ALGEBRAIC SURFACES WITH k*-ACTION
P. O R L I K University of Wisconsin, Madison, Wisconsin, USA
BY
and P. W A G R E I C H University of Illinois, Chicago, Illinois, USA (1)
Let k be an arbitrary algebraically closed field and let k* denote the multiplicative group of k considered as an algebraic group. We give a complete classification of all non-singular projective algebraic surfaces with k*-action. Moreover, for singular surfaces we provide an algorithm for finding an equivariant resolution. Explicit computations are given for hypersurfaces in pa.
The topologically significant case emerges when k = C, the complex numbers, when non-singular surfaces are smooth orientable closed 4-manifolds with C*-action. This was the case studied in our earlier papers [16, 17, 18]. I n order to facilitate the reading of the present paper for the reader whose main interest lies in these topological aspects, we have included geometric motivation for several of our algebraic constructions. I t is clear from the naturality with which these constructions can be extended to the general case t h a t it is appropriate to treat the problem from the point of view of algebraic geometry.
The paper is organized as follows. I n section I the necessary tools are introduced both from transformation group theory and algebraic geometry. We prove a number of lemmas about actions in general, needed later in the paper. I n section 2 we focus our attention on non-singular algebraic surfaces with k*-aetion. The fixed points of a k*-action are divided into three classes; elliptic, hyperbolic and parabolic (w A "topological"
classification of non-singular surfaces with no elliptic fixed points is obtained in w 2.5. The classification theorem states t h a t the fixed point set is F = F + U F - U {x 1 ... xr} , where the x, are hyperbolic fixed points and F + and F - are isomorphic complete curves. More- (1) Both authors received partial support from NSF. During part of this research the first named author was visiting the University of Oslo, Norway.
44 P . O R L I K A N D ]?. W A G R E I C H
o v e r t h e surface is o b t a i n e d f r o m a P l ( k ) b u n d l e o v e r F + w i t h k* a c t i o n in t h e fiber, b y b l o w i n g u p a s u i t a b l e n u m b e r of f i x e d p o i n t s of t h e a c t i o n . A c o m p l e t e set of " t o p o - logical" i n v a r i a n t s is i n c l u d e d in a l a b e l e d g r a p h F a s s o c i a t e d t o t h e surface V. A s s i g n a v e r t e x e a c h t o F + a n d F - w i t h labels [g] for t h e i r g e n u s a n d - c a n d - c ' r e s p e c t i v e l y for t h e i r s e l f - i n t e r s e c t i o n in V. A s s i g n a v e r t e x t o e a c h 1 - d i m e n s i o n a l o r b i t whose closure does n o t m e e t b o t h F + a n d F - , w i t h label - b ~ j for its s e l f - i n t e r s e c t i o n in V. F i n a l l y , let a n edge c o n n e c t t w o v e r t i c e s if t h e c o r r e s p o n d i n g c u r v e s m e e t ( t r a n s v e r s l y w i t h i n t e r s e c t i o n n u m b e r § 1).
T H E O R E M 2.5. Let V be a non-singular, complete algebraic sur/ace with lc* action, so that d i m F + = d i m F - = 1. Then we have:
(i) F + is a complete curve o/ genus g, isomorphic to F-;
(ii) the graph F o/ (V, k*) is o / t h e / o r m
(iii) b~j>~l and the continued/faction (see the Appendix /or this notation) [bt.i . . . b~.~] = 0 for i = 1 , ..., r;
(iv) i/ we define the integers c~ by
1/c~=[b~.~ .... ,b~.s~-l] for i = 1 . . . r then the/ollowing equation holds
C§ ~ ~ Ct.
|~1
I n section 3 we c o n s t r u c t t h e c a n o n i c a l e q u i v a r i a n t r e s o l u t i o n of t h e s i n g u l a r i t i e s of a surface V w i t h k*-action. B y t h i s we m e a n a n o n - s i n g u l a r surface W w i t h a k*-action
ALGEBRAIC SURFACES WITH k*-ACTION 45 and a k* equivariant proper map ~: W-~ V which is biholomorphic on the complement of the singular set of V and so that there are no elliptic fixed points on W. Note t h a t even if V is nonsingular, W will not be equal to V if there are elliptic fixed points on V. We define in w a marked graph Fv, starting with Fw and indicating which orbits in W are identified to points in V and which orbits in W are identified to each other in V. This marked graph contains all the "topological" information about V. I n particular if k = C, F v determines V up to equivariant homeomorphism. I n w 3.3 we define the Seifert orbit invariants (:r fl) of an orbit of dimension 1. Here a is the order of the isotropy group of a point v on the orbit and fl determines the action of the isotropy group on the tangent space at v. Section 3.4 is devoted to determining the behavior of orbit invariants under
"blowing u p " and we use this in w to prove a relation between the continued fraction expansion of ~/fl and the self-intersection of certain orbits on V. Explicit formulas for determining the orbit invariants and the graph of the resolution of an elliptic fixed point on a surface in k 3 are given. The corresponding action was called "good" in [16]. These results are an improvement over [16] since t h e y are valid for a field of arbitrary charac- teristic. Moreover, the formulas are in terms of the weights and degree of the defining polynomial. I n w 3.7 we indicate how to find the minimal equivariant resolution of a hyper- bolic or parabolic singular point, not treated in [16].
Section 4 is devoted to the diffeomorphism and algebraic classification problems. First we classify the (relatively) minimal surfaces with k*-action w and apply this to deter- mining which surfaces admit an essentially unique k*-action and which k*-actions extend to k* • k*-actions w 4.2. I n w 4.3 it is shown t h a t a complete surface with algebraic C*-action is diffeomorphic to precisely one of the following
(1) ( R g • ~ g>~0, k~>0,
(2) Na,
(3) CP 2,
where Rg is a compact Riemann surface of genus g and Ng is the non-trivial S 2 bundle over Rg, and CP ~ denotes CP ~ with the opposite of the usual orientation. This is applied to give examples of non standard S 1 actions on CP 2 # k C P ~, k ~>3 i.e. S 1 actions which do not extend to S 1 • S 1 actions (w Finally, we give a complete list of invariants of V up to equivariant algebraic isomorphism (w
Specific examples of the resolution of singularities of hypersurfaces V in p3 and calculation of Fv are given in section 5. Some facts about continued fractions have been collected in an appendix.
46 P. O R L I K A ~ D Y. W A G I ~ E I C H
1. Group actions o n varieties
1.1. L e t G be an algebraic group. We shall n o t assume t h a t G is compact. I n t h e applications G will be It*, t h e multiplicative g r o u p of a field /c.
L e t V be a algebraic variety. A n action of G on V is a m o r p h i s m of algebraic varieties a : G • V - ~ V
so t h a t a(s, a(t, v))=~(st, v) a n d (~(1, v ) = v . I f g E G a n d vE V it is c u s t o m a r y to d e n o t e t h e element g(g, v) b y gv. The action is e//ective if gv = v V v E V -*g = e, the i d e n t i t y element of G.
F o r vE V a induces a m o r p h i s m av: G-~ V defined b y av(g) =gv. W e define the i s o t r o p y subgroup G v of v to be t h e scheme theoretic fiber a~ l(v). If k is a field of characteristic 0 t h e n Gv = (g E G]gv = v}. T h e orbit of v is the s u b v a r i e t y defined b y G(v) = {w E V [ 3 g e G, gv = w}.
This induces a n a t u r a l equivalence relation on V, x . . ~ y ~ x E G ( y ) . T h e quotient V* = V/,~
equiped with the q u o t i e n t t o p o l o g y is called t h e orbit space of t h e action. E v e n if V is a complex manifold, V* can be r a t h e r unpleasant, as t h e examples below indicate, see also [8]. L e t F = { v e V I G , = G } d e n o t e the fixed p o i n t set a n d E = { v E V [ G , i s finite a n d G ~ ( 1 ) ) d e n o t e the exceptional orbits, i.e. orbits with finite non-trivial isotropy group.
1.2. L e t V = C 2 with t h e metric topology, and G = C* a n d ql, q~ integers. Consider t h e action a(t; vl, v2)=(tq'vl,
tq2V2).
