if the

set of all p oints v S where f is biregular ie S U is dense in S In this case

the restriction f jS denes a birational mapping from S onto the image f S U

Algebraic pregroups

Similarly to the algebraic groups we call the pregroups algebraic in order to

emphasize their algebraic nature As was p ointed out ab ove the algebraic pre

groups are obtained from algebraic groups via birational mappings The original

denitions of them byWeil make use of the language of generic p oints Here

we reformulate these denitions in the terminology of Our denitions coincide

with given in if all varieties are irreducible

The standard group axioms usually involve the unit Since wewishto

allow birational transformations the unit may disapp ear Therefore we start

with another set of group axioms namely with the asso ciativity condition and the

existence and uniqueness of the left and right divisions

Denition An algebraic pregroup is an algebraic variety V with a

rational mapping V V V written as v w vw suc h that

for generic u v w V V V b oth expressions uv w and uvw are

dened and equal generic asso ciativity condition

the mappings v w v vwandv w v wv from V V into itself

are birational generic existence and uniqueness of left and right divisions

Remarks

Of course an algebraic group is a sp ecial case of an algebraic pregroup

As noted ab ove an algebraic pregroup mayhave no unit eg V C n

We write v w v v w and v w v wv for the inverses of

the mappings v w v vwandv w v wv resp ectively These

birational mappings are the analogons of the left and right divisions

Condition in Denition implies that the pregroup op eration v w

vw is dominating

Warning If wv is dened it do es not mean that the inverse v is

dened Since in general an algebraic pregroup has no unit we cannot dene

the inverse elements directly

The following Lemma due to Weil intro duces the inverse mapping

V V v v as a birational mapping

Lemma There exists a birational mapping V V with fol lowing prop

erties

v w v w

wv wv

vww v

where v w V V is generic

Pro of We are lo oking for v in the form w vw This expression app ears as

a comp onent of the mapping f v w vw wvw fromV V into itself

The mapping f is birational as a comp osition of two birational mappings from

V V into itself v w vw wandu w u w u

We rst wish to prove that v w w vw is indep endentof w V

For this wemultiply b oth sides of the equality v wvw w by some u V

and use the asso ciativity condition Since for generic w u the righthand side

wu is dened the lefthand side v wvwu is also dened and one has

v wvwu wu

In order to apply the asso ciativity condition to wehavetoprove that the ex

pression v wvwu is also dened For this consider the birational mapping

g v w u f v wu vw v wu

from V V V into itself Then for ha b c bac the comp osition

h g v w uv wvwu is dened for generic v w u and one has the

asso ciativity

v wvwu v wvwu

The relations and imply

v wvwu wu

By the asso ciativity the expressions vwu and v wu are dened and equal for

generic v w u V V V Therefore implies

v wv wu wu

Since the mapping v w u v w wu is birational one can write

v wvt t

for generic v w t This implies

v w tvt v t

which means that v w is indep endentof w V Thus we can write v

v w

The prop ertyisnow implied by v vw w for generic v w The

latter relation means that the left multiplications by v and by v areinverse

to each other This implies the prop erty and therefore the birationalityof

The fact that the ab ove left multiplications are inverse to each other can b e also

expressed as follows

v v w w

For proving prop ertywe use similar calculations as for proving the indep endence