First assume t h a t ql, q~>0. Then F = ( O , 0) a n d E consists of two orbits: E1=(Vl, v ~ [ v l ~ 0 , v ~ = 0 ) with isotropy Zq, and E 2 = ( v l , v21vl=0, v2~=0 } with Zq,. T h e orbit space V* fails to be T o a t t h e image of t h e fixed point. H o w e v e r , if we let V 0 = V - F , t h e n V* is Hausdorff, in fact b y a t h e o r e m of H o l m a n n [8] it has a n a t u r a l complex s t r u c t u r e a n d is easily identified as p1. T h e situation is the same if ql, q~ < 0 -1.3. I n t h e example above, if q1:4=0, q2 = 0, t h e n F = (Vl, v 2 Iv 1 = 0} a n d E = {Vl, v 2 Iv 1 ~ 0 ) } so the action is n o t effective unless ql = - 1 . I n t h a t case E = O. Again, V* fails to be T O at F, b u t V 0 = V - F has H a u s d o r f f orbit space, V~ = C.
1.4. If we assume q l > 0 , q 2 < 0 , t h e n F a n d E are t h e same as in w 1.2, b u t even t h e orbit space of V0 fails to be Hausdorff. The images of E 1 a n d E 2 h a v e no disjoint neighborhoods in V~. (This holds even if ql = 1, q~ = - 1 . ) This p h e n o m e n o n is indeed one of the crucial differences between t h e actions of c o m p a c t a n d n o n - c o m p a c t groups. T h e r e m a r k s above a p p l y in the c a t e g o r y of algebraic varieties if we replace " H a u s d o r f f " b y
" s e p a r a t e d " .
A L G E B R A I C S U R F A C E S W I T H k * - A C T I O N 47 1.5. A m a p / : V - + W between G spaces V a n d W is equivariant i f / ( g v ) = g / ( v ) for all g EG, v E V. A G-space is linear if it is e q u i v a r i a n t l y isomorphic t o a v e c t o r space with linear G action. L e t H be a closed subgroup of G a n d S a linear H-space. W e form G • H S b y identifying in G • S (g, s)... (gh -1, hs) for all h E H. Call t h e equivalence class of (g,s), [g,s]. Given a G-space V a n d v E V we call S v a linear slice at v if it is a linear Gv-space a n d some G-invariant open n e i g h b o r h o o d of the zero section of G • is e q u i v a r i a n t l y isomorphic to a G-invariant n e i g h b o r h o o d of the orbit G(v) in V b y t h e m a p
[g, v]-*gv, so t h a t the zero section G/G v m a p s o n t o the orbit G(v).
I t is essential for the reader to keep in m i n d t h a t t h e s t a n d a r d tools of t r a n s f o r m a t i o n g r o u p s do n o t always a p p l y in our context. F o r holomorphic or s m o o t h action of a compact Lie g r o u p G, the existence of slices is a classical result. I n our situation with G = k*, a slice need n o t exist.
W e refer t o H o l m a n n [8] a n d L u n a [I0] for results on t h e existence of slices. These will only be used in the present paper in w
1.6. A k*-action on an affine v a r i e t y V = S p e e (A) is equivalent to a g r a d i n g of the ring A. The correspondence is defined as follows. I f V is an affine v a r i e t y with k* action we define a g r a d i n g on A b y letting
A~ = {~CA I/(tz) = t'/(z), for all zE V}.
T h e n one can verify t h a t A = (DtEzA~ a n d t h a t A ~ A j c A t + j. Conversely, if A is g r a d e d we can find homogeneous generators of t h e k-algebra A, s a y x 1 ... xn. T h e b o m o m o r p h i s m
~: k[X1, ..., X n ] ~ A defined b y ~0(X~)=xt defines an e m b e d d i n g of V in k n. L e t q~=
degree x~. T h e n define an action of k* on k" b y
t ( z 1 . . . Zn) = (t q' Z 1 . . . t q n Z n )
This action on k n leaves V invariant.
W e can use this correspondence t o get some i n f o r m a t i o n a b o u t the s t r u c t u r e of a k* variety.
L E ~ M A . I / k* acts e/fectively on V, then there exists an invariant Zariski open set U, equivariantly isomorphic to W x k* where
(i) W is a]/ine, and
(ii) k* acts on W • k* by translation on the second/actor.
Proo/. W e first consider t h e case where V = A ~ a n d t h e a c t i o n is given b y
4 8 P. ORLIK AND P. WAGREICH
~(t, (zl, ..., zn) ) =(tmlzl ... tmnzn). I n this situation let U = { ( z 1 ... zn)[z~=O, for all i}. We claim t h a t U is e q u i v a r i a n t l y isomorphic to W x k* with k* a c t i n g on the second f a c t o r b y multiplication. N o w
U = Spec (k[zl, ..., z~, zi -1 ... z~l]).
I n this case the grading is defined b y degree z~=m~. Since the a c t i o n is effective, g.c.d.
(m I ... m n ) = l . Therefore there exist integers al, a 2 ... a n so t h a t ~ _ l a , m , = l . L e t X = Z ~ ' ... Z~% a f o r m of degree 1. L e t R be t h e subalgebra of forms of degree 0. T h e n R is an algebra of finite t y p e over k a n d
k[Z 1 . . . Zn, Zl 1 . . . Zn 1] = R [ X , X - l ] .
L e t t i n g W = Spec (R), we h a v e p r o v e n our claim.
N o w consider the case of general V. W e m a y assume t h a t V is normal. T h e n a t h e o r e m of Sumihiro [23] asserts t h a t an open subset of V can be e q u i v a r i a n t l y e m b e d d e d in A~, with an action as above. T h u s we m a y assume t h a t V c A ~ a n d is i n v a r i a n t u n d e r g. L e t I c R [ X , X -1] be the (prime) ideal defining V. T h e fact t h a t V is i n v a r i a n t u n d e r ~ means t h a t I is a g r a d e d ideal. T h u s I = | where I j consist of the elements of I which are homogeneous of degree j. H o w e v e r / c I j ~ X - J / E I o ~ / = X j ( X - j / ) E X j l o . T h u s I = | XJIo a n d hence V = S p e c (R/Io) • Spec (k[X, X - l ] ) which is the desired result.
1.7. LEMMA. Let G be an algebraic group, V and W varieties with G action and U c V an invariant ZarisIci open set. I / t h e map ]: V ~ W is equivariant when restricted to U, then it is equivariant on V.
Proof. T h e following d i a g r a m with 1 d e n o t i n g t h e i d e n t i t y m a p G • I • 2 1 5
V / , W
is c o m m u t a t i v e because G • U is dense in G x V a n d the m a p s 0o(1 x ] ) a n d [oo" agree on G • U, hence t h e y agree on G • V.
1.8. L e t Ov d e n o t e t h e sheaf of rational functions on V a n d I c Ov an ideal sheaf.
The monoidal trar~[orm with center I is a pair (~, V') with ~: V ' ~ V a n d
(i) IOv, is locally principal, i.e. u t h e stalk (IOv,)w is generated b y one function,
ALGEBRAIC SURFACES WITH ~*-ACTI01~ 49 (ii) for every ~0: V0~ V satisfying "IOv, is locally principal" there is a unique
~/: Vo-~ V' with ~ro~/-~vr o.
The monoidal transform exists b y Itironaka [7] and it is unique b y (ii). If X is a subspaee of V and I x is the sheaf of functions vanishing on X, then the monoidal transform with center X is defined to be the monoidal transform with center Ix. I t is also called
"blowing up of X " . For a geometric description of the blowing up of a subvariety see [22].
Lv,~II~A. Let G be an algebraic group acting on an algebraic variety V and (zt, V') a monoidal trans/ormation with center a G-invariant shea/ o/ ideals I. Then there is a unique extension o] the action o/ G to V' so that zt is equivariant.
Pro@ We first show t h a t there is a unique map a': G x V'-* V' so t h a t the diagram below commutes.
G x v ' l X ~ , G x V
1
V' , V
For this it is sufficient to show that (oo(1 x~))* (I) is a locally principal sheaf of ideals in Oo• Now I is invariant under a so a * ( I ) = O o | and hence ( a o ( 1 x ~ ) ) * ( I ) = Oa| which is locally principal because z~*(I) is locally principal. To verify that 0' is an action, let U= V - s u p p o r t ( I ) a n d U' =~-I(U). Since U' is isomorphic to U, the follow- ing diagram is commutative
G x ( G x U') m x 1 G x U'
G x U ' , U'
where m is the multiplication of G. An application of w 1.7 completes the proof.
1.9. PROPOSITXON. Suppose V is a non.singular algebraic sur/ace with k*-action and C ~ V is a complete curve with negative sel/-intersection number. Then C is invariant under the action.
Pro@ Let Ct = {tx I x E C}. Then Ct is rationally equivalent (homologous if k = C) to C for all t. If C is not invariant then (C. C)=(C.C~)>0, which is a contradiction.
4 - 7 7 2 9 0 2 Acta mathematica 138. Imprim6 le 5 Mai 1977
50 P . O R L I K A N D P . W A G R E I C H
2. N o n - s i n g u l a r surfaces with k*-ar
2.1. F r o m now on we shall concentrate on the case G = ]r the multiplicative group of a field, and V an algebraic surface over t h a t field. I n this section we shall determine the topological t y p e of a non-singular algebraic surface with k*-aetion, provided it has no elliptic fixed points.
De/inition. A m o r p h i s m / : X--> Y of algebraic varieties is said to be birational if there is a closed subvariety Z = Y so t h a t / : X - / - I ( Z ) ~ Y - Z is an isomorphism.
LEMMA. Suppose/: X--* V is a G-equivariant birational proper morphism o/non-sigular sur/aces. Then there exist equivariant morphisms /~: X ~ X ~ _ I , i = 1 .... , n so that X 0 = V, X n = X and/~ is a monoidal trans/orm with center at a /ixed point o/ Xf_ 1.
Proo/. B y [9, w 26] the set of points S w h e r e / - 1 is not defined is finite. Moreover if y E S , then y is a fixed point a n d / - l ( y ) is a divisor. Now let S = ( y o , Yl ... Yr} and define h I to be the blowing up of the points Y0 ... Yr- Then since /-l({y 0 ... Yr}) is locally principal we have a factorization X gl , X1 h~ , y . Repeating the process, b y [9] we reach a point so t h a t /=hloh2o...ohr, where h,: X ~ X ~ _ 1 is a monoidal transform with center at a finite n u m b e r of points. Factoring each h~ gives the desired result.
2.2. De/inition. V is normal at v E V if the local ring Ov at v is integrally closed. For a n y variety V there is a v a r i e t y l~ and a proper morphism ~: l~-~ V called the normaliza- tion of V characterized by the fact t h a t for every affine open subset U of V, F(~-I(U), O~) is the integral closure of F ( I U , Ov) in its field of quotients [4, I I . 6.3]. If a: VI-~ V2 is a birational morphism there exists a unique ~: ~1-~ l~ so t h a t
commutes.
V1
1.1
' V2Definition. V is complete if the morphism V-~Spec (k) is proper. I f k - - C this is equivalent to V being compact in the metric topology.
THEOREM. Suppose V is a complete, normal algebraic sur/ace with k* action. Then there is a complete non-sinilular curve C, a complete sur]ace Z with k* action and morphisms
ALGEBRAIC SURFACES W I T H k*-ACTION 51
V C x p1
so that
(i) the action on C • pl(k) is given by
t(x, [Zo: zl]) = (x, [tZo: zl]), (ii) q~x and q~2 are equivariant,
(iii) q9 2 is a composite o] monoidal transforms whose centers are fixed points of the k*-action,
(iv) ql is a composite o] maps, each o] which is either a normalization or a blowing up o] a ]ixed point. I] V is non-singular, then each o / t h e maps is a blowing up.
The variety o] Z constructed here is not canonical.
Proo]. I t follows from w 1.6 t h a t there is an open invariant subset U = V equivariantly isomorphic to C o • k*, where C o is a non-singular curve. Thus there is an equivariant open embedding ]: U ~ C • where C is the (non-singular) completion of C 0. Now ] defines a rational m a p ]: V ~ C • and it follows from [9] t h a t the set S of points where ] is not defined is finite. I f x E X is not a fixed point then ] is defined at x, since for some gEk*, ] is defined a t gx and 1(x)=g-1](gx). Thus S consists of fixed points. L e t V 0 = V. Define inductively It: Vt -+ Vt-1 to be the normalization of the monoidal transform with center at the singular points and points of indeterminacy of ]O]lO...o]t_ 1. B y [9, (26.2)] and w there is an 8 so t h a t Vs is non-singular and /tOtlO...o]8 is defined on Vs. L e t Z = Vs,
~l=]lO...o]s and ~ = ] O ~ r The ]~ are equivariant b y L e m m a 1.8 and the funetoriality of normalization. Moreover, q2 is equivariant b y w 1.7. Assertion (iii) follows from w
2.3. L e t V be a non-singular v a r i e t y and v E V a fixed point of a k*-action. The t a n g e n t space T~ at v has a local coordinate system in which the induced action of k* is linear and given b y
tCz. . . . z,,,) = (tq'zl . . . t q" Zm)
for integers ql, -.., qm [2]. Denote b y N+(v) the dimension of the positive eigenspace of this action # {q~ > 0}, N-(v) the dimension of the negative eigenspace and N~ the dimension of the subspace fixed under the action in Tv.
We call v an elliptic ]ixed point if r e = d i m T~=N+(v) or m = N - ( v ) . I t is a source in the former case and a sink in the latter.
52 P . O R L I K A N D P. W A G R E I C H
We call v a
parabolic fixed point
if N ~ and eitherm=N+(v)+N~
orm=N-(v)+N~
The former is a parabolic source, the latter a parabolic sink.All other fixed points are called
hyperbolic.
We can extend the definition of an elliptic fixed point to the singular case. Assume v E V is normal. We say v is an elliptic fixed point if there exists an invariant neighborhood U containing v so t h a t v is in the closure of every orbit in U. Recall t h a t we denote b y F the fixed point set of V. Let
F + = { v e F I N - ( v ) = O }
andF-={veFIN+(v)=O}.
I t follows from [2] that if V is non-singular then F+ and F - are irreducible, connected components of F. If V is complete, then both F+ and F - are non-empty.2.4. Let V be a non-singular, complete surface with k*-action and assume t h a t dim F + - ~ d i m F - = l . A 1-dimensional orbit O is called
ordinary
if 0 f l l V + ~ and (~ n F - ~ , where 0 is the closure of 0. According to Theorem 2.2 the odinary orbits form an open set in V and since V is compact, there are only a finite number of 1-dimen- sional orbits El, E~ ... Em which arespecial,
i.e. not ordinary. We define theweighted graph
F of (V, G) as follows:(i) its vertices are /+ for the curve F +, / - for F - and e 1 ...
em
for the closures(ii) each vertex carries a weight [g], referring to the genus of the curve it represents, (if
g=O
this weight is omitted),(iii) each vertex carries a weight n, representing the self-intersection in V of the curve it represents,
(iv) two vertices are connected b y an edge if and only if the respective curves intersect in V.
Clearly each point of intersection between different orbit closures is fixed under the action. I t follows from w 2.2 t h a t if a n y two of the curves above intersect, they intersect transversely.
2.5. T~EOREM.
Let V be a non-singular, complete algebraic sur/ace with k*-action so that
dim F+ = dimF - = 1. Then we have
(i)
F + is a complete curve o/genus g, isomorphic to F-,
(ii)the graph F o/
(V, k*)is o/the/orm
ALGEBRAIC SURFACES WITH ~*-ACTION 53
(iii)
{b~,l, bt, ~ . . . b~,~} is admissible (see Appendix) and the continued/raction [bt.,,b~.~ .... b ~ . ~ ] = 0 for i = l . . . r,(iv) with c~ = 1/[b~, 1 . . . b~.~_l] we have
Proo/. A p p l y i n g T h e o r e m 2.2 we see t h a t V can be o b t a i n e d f r o m C • p1 b y b l o w i n g u p f i x e d p o i n t s a n d t h e n b l o w i n g d o w n curves. N o f i x e d c u r v e s are b l o w n d o w n since d i m F + = d i m F - = I . T h e t h e o r e m c l e a r l y h o l d s for V = C •162 since F is of t h e f o r m
Q Q
where c = c ' = 0, a n d F + = F - = C. H e n c e i t is s u f f i c i e n t t o show t h a t if V is a surface w i t h d i m F + . = d i m F - = 1 a n d g : V-+ W is a m o n o i d a l t r a n s f o r m w i t h c e n t e r x a t a h y p e r b o l i c or p a r a b o l i c f i x e d p o i n t of V, t h e n t h e t h e o r e m h o l d s for V if a n d o n l y if i t h o l d s for W.