of v wofw V Here wemultiply by u V on the left

uv v w uw

By similar arguments we see that for generic v w u the asso ciativitycondition

can b e applied and we obtain

uv v w uw

This can b e reduced to

uv v u

This means that the rightmultiplication by v isinverse to that by v which

implies prop erty

Finally prop erty is obtained by the following calculation

w v vwu w v v wu w wuu

where all expressions are dened for generic v w u

Remark We write v v By prop erties and the expressions v w

and wv are then welldened If v and these pro ducts are dened they are

also dened in the previous sense but the converse may not b e true When saying

that such expressions are dened we understand them in the previous sense

The morphisms in the category of algebraic pregroups are dened as fol

lows

Denition A rational resp birational homomorphism between

two algebraic pregroups V and W is a rational resp birational mapping

f V W such that the expressions f uv and f uf v are dened and equal

for generic u v V V

Remarks

Since the mapping u v uv is dominating the expression f uv isalways

dened for generic u v V V The expression f uf v on the contrary

maybenowhere dened in general

Of course if f V W is birational the second expression is also dened

for generic u v V V In this case the inverse mapping f W V is

also a birational homomorphism

If f V V g V V are two rational homomorphisms and f is domi

nating than the comp osition g f V V is dened and is also a rational

homomorphism

For algebraic groups such morphisms coincide with the usual ones

Prop osition Let f V W bearational homomorphism between two

algebraic groups V and W Then f is a regular homomorphism

Pro of Let U V b e the op en dense subset where f is regular and v V an

arbitrary p oint Since V is an algebraic group one has u w v U for generic

w U By the homomorphic prop erty

f wu f w f u

for generic w u V V Sincew u U and W is a group the righthand

side is dened for u u Then the lefthand side f wuf v is also dened

V is arbitrary f is regular Since v

On the contrary to the algebraic group structures the algebraic pregroup

structures can b e transformed via birational mappings which follows directly from

the denition

Prop osition For every birational mapping f from an algebraic pregroup

V into an algebraic variety W there exists unique algebraic pregroup structure

on W such that f V W is a birational homomorphism with respect to this

structure

We call this algebraic pregroup structure induced by f If an algebraic

pregroup obtained in suchaway is a group it is called the regularization

Denition A regularization of an algebraic pregroup V is an algebraic

group V with a birational homomorphism V V

One of the main results of is the existence and uniqueness of such

regularizations We reprove and generalize this result for algebraic pregroups

with several irreducible comp onents

Theorem For every algebraic pregroup V there exists a regularization V

which is unique up to isomorphisms

The pro of will b e given in section Nowwecharacterize the p oints of V

where the regularization homomorphism V V is biregular

Denition Let V b e an algebraic pregroup A p oint v V is called a

p oint of regularity if the following conditions are satised

the mapping w wv from V into itself is birational

the mapping v uv from V into itself is biregular at v v for generic

u V

Lemma The set of points of regularity of an algebraic pregroup V is an

open dense subset of V

Pro of By denition of an algebraic pregroup the rightmultiplication v w

v wv is birational It is biregular on an op en dense subset U V V

Condition is satised if the intersection U fv g V is dense in fv gV

V V U where This is valid for all v from the op en dense set V

j j

V s are the irreducible comp onents of V and V V V is the pro jection on

the rst comp onent

The left multiplication u v u uv is also birational Again denote

by U V V the biregular set of it Condition is satised if the intersection

U V fv g is dense in V fv g Thisisvalid for all v from the op en dense

set V V V U where V s are as ab oveand V V V is the

j j

pro jection on the second comp onent

The set of p oints of regularityisnow V V which is op en dense as an

intersection of two op en dense subsets

Prop osition Let V be an algebraic pregroup and V V be its

regularization A point v V is a point of regularity if and only if the mapping

is biregular at v

Pro of Let b e biregular at v Then the mapping w wv can b e decomp osed

as a comp osition of the following three birational mappings w w

w v w v Indeed by denition of a birational homomorphism

w v wv ie w v wv for generic w v V V If

v v b oth sides dep end birationally on w Thus for generic w V they are

dened and equal This proves that the mapping w wv is birational

For generic v u V V the second mapping v uv can b e de

comp osed as follows v v uv uv Here the rst

mapping is which is biregular at v v by the assumption The second one

is biregular as a rightmultiplication in the algebraic group W Finally the third

mapping is biregular at uv for generic u V Thus the required mapping

v uv is biregular at v v for generic u V as a comp osition of biregular

mappings

Supp ose nowthat v V is a p oint of regularity By denition of a birational

homomorphism wv w v for generic w v V V Since V is

an algebraic group v w wv and the mapping v v can

b e decomp osed as the comp osition of the following three birational mappings

v wv wv w wv Since v is a p oint of regularity the rst

mapping is biregular at v v if w V is generic Since the mapping w wv is

birational is biregular at wv for generic w V Since V is an algebraic group

the last mapping V Vu w u is biregular whenever w is dened

For generic w V this is the case Then is biregular at v as a comp osition of

biregular mappings

The regularization satises also the following functorial prop erty

Prop osition Let f V W bearational homomorphism between two

aic pregroups V and W such that f v W is a point of regularity for algebr

generic v V Let V V W W be their regularizations Then

V W

there exists unique homomorphism f V W such that the fol lowing diagram is

commutative

V W

y y

V W

W V

Remark The condition that f v is a p oint of regularityisessential This

is shown on the example of the inclusion f V W where V fg

W C Here V V and W C

Pro of By Prop osition the mapping is biregular at f v W for generic

v is dened for generic v V Therefore the comp osition f v f

v V Moreover it is a rational homomorphism as a comp osition of rational

homomorphisms By Prop osition f V W is a regular homomorphism

Algebraic pretransformation spaces

Together with the pregroups A W eil intro duced also their birational

actions on algebraic varieties which he referred as pretransformation spaces

Again we reformulate this notion where several irreducible comp onents are also

allowed

The standard axioms for the group action involve the unit of the group

Since an algebraic pregroup mayhave no unit we pass to another set of axioms

namely to the asso ciativity condition and the existence and uniqueness of the left