T h i s is o b v i o u s for (i) a n d (ii). F o r (iii) we h a v e
[b, . . . . , b~, b~+ 1 .. . . . bs] = [b, . . . b~_l, b t + l , 1, b~+l+ 1 , bt+ ~ .. . . . b~]
a c c o r d i n g t o A.7. F i n a l l y , t h e e x p r e s s i o n s of t h e f o r m u l a of (iv) r e m a i n u n c h a n g e d if x is n o t on F ~ or F ~ . A s s u m e w i t h o u t loss of g e n e r a l i t y t h a t t h e c e n t e r of ~ is on F ~ . T h e n Cw=Cv a n d Cw =C'v+ 1. I f x is i n t h e closure of a n o r d i n a r y o r b i t , t h e n Fw is o b t a i n e d f r o m F v b y a d d i n g one new a r m [1, 1] as follows
54
P. ORLIK AND P. WAGREICH
This clearly increases both sides of (iv) by 1, as required. If x is not on the closure of an ordinary orbit, then W is obtained from V by blowing up a point on F - which is in the closure of a special orbit. This corresponds to adding a new vertex to the ith arm of Fv.
Omitting the first index for convenience, it changes as shown:
9 eee 9 9 9 ee 9
i "
i i
.Finally, 1 +
1/[b I
. . .bs_l] = 1/[bl
. . .bs-1, bs+
1] follows from Lemma (A.8) to complete the argument.2.6. PROPOSITIOn.
Suppose k is an algebraically closed/ield. A n y graph F satis/ying
(ii)-(iv)o] w 2.5 arises/tom a complete algebraic sur/ace with k*-action.
Proo/.
Let V 0 = C x p1 where C is a non-singular complete curve of genus g and let k*act on the second factor. B y A.3 and A.7 we can obtain a surface V 1 with the desired
A L G E B R A I C S U R F A C E S W I T H k * - A C T I O N 55 weights on the arms b y a sequence of monoidal transforms with centers at fixed points.
The graph of V1 m a y not have the desired weights c and c' f o r / - a n d / % L e t elm(V1) be the v a r i e t y obtained from V 1 b y blowing up a point x of F + which lies in the closure of an ordinary orbit (~, a n d t h e n blowing down the transform of 0. This surface has the same graph as V 1 except t h a t the weight o f / - is increased b y 1 and the weight o f / + is decreased b y 1. Similarly the weight o f / - can be decreased. Thus in a finite n u m b e r of steps we can obtain a surface V~ with F + having self intersection - c. Now
c'
is determined b y the relation (iv) hence V2 has the desired graph.3. Resolution of singularities
3.1. We shall now consider
singular
surfaces V with k*-action. B y anequivariant resultion
of the singularities of V we m e a n a non-singular surface W and a k* equivariant m a p ~: W ~ V which is proper and birational. The existence of an equivariant resolution was demonstrated in w 2.2. I n this section we construct acanonical
equivariant resolution.3.2.
The canonical equivariant resolution.
Suppose V is an irreducible normal surface.We construct the canonical equivariant resolution locally and then p u t the pieces together. Since V is normal, the singular set consists of a finite n u m b e r of points, each of which must be fixed. Suppose v is an elliptic fixed point (singular or not). B y [23], the normality of V implies there is an affine invariant neighborhood U of v and an equivariant embedding ]:
U-~k n,
where the action on k n is given b yt(Z 1 . . . Zn) = (tq~Zl . . . tq"Zm).
Moreover we can assume q t > 0 , for all i, if v is a source and q~<0, for all i if v is a sink.
Assume t h a t v is a source. Now U = S p e e (A) and
A =k[X 1
...Xn]/I,
where I is the ideal of functions vanishing on U. The invariance of U under the action is equivalent to the fact t h a t A is a graded ring, the grading being induced b y letting degreeX~=q~, w
1.6. The quotient spaceX = U-~v}/k*
is t h e algebraic v a r i e t y Pro] (A) [4, I I , w 2]. We shall construct a canonical varietyUx
and morphisms h: Ux-~ U and ~:U x ~ X
.Geometrically1,: U x ~ X
is the Seifert k* bundle with fiber k associated to the principal Seifert k*-bundleU - ( v } ~ X .
The m a p h eollapes the zero section to the point v.L e t
AEn j = | A~ and A ~ = | AE~ ].
m>~ n n>~O
I f we consider A as an ungraded ring then A~ is a graded A algebra under the multiplica-
56 P . 0 R L I K A N D P, W A G R E I C H
tion which makes
A[nl A[,~j= A[n+m ].
Let U x = P r o j (Ate). Then b y [4, I I , 8.3 and 8.6.2] Ux is a projective variety over U = Spec (A) and there are natural morphisms
vx " u - (v)
X so t h a t
(1) U x - e ( X ) is isomorphic to U - (v) and v restricted to U x - ~ ( X ) is the orbit map (2) ~oe=idx.
If vE V is a sink and U = S p e c (A) is a neighborhood of v as above, then A is generated b y forms of negative degree. We can define a new grading on A b y letting new degree x - - ( o l d degree x). Then we can perform the same construction as above and h: U x ~ U will again be equivariant.
Now there is a unique proper birational morphism :~1:V1 ~ V so t h a t re 1 agrees with the above in the neighborhood of every elliptic fixed point and zt 1 is an isomorphism elsewhere. Moreover one can show that V x is normal (A.10). The only fixed points on V 1 are hyperbolic or parabolic. Note t h a t the definition of g: V I ~ V makes sense for a variety with k*-action of a n y dimension.
LEMMA. I / VE V is a singular point on a normal sur/ace with k*-action and ~: ~ ~ V is the minimal resolution o/the singularity, then there is a unique action ~ on ~ so that ~ is equivariant.
Proo/. B y w we know t h a t there is an equivariant resolution p: W-~ V 0. The minimal resolution is obtained stepwise by collapsing rational curves on W having self- intersection - 1 . By w 1.8 these curves are invariant under the action, hence there is an induced action on the minimal resolution.
Now define ~ : ]~-+ V 1 to be the minimal resolution of the singularities of V 1. The composite :z = ~ o : z x : l ~ V is called the canonical equivariant resolution o/ V.
If V is not normal we let :~0: V0-* V be the normalization of V and :~1: ~-~ V0 the canonical equivalent resolution of V 0. Then ~r =g0O:~l is the canonical equivariant resolution o / V .
ALGEBRAIC SURFACES WITH k*-ACTION 57 3.3. Orbit invariants. Suppose V is a v a r i e t y with k*-action a n d v E V is a simple point.
I f v is n o t a fixed p o i n t t h e n Gv is a proper subgroup scheme of k*. All such subgroups are of the f o r m / ~ = S p e e ( k [ T ] / T ~ - I ) , for some a>~l. I f t h e characteristic p of k does n o t divide a, t h e n this is t h e g r o u p of a t h roots of u n i t y . I f p divides a t h e n t h e scheme #a has nilpotent elements. T h e action of k* on V induces an a c t i o n of G v on t h e t a n g e n t space T~. E v e r y such action is linear a n d b y [3, I I I 8.4] we can choose coordinates (x 1, x.,, ..., xn) so t h a t t h e t a n g e n t space to the orbit is the x. axis a n d t h e action is defined b y
t(Xl . . . Xn-1, ~n) = (t~'l~l . . . tYn- lxn-1, ~n)"
I f V is a sur/ace t h e integer ~21 is well defined.
De/inition. T h e Seifert i n v a r i a n t of t h e point v is the pair (a, 8) where Gv = / ~ a n d
~1 fl ---- 1 ( m o d ~) 0 < f l < g , g.c.d. ( ~ , ~ ) = l .
Warning. T h e definition of fl a d o p t e d here corresponds to [13] a n d is the negative of t h a t given in [16].
I n [16] we calculated the g r a p h of the resolution of an elliptic p o i n t f r o m t h e Seifert invariants a n d other data. W e shall generalize t h a t result in our current context. F o r this we m u s t first analyze h o w t h e orbit i n v a r i a n t s are affected b y "blowing u p " .
3.4, P R O P O S I T I O N . Suppose V is a non-singular sur/ace with k*-action and vE V is a hyperbolic fixed point.
Suppose the action o/ k* on T~ is given by t(x, y) = (t-ax, tby)
where a, b>~O and ( a , b ) - - 1 . Let re: V ' o V be the monoidal trans/orm with center v and X=re-l(v). Then
(i) X is invariant under the action and there are two /ixed points x o and xoo on X , where x o is a sink and x~o is a source o / t h e action on X ,
(ii) i/ x E X and x ~ x o, x~ then the Sei]ert invariants (a, fl) o / x are given by a = a +b and b f l - 1 (mod oc),
(iii) there are coordinates x, y / o r Tv. x~ (resp. T v . z~) so that the action o] k* is given by t(x, y) = (t-(a+b)x, tby) (resp. t(x, y) = (t-ax, ta+by)).
Proo/. This is easy t o see in t h e case k---C. Near x 0 we have coordinates (x, y) so t h a t re(x, y)-~(xy, y) a n d near x~o we have coordinates x, y so t h a t re(x, y ) = ( x , xy). This
58 P . O R L I K A N D P . W A G R E I C H
i m p l i e s (iii) a n d h e n c e (ii). To c o n s i d e r t h e g e n e r a l (algebraic) case we r e p l a c e V b y a s u i t a b l e affine n e i g h b o r h o o d of v so t h a t we m a y a s s u m e t h a t V = S p e c (A) a n d m, t h e m a x i m a l i d e a l of v, is g e n e r a t e d b y a h o m o g e n e o u s s y s t e m of p a r a m e t e r s x, y i.e. x a n d y a r e h o m o g e n e o u s a n d g e n e r a t e m/m S. T h e n V' = S p e e (BI)tJ S p e c (B2) w h e r e B l = A [ x ] y ] a n d B2=A[y/x]. L e t x l = x / y , y l = y , x~=x, y2=y/x. B y (1.6) t h e a c t i o n o n V' is d e t e r - m i n e d b y t h e g r a d i n g of t h e r i n g s B~. T h e s e r i n g s a r e g r a d e d i n t h e n a t u r a l w a y i.e.
d e g r e e x 1 = d e g r e e x - d e g r e e y = - a - b, d e g r e e Yl = d e g r e e y = b, d e g r e e x z = d e g r e e x = - a, d e g r e e Y2 = d e g r e e y - d e g r e e x =b § N o w X is i n v a r i a n t since v is a f i x e d p o i n t . M o r e o v e r x o is t h e p o i n t i n Spee (B1) g i v e n b y x 1 =Yl = 0 a n d x ~ is t h e p o i n t in Spee (B2) g i v e n b y x z = y 2 = 0 . A s s e r t i o n (iii) follow f r o m t h e g r a d i n g given a b o v e a n d i n a d d i t i o n we see t h a t x o is t h e source o n X a n d x ~ t h e sink. T o v e r i f y (ii) we n o t e t h a t t h e (a, fl) g i v e n a b o v e a r e t h e Seifert o r b i t i n v a r i a n t s of a n y p o i n t on t h e l i n e y 1 = 0 in Tv.x0. T h e i n c l u s i o n k[xl, y l ] ~ B 1 i n d u c e s a n e q u i v a r i a n t m a p /: Spec (B1)-~Spee (k[xl, Yl]) ~ Tv.x~ T h i s m a p is e t a l e a t x 0 since xl, Yl is a s y s t e m of p a r a m e t e r s a n d hence i t is dtale in a n e i g h b o r h o o d of x 0. I f we choose a p o i n t x on X n e a r xo, t h e n t h e Seifert i n v a r i a n t s of x a n d ](x) a r e t h e s a m e h e n c e x, a n d t h e r e f o r e a n y p o i n t in G(x), h a s i n v a r i a n t s (~, fl)
3.5. Isotropy groups and the isotropy representation. (1)
P R O P O S I T I O N . Suppose V is a non-singular surface with graph F. Given an "arm"
of the graph
(3-o-o- -o-Q
/ f-
source sink
let X , be the curve corresponding to the i t h vertex on the arm, i.e. (Xt, X , ) = - b , a n d /et [b 1 .. . . . b,] =pJq,, where ( p , q t ) = 1.
(1) I / v E X , N X~+I, O~i<~r~ then we can choose coordinates (x, y) in T , so that the induced action of k* on Tv is o / t h e form
(t-~- 1 x, troy).
(~) For the terminology of this section, see the appendix.
ALGEBRAIC SURFACES WITH ~*-ACTION 59 (2)
I / v E X t + l is not a/ixed point, then the Sei/ert orbit invariants at v are
(~,/7),where
r and/ip~_l = - 1 (modp~),O < i < r - 1 .
(3)
Suppose b 1, b2, ..., bi>~2. I] the Sei/ert orbit invariants o/ a non-/ixed point veXi+ 1 are
(g,/7),then the equation
[b I ... b i ] = a / ( a - / 7 )determines the bi-uniquely. More- over
[b~, b~-i ... bl+~] =g//7.Proo/.
B y i n d u c t i o n on r. If r = 1 t h e n b~ = 0, p_~ = 0, P0 = 1 a n d p~ = 0. On t h e other h a n d X 0 = F +, X 1 is an o r d i n a r y principal orbit a n d X 2 = F - . T h e assertions are easily verified. N o w suppose the s t a t e m e n t s are true for an a r m with fewer t h a n r vertices. B y (A.3) a n d (2.5) one of t h e integers bj+ 1 m u s t be equal t o 1. L e t :t: V ~ V 0 be t h e m a p which collapses Xj+ 1. T h e n ~ m a p s Xf isomorphically o n t o a curveX'i
for i~=~+1. L e tp;lq~
= [hi . . . b j - 1, bj+2 - 1 . . . b~+~]as in t h e appendix. T h e n b y A.6
p'~
=p~,q'~ =q~
for i < 7 " - 1 a n d p~ =p~+l,q'~ =q~+~
for i >~7".N o w one can easily verify t h a t (1) holds for i#7", j + l a n d (2) holds for i # j . L e t v be the point of intersection of
X'~
a n d X'j+9. B y (1) we m a y choose coordinates (x, y) in Tv so t h a t the action of k* on Tv is of t h e formt(x, y) = (t r -~x, tCJy)
and
P~I-1 =Pt-1
a n dP's = ( b j -
1)p j_ 1 - P s - z = - P J - P ) - I =P/+I. N o w b y w 3.4, X~ has isotropy group /xa wherea=p's_l+p'j=pr
Again b y w there are coordinates at X j r / X j + I a n d Xj+lf~ Xj+~ so t h a t t h e a c t i o n is of t h e f o r m in (1). B y w (ii) we h a v e t h a t for a n y x E X j+l, the Seifert i n v a r i a n t s of x are (zr where PJ+I/7 = 1 (rood p j). N o w PJ+I=PJ - P j-1
so we get t h e desired result for (2). To verify (3) first note t h a t:r
since b o t h are between 0 a n d :r a n d ( ~ - / 7 ) p t _ l = l ( m o d p , ) b y (2) a n dqtp~_l=-I
( m o d p t ) b y A.1.T h e uniqueness when bl, ..., b~ >~ 2 is easily verified. Finally, t h e last e q u a t i o n follows f r o m considering t h e inverse action a n d using the a b o v e a r g u m e n t .
3.6.
Invariants o / a n elliptic singular point.
T h e d e t e r m i n a t i o n of t h e invariants of an affine v a r i e t y V with k*-action ( k = C ) , h a v i n g an elliptic singular point
v E V
was the m a i n object of [16]. T h e orbit space V* = V - {v}/k* of the action is a non-singular complete algebraic curve of genus g (i.e. a R i e m a n n surface if k = (3). I t has a finite n u m b e r of orbits with non-trivial isotropy a n d Seifert invariants (at, fit), i = 1 .... , r. There is a n additional integer i n v a r i a n t b, related to the " C h e r n class" of t h e Seifert bundleV - ( v } ~ V * .
W h e n k = ( ] t h e i n v a r i a n t s60 P . O R L I K A N D P . W A G R E I C H
{b; g; (a D ill) ... ( a , fir)} determine the neighborhood b o u n d a r y K of v up to orientation preserving S 1 equivariant diffeomorphism. The main result of [16] states t h a t the canonical equivariant resolution of the singularity has weighted graph
where the b t . j ~ 2 are obtained from the continued fractions a J ( a ~ - f l , ) = [ b t . 1 . . . b~.s,], i = 1 ... r.
Note t h a t the definition of b here is t h a t of [13] and differs from [16].
Thus in order to find the resolution of V at this fixed point we have to find all orbits with finite isotropy and obtain their Seifert invariants (at, fit) from the action of the isotropy group in the t a n g e n t space; the genus of the orbit space, and the integer b.