divisions by group elements

Denition Let V be an algebraic pregroup An algebraic pre

transformation V space is an algebraic variety X with a rational mapping

V X X written as v x vx suchthat

for generic v w x V V X b oth expressions vwx and v wx are

dened and equal generic asso ciativity condition

the mapping v x v vxfrom V X into itself is birational generic

existence and uniqueness of left divisions

Remark Condition implies in particular that the op eration v x vx is

dominating

In Lemma weintro duced the inverse mapping v v Thus for

generic v x V X the expression v x is dened

Lemma The mapping v x v v x from V X into itself is the

inverse of the operation v x v vx

Pro of Both mappings are birational Therefore it is enough to prove that

v v xx for generic v x V X For generic w v x V V X one

has the following relations

w v v x wv v xwv v x wx

Since for generic w V the mapping x wx from X into itself is birational

implies the required relation v v x x

Lemma Let V be an algebraic pregroup and X an algebraic pre

transformation V space Let v V beapoint of regularity Then the mappings

w vw from V into itself and x vx from X into itself arebirational

Pro of Since V is a pretransformation V space with resp ect to the left multipli

cation it is sucienttoprove the birationality of the second mapping f x vx

By denition of a pretransformation V space the mapping f X X x ux

is birational for generic u V Let U V b e the op en dense subset of such

elements u We express the required mapping f as a comp osition f f

v uv

Since the mappings u u and u uv are birational one has u U uv U

f X X are birational for for generic u V Bythechoice of U b oth f

generic u V

By the asso ciativity condition

f f xu uv xu uv x v x

for generic u v x V V X Ifu x V X is generic the lefthand side

is dened for v v Then the righthand side vx f x is also dened and

f f f This proves that f is birational

v uv v

Denition Let V b e an algebraic pregroup and X and Y two algebraic

pretransformation V spaces A rational resp birational mapping f X Y is

called V equivariant if the expressions f vxand vf x are dened and equal

for generic v x V X

Remarks

Since the mapping v x vx is dominating the expression f vx is dened

v on the contrarymay for generic v x V X The expression f uf

be nowhere dened in general

Of course if f X Y is birational the second expression is also dened

for generic v x V X In this case the inverse mapping f Y X is

also V equivariant

If f X X g X X are two V equivariant rational mappings and f

is dominating than the comp osition g f X X is dened and is also

V equivariant

Similarly to algebraic pregroups the structure of algebraic pre

transformation spaces can b e transformed via rational mappings

Prop osition For every birational mapping f from an algebraic pre

transformation V space X into an algebr aic variety Y there exists unique

algebraic pretransformation V space structureon Y such that f X Y

is V equivariant with respect to this structure

For every rational homomorphism between two algebraic pregroups V

and W and an algebraic pretransformation W space there exists unique

algebraic pretransformation V space structureon X suchthat v x vx

for generic v x V X

This follows directly from denitions We call such algebraic pre

transformation V space structures induced by f and resp ectively For in

stance the regularization V V induces an algebraic pretransformation V

space structure

Similarly to algebraic pregroups wewould liketointro duce regularizations

of pretransformation V spaces X However if we wish the uniqueness it is not

sucient to require the induced op eration V Y Y to b e regular For example

let V C and X C C where V acts bymultiplication which is already

regular The inclusion into Y C and restriction to Y C induce other

regular actions

Therefore we call regularizations only some sp ecial regular actions They

should not lo ose the go o d p oints as in the ab ove example with Y These

go o d p oints will b e called the p oints of regularity The denition of them is

analogous to the second condition in the corresp onding denition for algebraic

pregroups see Denition

Denition Let V b e an algebraic pregroup and X an algebraic pre

transformation V space A p oint x X is called a point of regularity if the

mapping x ux from X into itself is biregular at x x for generic u V

Remark If V W is a birational homomorphism into another algebraic

pregroup W the sets of regularityof X with resp ect to the actions of V and W

coincide

Similarly to Lemma one obtains the following

Lemma The set of points of regularity of a pretransformation V space

X is an open dense subset of X

Denition Let V be an algebraic pregroup and X an algebraic

pretransformation V space A regularization of X is an algebraic pre

transformation V space Y together with a V equivariant birational mapping

X Y such that

if v V is a p oint of regularity vy is dened for all y Y