This was described in principle for hypersurfaces and complete intersections in [16, 18, 13]. If V = S p e c (A) is an affine v a r i e t y with k*-action, there is a k*-action on k n defined b y
t(zl . . . zn) = (tqlZl . . . tq'Zn)
and an embedding of V in k n so t h a t V is invariant under this action. If v E V is an elliptic fixed point we can choose the embedding so t h a t q , > 0 for all i. The ideal of functions I vanishing on V is generated b y polynomials which are homogeneous with respect to the grading given b y degree (X,)=qt. Thus we can choose g e n e r a t o r s / , for I so t h a t
l~(t ~ z l . . . t ~ z ~ ) = t d ' l ~ ( z l . . . z ~ ) .
ALGEBRAIC SURFACES W I T H ~*-ACTION 61 We s a y t h a t a polynomial satisfying the above is weighted homogeneous o/ degree d~
relative to the exponents ql ... qn. T h e rational n u m b e r s w~=dJq~ are called t h e weights of/~.
A n explicit c o m p u t a t i o n of the Seifert i n v a r i a n t s of an isolated elliptic singularity of a hypersurface in C a defined b y a single weighted h o m o g e n e o u s polynomial was given in [16, 18, 13]. U n f o r t u n a t e l y , there is an omission in [16], where it is claimed t h a t if t h e singularity of a weighted homogeneous polynomial in three variables is isolated, t h e n it is essentially one of six classes:
(i) z~ ~ + z[' + z2 ~',
(ii) z~ . . . + 2:1 + Z 12:2 , a~ > 1
(iii) z ~ ~ a l > 1 , a 2 > 1 ( i v ) Z~ ~ § a, + z l z 2 , a2 a o > 1
(V) Z~~ -~ Z~'Z 2 -~ ZoZ 2 , as
(vi) z~ ~ + z 1 z 2.
The correct description of weighted homogeneous polynomials in general [14] shows t h a t there are two m o r e classes t o consider (see also A r n o l d [1]):
(vii) z~ . . . +ZoZ 1 +z0z2 +z~'z~', (a o - 1 ) ( a l b 2 + a 2 b l ) / a o a l a ~ = l
(viii) z~~ +zoz~ ' +zoz~'+z~'z~" , (a o - 1 ) ( a l b ~ + a~bl)/a2(aoa 1 - 1 ) = 1.
We could a m e n d the discussion of [16, w 3] to give the Seifert i n v a r i a n t s of the corresponding classes. However, we now h a v e a new m e t h o d of o b t a i n i n g these Seifert i n v a r i a n t s directly/rom d and the q~, m a k i n g it unnecessary to list all t h e classes above.
F o r integers a 1 ... a, let <a 1 ... a~> d e n o t e their least c o m m o n multiple a n d (a 1 ... a,) their greatest c o m m o n divisor.
L e t /(z o, Zl, z~) be a weighted homogeneous polynomial whose locus is V in k 3. I n order for a point (z o, z 1, z~) E V to h a v e non-trivial isotropy, a t least one of the z~ m u s t be zero. T h e n u m b e r of orbits so t h a t s a y z 0 = 0 is equal to t h e n u m b e r of factors of t h e polynomial /(0, z 1, z2). The following l e m m a lets us d e t e r m i n e this n u m b e r easily.
LEMMA. Suppose g(z 1, z~) is a weighted homogeneous polynomial so that g(tQ~zl, tq~z2) = Vg(z 1, z2). Then the irreducible/actors g~ o/ g satis/y
gt(tq'zl, tq'z2)=t'~g~(zl,
z2) and each gt is one o/ the /ollowing /orms(i) g~(zl, z2) =ezl, e=4=O (ii) g~(zl, z~)=cz~_, c # O
(iii) gt(zl, z~) - ~ zv~+c 1 ~ 2 where elC2#O and P t = q J ( q l , q2).
I n particular, i/ z 1 and z 2 do not divide g, then g has r(ql, q~)/qlq2 =r/<ql, q2> /actors.
62 P . O R L I K A N D P . W A G R ~ I C H
Proo/.
The fact t h a t the g~ are weighted homogeneous is a general fact a b o u t factoriza- tion in graded rings. I f g~ is irreducible t h e n the v a r i e t y{9~(zi, z2)=0}
m u s t be the closure of an orbit of the actiont(z~, z~)
= (t q~ Zl, tq'z2). This action has the same orbits as the effective actiont(z~, z~)=(tv'zl, tv'z2).
The closures of the orbits are precisely {z~=O}, {z~=O} and(z~' + cz~'
= 0}, where c =V O.We a p p l y the t e m m a above to show t h a t in the case of isolated singularities the orbit invariants are determined b y d and the q~. L e t
wt=d/q~=uJv,
where g.c.d. ( u , v i ) = l and u , v~ >/1.P~OPOSXTIO~ 1.
Suppose V has an isolated singularity at O. Assume 1 <.vo <~v~ <v ~.
Then the table below indicates the number o/orbits o] each type.
II l~ II
1 = v 0 = v 1 = v 2 d/<q~, ~> d/<q o, q~> d/<q o, q~>
I =vo=v~<v., (d-qx)/<q~, q~> (d-qo)/< %, q~> d/<q o, q~> 1
1 = v o < v ~ <~v 2(d-ql-q~)/<qp q~> (d-qo)/<qo, q~> (d-qo)/<qo, q~> 1 1 1 <vo<<-v ~<v~ (d-ql-q~)/<qx, q~> (d-qo-q~)/(qo, q~> (d-qo-q~)/<%, q~> 1 1 1 The blank entries are zero i / q i does not divide qj /or i # i . I/q~lqJ, then
(q(,qj} =qt and we list those orbits under the column headed (ql, qJ).
Proot.
The exceptional orbits are in the hyperplanes z t = 0 . Supposev o = v l = v 2 = l .
The exceptional orbits contained in{zo=O )
are given b y {t(0, z,, z~)=0} We writee l e l $
](O, zl, z:)=z I
z,, 1-it=lgt(zl, z2) a product of irreducible factors.The
factors m u s t be distinct since otherwise the curvez~==gi(zl, z2)=0
would be a singular curve on V. By L e m m a 3.6; each 9t has weighted degreeqxq2/(qx,
q2)=<qx, q2> so we haveex ql + e2q~. + s<ql, q~> = d
(1)where e~=0 or I. I f
el=~=-O
t h e ns==d/<ql, q~.>
which is the desired result. If say e l = l then ql = 0 rood q2 so t h a t q~ = (q~, ql). One can easily verify t h a t the n u m b e r of orbits in (z o = 0) isd/q t =d/<ql, q~>.
Suppose 1
= % = v l < v 2.
Thenqo[d, q, ld
andq2[d.
I f we write[(O,z~,z~)
as a p r o d u c t of irreducible factors as above, then the equation (1) gives us e 1 ~ 0 rood q2, hence e~ = 1.Thus there m u s t be an orbit
{Zo=z~=O }
with isotropy q2>(ql, qa). I fe2=l
t h e n q~=0 (rood ql) so<ql, q~> =q~.
Thus the orbits contained in {zo---0 ) with isotropy precisely (ql, q~) correspond to the factors ofz~' I-I~-1 9~(zx, z2).
I f e~=0 thens=(d-qx)/<q 1, q~).
I f e~ = 1 then
s + 1 = (d-ql)/qz
which is the desired result.ALGEBRAIC SURFACES WITH ~*-ACTION 63 Suppose 1 ~ v 0 < v 1 ~<v2. T h e n again we h a v e t h e faetorization of [(0, gl, Z2) which gives us the e q u a t i o n (1). N o w
elq~ =-- d ~ 0 m o d q~
e, q2 =- d ~- 0 m o d ql
hence e l = e ~ = l . This gives us s = ( d - q l - q ~ ) / ( q ~ , q 2 ). T h e calculation of exceptional orbits c o n t a i n e d in {z~=O}, i = l , 2 is similar to those above.
P R O P O S I T I O N 2. Under the hypotheses o/ Proposition 1, the values o/ fl are computed as ]ollows.
2'or ~ = ( q l , q2), qofl =-1 rood ~, O < f l < a .
For an orbit o/the ]orm {zo=Zl=O} we have ~=q~. I n this case (i) i/there is an x E Z + so that
t h e n qlfl = - 1 (rood a)
(if) i/there is a y E Z + so that
then q o f l - 1 (mod a).
l + x 1, Wo w2
1 ~ Y = I ,
wl w2
Cyclic permutation o[ the indices 0, 1, 2 permits calculation o / a l l the required ft.
Proo/. F o r a = ( q l , q~) the action o f / l ~ on k a is ~(z0, Zl, z~)=(~q~ Zl, z2). This implies t h a t the action o f / ~ on the t a n g e n t plane, an affine subspace of k a, m u s t h a v e v ~ qo (mod a).
F o r an orbit of t h e f o r m {z o = z 1 = 0} we calculate t h e action of /1 a on t h e t a n g e n t space Tv, with v=(O, O, 1). T h e action of #a on k a is ~(z o, z I, z~)=(~qOZo, ~q~z 1, z~). T h e t a n g e n t plane at v is (a//~Zo)(V)Zo+(~//~zl)(V)Zl=0, since (a//~z~)(v)=O. I f (~]/~Zo)(v)~0 t h e n t h e a c t i o n of #~ on t h e t a n g e n t plane m u s t have v-=ql (rood a). If (all,z1) (v):t:O t h e n the action of/~a on the t a n g e n t plane m u s t h a v e v--qo (mod a). T h u s q l f l = l (rood a) in t h e f o r m e r case a n d qofl- 1 (rood a) in t h e latter. N o w if (~]]~zo) (v) :4:0 there m u s t be a monomial of t h e form ZoZ~ in [, hence (l/w0)+ (x/w2)= 1. If (~//~zl)(v):4:O t h e n there is a in ] so ( l / w 1 ) + (y/w~)= I. Finally, if b o t h ( I / w o ) + (x/w2)= 1 a n d m o n o m i a l of t h e form z 1 z2
(l/w1) +(y/we)=1 then we claim t h a t qo-ql (rood ~). F r o m t h e two equations we obtain x = (u o-vo)uJuov 2 hence u 0 - v 0 rood v2,
y = (u 1 - v l ) u J u l v 2 hence u 1 - v 1 rood v s.
64 P . O R L I K A N D P . W A G R E I C H
Also, u 0 a n d u 1 divide u 2 a n d d = < u o , ul, u2) so d = u 2 a n d ot=q2=v ~. B y definition qo = dvo/uo, ql = dVl/Ul so q0 ~ ql rood v ~ u 0v 1 = u 1 v 0 m o d v~ a n d the Iatter follows f r o m t h e a b o v e congruences.
COROLLARY. For an orbit o / t h e /orm { z 0 = z l = 0 } we have a=q~. I n this case
(i) i~
(if) i/
aJ-] (0, 0, 1 ) # 0
~ z 0
~!(o, o, 1).o
~ z o
then ql fl ~ 1 rood a.
then q0 fl = 1 m o d ~.
P R O P O S I T I O N 3. The /ormulas /or g and b are given by
2~, d~ d(qo, ql) d(ql, q2) d(q~, qo) + . . . (d, %) + (d, ql) _~ (d, q2) 1
qo ql q~ qo ql ql q2 q2 qo qo ql q~
q0ql q~ ~ a~
Proo]. A formula for t h e genus is given in [17, 5.3]. T h e proof there is valid u n d e r t h e hypotheses of this paper. T h e f o r m u l a reduces to the formula a b o v e once we verify t h a t for a n y hypersurfaee in k 3 with an elliptic isolated singular point, for each i either d = 0 (mod qt) or d=-qj (rood q~) a n d (qt, qk) = 1 for some j : # k # i , see [17, 5.4]. To verify this for i = 0 we first note t h a t (q,, qj)ld for i~=j because there m u s t be some m o n o m i a l in / which does n o t involve zk. Moreover there m u s t be come m o n o m i a l M - ~ ~ ~ - ~ o ~ ~z in / so t h a t a # 0 , i l + i ~ < l , since otherwise z l = z z = 0 would be a singular line on t h e surface.
If i 1 + i 2 = 0 we get d = 0 (mod q0) a n d if, say i 1 = 1 we get i o % + q x = d so d=qx (mod q0) a n d (%, q~) = 1. B y s y m m e t r y , we get t h e result for i = I, 2. T h e f o r m u l a for b was p r o v e n in [16, 3.6.1]. T h e proof there generalizes t o our case.
This completes t h e c o m p u t a t i o n of t h e weighted g r a p h of t h e resolution of V f r o m t h e integers d, q0, ql, q~. I n t h e case t h a t k = (3, {b; g; (~1, ill) ... (at, fir)} are t h e Seifert invariants of the n e i g h b o r h o o d b o u n d a r y K, a n d hence K is d e t e r m i n e d up to Sl-equi- v a r i a n t diffeomorphism b y these i n v a r i a n t s [13].
3.7. Non-elliptic singular points. Suppose v E V is a n o r m a l non-elliptic singular p o i n t on a v a r i e t y V with k*-aetion. I f zt: ~ V is t h e canonical e q u i v a r i a n t resolution ( = m i n i m a l resolution) of V, t h e n n o n e of t h e curves in zt-l(v) can be fixed curves, otherwise v would
A L G E B R A I C SURFACES W I T H ~*-ACTION 65 be elliptic. The graph F of the resolution m u s t be a connected subgraph of the graph in w 2.5, since 17 has a non-singular equivariant completion [23]. Thus P m u s t be of the form
L e t A (resp. B) be the orbit on l~ (and V) which is not contained in ~-l(v) so t h a t ,~I inter- sects the closure of the orbit corresponding to the first (resp. rth) vertex. The Seifert orbit invariants of the orbits, (al, ill) for A and (~2, f12) for B determine the n~ as follows.
With the notation in the appendix, al/fll=[mx, m 2 ... ms], m~>~2. L e t ms+a be the self intersection of ~ . B y w ~ / f l 2 = [ m l , m 2 .. . . . ms, ms+l, nl ... at]. Now n~>~2 and ms+l >~ 1, hence b y A.9 the n~ and ms+ 1 are uniquely determined.
4. Difleomorphlsm and algebraic classification
4.1. M i n i m a l models. A non-singular surface V with k*-action is said to be relatively minimal if there is no curve X c V so t h a t ( X . X ) = - 1 and X is isomorphic to p1. I f such a curve does exist it must be invariant under the action (1.8) and hence there is a contraction of X, z~: V ~ V 0 which is equivariant.
PROPOSITION. 1] V is a non-singular complete relatively m i n i m a l sur/ace with k*- action, then V is equivariantly isomorphic to one o/ the /ollowing
(i) a Pl-bundle o/ degree d over a complete curve o/ genus g (the degree is the sell- intersection of the /ixed curve F+), g > 0 .
(ii) a Pl-bundle E o/degree d~= + 1 over p1 with an action subordinate to the standard k* • k* action a 1 x a2 on V i.e. ffl acts in the /iber and a 2 acts on the base.
(iii) p2 with an action subordinate to the standard k * x k* action on V O'((t 1, t2), (Zo: Zl: 2:2) ) : (ZO: t12:l: t22:2).
P r o @ If there is no elliptic point on V then w 2.5, A.3 and the minimality of V imply t h a t Fv is of the form
Q Q
Hence V is a p1 bundle over a complete curve of genus q, and if g = 0 then c and - c ~ - 1.
5 - 772902 Acta mathernatica 138. lrnprim~ le 5 Mai 1977
66 P . O R L I K A N D P . W A G R E I C H
I f there is an elliptic point on V, then the non-singularity of V implies g = 0. B y a theorem of N a g a t a [6] a relatively minimal rational surface is isomorphic to p2 or a p1 bundle E~
of degree d ~ • 1 over p1. I t remains to show t h a t a n y k*-action on p2 or Ea can be extended to the standard k* x k*-action on P~ (resp. Ea). :Now a k*-action on V is equivalent to an injective algebraic group homomorphism k*-+Aut (V). I n our case Aut (P~)=
PGL (3, k) and Aut (Ea)=PGL(2) • Each of these (linear algebraic groups) has a maximal torus of dimension 2, i.e. every subgroup isomorphic to k* is contained in a subgroup isomorphic to k* x k* and a n y two subgroups isomorphic to k* • k* are conjugate [3, IV. II.3]. This implies the desired result.
4.2. Definition. Call a k*-action a on V essentially unique if the only other k*-action on V is ~(t, v)=a(t -1, v).
PROPOSITION. Suppose V is a non-singular sur]ace with k*.action ol. Then the ]ollowing are equivalent.