if x X is a p oint of regularity is biregular at x

Remark If V is an algebraic group it follows from condition that the action

V Y Y on the regularization is regular

The following is a generalization of a result of A Weil

Theorem

For every algebraic pregroup V and algebraic pretransformation V space X

there exists a regularization X X

The pro of will b e given in section The set of regularity is exactly the

set of p oints where a regularization is biregular

Prop osition Let X X bearegularization which is biregular at

some point x X Then x is a point of regularity

Pro of For generic v x V X the mapping x vx can b e decomp osed as

follows x x vx vx The rst mapping is biregular

at x x by the assumption The second one is biregular for generic v V by

condition of the denition of a regularization Finally the third mapping

is biregular at vx for generic v V Thus the required mapping x vx is

biregular at x x for generic v V as a comp osition of biregular mappings

Unfortunately such regularizations are still not unique as the ab oveexample

with Y shows To obtain the uniqueness weintro duce some sp ecial regularizations

which are in some sense minimal

Denition A regularization X X of an algebraic pre

transformation V space is called minimal if there are no prop er op en subsets

of X which are also regularizations of X with resp ect to the induced algebraic

pretransformation V space structure

Theorem For every algebraic pregroup V and algebraic pre

transformation V space X there exists a minimal regularization X X which

is unique up to isomorphisms

V b e the regularization of V and X X b e an arbitrary Pro of Let V

regularization of X as a V space Further let U X b e the set of regularity

where is biregular Then the image U is op en dense in X and the union

X VU vU

v V

is also op en dense Since V is an algebraic group X is also a V regularization

and therefore a V regularization of X

We claim that X is a minimal V regularization Assume that there exists

a smaller one X X By denition it contains the image U of the set of

regularity U Let W V b e the set of all p oints of regularity By condition of

Denition W U X and

W U fw w j w w W g X

This is equivalentto

W U X

Since W is op en dense in the algebraic group V one has

W V

This implies X VU W U X Thus X X and the minimality

is proven

It remains to prove the uniqueness Let X X i betwo

i i

minimal regularizations Then is a birational mapping b etween X

and X Since b oth and are biregular at U is biregular at U We

have seen that W U i are also regularizations which coincide with

X s b ecause of minimality Since b oth op erations on X and X are regular

the mapping extends to a biregular mapping from W U X onto

W U X by setting

w w x w w x

Thus X and X are isomorphic

Wehave proven in fact the following

Corollary Let X X be the minimal regularization If U X

W V are the sets of regularity one has W U X IfV is an algebraic

group VU X

Construction of regularizations

Our goal here is to prove Theorems and We rst reduce the general

algebraic pregroups and pretransformation spaces to the case where all p ointof

them are p oints of regularity

Let V b e an algebraic pregroup and X an algebraic pretransformation

V space Let V V and X X denote the sets of p oints of regularityBy

Lemmas and these sets are op en dense Then V and X can b e considered

as an algebraic pregroup and an algebraic V space with resp ect to the induced

structures

Prop osition All points of V and X arepoints of regularity

Remark The statementisessentially that the concept of p oints of regularityis

invariant under passing to op en dense subsets

Pro of Let x be a point from X ie a p oint of regularityof X This means

that the mapping x ux from X into itself is biregular at x x for generic

u V Wehave to prove that it is also biregular there as mapping from X into

itself For this it is sucienttoshowthat ux X ie y ux is again a p oint

of regularity for generic u V

This means by denition that the mapping y vy from X into itself is

biregular at y y for generic v V For generic u v y V V X this

mapping can b e decomp osed as

y u y vuu y

which is implied by generic asso ciativity Since x isapoint of regularity the rst

mapping is biregular at y y for generic u V By the same reason the second

mapping x vux is biregular at x x Then the comp osition y vy is

biregular at y y

We can apply it to the case X V where This proves the statementfor X

V acts by the left multiplications Then condition in Denition is satised

for all v V To prove condition we decomp ose the mapping f w wv from

V into itself as i f i where f is the same mapping from V into itself and

i V V denotes the birational inclusion Then f is birational as a comp osition

of three birational mappings Thus condition is also satised

Prop osition reduces the general case to the case where all p oints of V

and X are p oints of regularityThus we take this case for granted in the following