(i) g =0 and any/ixed curve intersects at most two invariant curves which have negative sell-intersection.
(ii) the action (~1 extends to an eHective k*• k*.action 01 • (~.
(iii) the action ol is not essentially unique.
Proo/. The group of automorphisms of a projective v a r i e t y Aut (V), is a reduced algebraic group [11 ], and a G-action on V is equivalent to a h o m o m o r p h i s m a: G-+ Aut (V).
Hence it is sufficient to show t h a t if (i) holds the maximal torus of Aut (V) is k* • k*, and if (i) does not hold then Aut o ( V ) = k*. Here Aut0(V) denotes the component of the identity in Aut (V).
We m a y assume V~=P ~. Then there is a birational equivariant morphism ]: V-+ V o so t h a t V0 is a p1 bundle over a curve X (4.1). B y an a r g u m e n t analogous to t h a t in w 1.9 we can show t h a t a n y element of Aut0(V ) leaves a curve with negative self-intersection invariant. Hence there is an injective homomorphism / . : Aut0(V)-+Aut0(V0). Moreover Aut ( V 0 ) = A u t (X) • Now if g > 0 , t h e n Aut (X) is finite a n d therefore A u t o ( V ) = Aut o (V0)=k*. I f g = 0 then a n y element of Aut0(V0) which is in the image of Aut0(V), leaves the points of X which lie on curves with negative self-intersection, fixed. Thus Aut o ( V ) = k* if there are more t h a n two such points, and Aut0(V ) has a m a x i m a l torus iso- morphic to k*x k* otherwise.
4.3. THEOREM. A complex algebraic sur/ace V with C* action is di//eomorphic to one, and only one, o/ the following
ALGEBRAIC SURFACES WITH k*-ACTION 67 (i) M g . k = ( R ~ x S 2 ) # k C P ~ g>tO, k~>0,
(ii) No, the non-trivial S 2 bundle over Rg g >~ 0, (iii) CP 2,
where Rg is a compact Riemann sur/ace o / g e n u s g and CP 2 denotes CP 2 with the opposite orientation.
Proo/. Blowing up a p o i n t on a non-singular complex surface is equivalent to t a k i n g connected s u m with CP 2. H e n c e t h e proposition implies t h a t V is diffeomorphic to W # k C P 2 where W = CP 2 or a CP i b u n d l e Ea of degree d over a R i e m a n n surface Rg.
N o w Ed ~ CP 2 is diffeomorphic to Ed+z # CP 2. I n fact, one can c o n s t r u c t an e q u i v a r i a n t diffcomorphism b y blowing u p a p o i n t on F + in E~ a n d a p o i n t on F - in E~+ 1. B o t h give us a surface with g r a p h
W e can conclude t h a t if W = E ~ a n d k > 0 t h e n V is diffeomorphic to Mo.k. N o w suppose V = E~. There are two S a bundles over a c o m p a c t R i e m a n n surface, t h e trivial one a n d a non.trivial one. T h u s Ed is diffeomorphic to E~ where 0 ~ < 1, is t h e residue of d m o d u l o 2.
E 0 is n o t diffeomorphic to E 1 since the former has an even q u a d r a t i c f o r m (intersection form) a n d t h e latter does not. Finally suppose V = C P ~ # k - C P 2. I t is well k n o w n t h a t CP 2 # CP 2 is diffeomorphie to t h e CP ~ bundle of degree 1 over CP 1.
T h u s V=Mo.k, if k~>2 a n d V = N o if k = l . T h e manifolds Mg.~, Ng a n d CP 2 can be shown to be distinct b y c o m p a r i n g t h e ranks of their h o m o l o g y groups, H z a n d He, a n d t h e p a r i t y of t h e q u a d r a t i c form given b y cup product.
4.4. SZ-actions on 4-mani/olds. I t is an open question w h e t h e r all S~-actions on CP ~ are orthogonal i.e. of t h e f o r m
t(Zo: z l : z~) = (t q~ z o: t '~' z l : t q' z2)
for t E S i. Orlik a n d R a y m o n d [15] classified T 2 = S i x S i actions on 4-manifolds a n d h a v e p r o v e n t h a t every s m o o t h T 2 action on CP ~ is orthogonal. T h u s we can rephrase t h e question to ask w h e t h e r e v e r y s m o o t h effective S z action e x t e n d s to a s m o o t h effective T~ action. W e h a v e the following related non-extension result.
THEOREM. I / k>~3 there is a smooth e//ective S i action on CP 2 ~kC-P 2 which does not extend to a smooth e//ective $ 1 • S 1 action.
68 P . O R L I K A N D P . W A G R E I C H
Proo]. I t follows f r o m [15, w 5] t h a t if t h e S 1 a c t i o n e x t e n d s t o a T ~ a c t i o n t h e g r a p h F V m u s t h a v e ~<2 a r m s . T h e surface V 1 w i t h g r a p h
is d i f f e o m o r p h i c to C p 2 # 4 C P 2 a n d h a s 3 arms, t h u s t h e C*-action i n d u c e s a n S l - a c t i o n 01 w h i c h d o e s n o t e x t e n d t o a T 2 a c t i o n . I f we collapse t h e c u r v e c o r r e s p o n d i n g to t h e l e f t h a n d v e r t e x we g e t a s u r f a c e V z d i f f e o m o r p h i c t o CP ~ # 3 C P 2. I f t h e S 1 a c t i o n 01 e x t e n d s t o o 1 • as, a2 m u s t l e a v e t h e f i x e d p o i n t s of o I f i x e d a n d h e n c e 01 • 02 e x t e n d s to V1.
T h i s is a c o n t r a d i c t i o n , hence t h e a c t i o n does n o t e x t e n d . W e can g e t n o n - e x t e n d a b l e a c t i o n s on C P ~ # k C P 2, k ~ 4 , b y b l o w i n g u p f i x e d p o i n t s on V 1.
W e d o n o t k n o w if such a c t i o n s c a n be o b t a i n e d for /c ~ 2 .
4.5. W e c a n n o w l i s t i n v a r i a n t s w h i c h g i v e a n e q u i v a r i a n t a l g e b r a i c c l a s s i f i c a t i o n of n o r m a l surfaces V w i t h k * - a c t i o n . L e t ~: l~-~ V be t h e c a n o n i c a l e q u i v a r i a n t r e s o l u t i o n of V a n d 1" ~, t h e g r a p h of l~. W e w o u l d like t o d e s c r i b e t h e t o p o l o g y of V u s i n g I ' ~ a n d s o m e a d d i t i o n a l d a t a . T h e m a p ~ i n d u c e s a n e q u i v a l e n c e r e l a t i o n in l~. T h e r e are t w o k i n d s of i d e n t i f i c a t i o n s ; c e r t a i n c u r v e s a r e i d e n t i f i e d t o p o i n t s a n d c e r t a i n curves a r e i d e n t i f i e d t o each other. I f a collection of c u r v e s is m a p p e d to a single p o i n t b y z we enclose t h e c o r r e s p o n d i n g v e r t i c e s of F b y a solid line. I f c u r v e s X 1 .. . . . X r on ~ a r e all m a p p e d o n t o t h e s a m e c u r v e Y in V a n d d~ is t h e d e g r e e of ~ ] X ~ t h e n we enclose t h e c o r r e s p o n d i n g vertices w i t h a d o t t e d line a n d i n d i c a t e t h e d e g r e e d~ a b o v e t h e v e r t e x . T h i s d i a g r a m is d e n o t e d b y Fv. See w 5 for e x a m p l e s .
S u p p o s e V is a c o m p l e t e n o n - s i n g u l a r surface w i t h k*-action a n d n o e l l i p t i c f i x e d p o i n t s , F v is t h e g r a p h of V, a n d t h e a r m s of F v are n u m b e r e d 1, 2 . . . r. D e f i n e t h e con- t r a c t i o n of V, ]: V - ~ c o n t (V), t o be t h e c o n t r a c t i o n of t h a t o r b i t X w h i c h lies on t h e r t h a r m , h a s a self-intersection - 1 a n d is t h e o r b i t closest t o F + h a v i n g t h e f i r s t t w o p r o - p e r t i e s (i.e. t h e v e r t e x c o r r e s p o n d i n g t o X is closest t o /+). I f we a p p l y t h e c o n t r a c t i o n