Prop osition and Lemmas

Prop osition There exists an algebraic pretransformation V space X

with a V equivariant open dense embedding X X such that

the inducedoperation V X X is regular

VX fvx j v V x X g X

For the pro of we need several lemmas Since every v V is a p ointof

regularity the mapping f X X is birational by Lemma

Lemma Let v V be arbitrary and X X be the closedgraph of

the rational mapping f x vx Let a b be an arbitrary point Then f is

v v

biregular at a and va b

Remark Lemma implies that if X is complete the action of V on X is

already regular

Pro of By denition of p oints of regularity the mapping w wv is

birational By Lemma b oth mappings V V w vw and X X x vx

t v x are are also birational Then the mappings t x tv x and t x

birational and we can comp ose them with the action Together with asso ciativity

this implies that the expressions tv x and tvx are dened and equal for generic

t x V X

By condition in the denition of p oints of regularity the expressions ta

and tb are dened for generic t V Since the mapping t tv is birational the

expression tv a is also dened for generic t V

The relation tvx tv x can b e written as ty tv x where y vx It

is valid for generic t x V X or equivalently for generic t x y V

Both sides are dened at x y a b as rational mappings on Since

they are equal for generic t x y V they are also equal for x y a b

tb tv a

Since by condition of the denition of p oints of regularity t tbis

dened and equal to b for generic t V one has

b t tv a

Consider the mapping t v a t tv a It follows by asso ciativity that it

is equal to t v a v a for generic t v a V V X But weknowby

v v and a a and is equal to b there that t tv a is dened for

Then the equal rational mapping t v a v a is also dened for v v and

a a and is equal to b there Thus f is dened at a and va b

To prove that f is biregular at a we apply the same considerations to the

relation

a tv tb

whichisvalid for generic t V b ecause of relation tv a tb condition of the

denition of p oints of regularity applied to a and the fact that the mapping t tv

is birational The mapping t v b tv tb is equal to t v b v b

They are dened for v v b b and equal to a there Thus f is biregular at

Lemma Let A B be algebraic varieties and U A B an open subset

with nonempty bres U fb B j a b U g a A Then there exists

a nite col lection of points b B i I such that for every a A one has

b U for some i I

i a

Pro of Without loss of generality A is irreducible We prove the statementby

induction on dimension of A IfdimA the statementisobvious Assume it

to b e proven for dim Ad Let a b U b e an arbitrary p oint The set A of

p oints a A suchthat b U can b e expressed as

A fa A j a b U g

This is an op en subset of A Since A is assumed to b e irreducible dimA n A

dim A d

Nowwe use induction for all irreducible comp onents of the complement

A n A This yields a numb er of p oints b i I These p oints together with b

satisfy the required condition

Denition Let t V i I be any nite collection of p oints Let

t X X i j I denote the comp osition of two birational mappings z t z

ij j

z A trace of X generated by these p oints is the quotient and z t

X ft gX X I where x y j resp x i y j if the

mapping z t z resp z t z from X into itself is biregular at z y and

j ij

x t y resp x t y

j ij

Warning If the mapping t is biregular at y the mappings z t z

ij j

z t z may not b e in general biregular at corresp onding p oints

Lemma The relation isanequivalencerelation

Pro of The symmetrical prop erty has to b e checked only for x iandy j

where x y X i j I In this case it follows from the fact that the mappings

and t are inverse to each other as birational mappings t

ij ji

The transitivity has to b e checked for the four dierent cases x y j

z x y j z k x i y z k x i y j z k Consider

birational mappings t t z t z The transitivity follows from the relations

ij i i

t t id t t t t t t t t t resp ectively

j j jk k k ik ij jk ik

j i

Lemma Let V and X satisfy the conditions of Proposition Then

the trace X ft g is an algebraic variety The restriction X X ft g of the

i i

canonical projection X X I X ft g is an open dense embedding

Pro of Since the relation isanequivalence relation the set Y X ft g

makes sense and satises the denition of an algebraic prevariety see It is a

variety if the hausdor prop erty of the closedness of the diagonal Y Y Y is

satised Weidentify the sets X and X fig with their images under the canonical

pro jection X X I X ft g Wehave the following decomp osition

Y Y X X X fig X

S S

X fig X fj g X X fj g

ij j

The intersection of the diagonal Y with the sets X X X fig X

X X fj g and X fig X fj g is equal to the graphs of the mappings

id x t x x t x x t resp ectively By Lemma these graphs are

i ji

closed Thus the diagonal Y is closed This proves that the trace Y is a

variety

The variety X considered as a subset of Y intersects every X fig along

some op en dense subsets U X fig where the mappings z t z are biregular

i i

X fig and therefore coincides This implies that the closure of X contains every

with Y Thus X is op en dense emb edded into Y

Pro of of Prop osition We are lo oking for the required pretransformation

space X among the traces X ft g Weprove that the induced op eration V X

X is regular for an appropriate choice of the p oints ft g

j j I

The regularityforsuch traces means that tx is always dened for t V x

X ft g We x an elementof X X I which represents the class of x in the

quotient X ft g and denote it also by x X or byx i X I

We wish to nd some j I such that the induced op eration V X

X fj g resp V X fig X fj g isdenedatt x resp at t x i

The induced op eration can b e obtained by passing to equivalent elements in X

applying the given op eration there and passing to the equivalent elements in the

required comp onent V j Itisgiven by

t x tx t txj

in the rst case and by

t x i t t x tt x t tt xj

i i i

in the second

The regularityfollows now from the following statement

Statement There exists a collection of p oints t V i I such that for all

t t V x X each of the expressions t tt x and t tx is dened at least

j j

for one choice of j I

Pro of of the Statement Since by Lemma the mappings v vx from V

into X are rational they are dened on op en dense subsets U V x X Then

the family U X V of these subsets is op en dense as a set of denition of the

rational mapping x v vx from X V into X

By the denition of p oints of regularity the mappings v vt from V into

itself are birational and therefore dened on some op en dense subsets Moreover

one has vt U for all v from op en dense subsets U V t V x X The

t x

family U V X V of these subsets is op en dense as the preimage of the

family U under the rational mapping t xv x v t from V X V into

X V

The mappings v v t from V into itself are also birational and therefore

for all v from dened on some op en dense subsets Moreover one has v t U

op en dense subsets U V t t V x X The family U V V X V

tt x

of these subsets is op en dense as the preimage of the family U under the rational

mapping t t xv t t xv t from V V X V into V X V

We constructed an op en dense family U V V X V such that

U V is op en dense and v tt x is dened for all t t V x X Now

tt x

we set in Lemma A V V X B V U U Then the conditions of

Lemma are satised and we obtain a nite collection of p oints t b V i I

j j

tt x is dened at least for such that for all t t V x X the expression t

one j I

By the same considerations weprove that for all t V x X the

expression t tx is also dened at least for one j I This nishes the pro of of

the ab ove statement

By the construction of the trace X ft g wehave the prop erty t X

i i

X fig for all i I This implies the required prop erty VX X

Pro of of Theorem Let V b e an algebraic pregroup and V the op en dense

subset of all p oints of regularity Then V is an algebraic pretransformation

V space which satises conditions of Prop osition by Prop osition Prop o

sition yields an op en dense V equivariantembedding V V

We wish to prove that V is the required regularization For this we need

to show that the op eration V V Vv w vw and the inverse mapping

V Vv v are regular

The op eration is regular for v V b ecause of prop erty in Prop osi

tion In general case by prop ertyevery v V is a pro duct of v v V

By the generic asso ciativity

v v w v v w

for generic v v w Since v v V the righthand side is regular at

v v w v v w Then the lefthand side is also regular there which means

is dened that vw

To prove the regularity of the inverse mapping we establish the birationality

of the mapping f w wv from V into V Since V consists of p oints of

regularity f is birational for all v V Again in general case v v v

v v V andby the generic asso ciativity

w v v wv v

The righthand side is equal to f f w which is a birational mapping in the

v v

variable w for v v v v Thus the lefthand side f w is also birational

By prop erty in Lemma wv v w ie v wv w for generic

v w V Wewishtoprovethat v is dened for all v V Since the

op eration V V V is regular it is enough to show that wv is dened for

generic w V Wehave just proven that the mapping f w wv from V

w wv is the comp osition into V is birational Then the required mapping

of f and the inverse u u which are birational Thus the inverse mapping is

also regular on V

Pro of of Theorem Let V b e an algebraic pregroup and X an algebraic

pretransformation V space Let V V and X X b e the op en dense subsets

of regularity By Prop osition V and X consists of p oints of regularity

Therefore we can apply Prop osition and obtain an op en dense V equivariant

emb edding X X The conclusions of Prop osition imply that this is the

required regularization

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Received July

and in nal form Novemb er