FJ "rIC/95/312

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

ON RATIONALITY OF LOGARITHMICQ-HOMOLOGY PLANES - I

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

C.R. Pradeep

and

Anant R. Shastri

Ml RAM A RE-TRIESTE

. •— A , !»«••<•» ,»- —

IC/95/312

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON RATIONALITY OF LOGARITHMIC Q HOMOLOGYPLANES - I

C.R. PradeepDepartment of Mathematics, Indian Institute of Technology, Bombay, India

and

Anant R. Shastri1

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

September 1995

'Permanent address: Department of Mathematics, Indian Institute of Technology,Bombay, India,

1 Introduction

Let V be a normal surface denned over (E. Following [G-M], we say V is logarithmic if

all its singularities are of quotient type. It is called a Q-homology plane if its reduced

homology groups with rational coefficients all vanish. Let E = {pi,... ,p,} denote the set

of singularities of V. Then recall that the logarithmic Kodaria dimension of V is denned

to be the logarithmic Kodaira dimension of V \ E. We propose to write a sequence of

three articles, and prove the following theorem:

Theorem 1.1 AH logarithmic Q>-homology planes with logarithmic Kodaira dimension 2

are rational.

All logarithmic (?-homology planes with logarithmic Kodaira dimension < 1 are knownto be rational (see [G-M], [F], [M-S]). Combining this with the above, we obtain:

Theorem 1.2 All logarithmic Q>-homology planes are rational.

Intact, we propose to prove:

Proposition 1: Let X be a smooth pwjective surface defined over (E. Suppose there is areduced effective divisor A on X such that

i) the invducible components of A generate the Pic{X) ®Q>;

ii) Each connected component of A is simply connected;

Hi) K(X, K + A) = 2.Then X is a rational surface.

Givwi the logarithmic Cf-homology plane V as above, let ip : U —> V be the minimalresolution of singularities, S — i/>~'(£), X be a smooth minimal completion of U, andD = X \ U U S. Below we recall some basic facts about logarithmic (J-homology planesfrom [M-S].

Lemma 1,1 For a logarithmic Q-homology plane V, the following holds:(a) V is an. affine surface.(b) A smooth projective completion X of U can be chosen such that D has at worst

ordinary double point singularities, and minimal with respect to this property.

(c) All the connected components of D are simply connected and the irreducible compo-

nents of D generate PicX®Q). Also, X is simply connected.

(d) The irregularity and the geometric genus ofV (and hence that of X) vanish.

(c) The intersection form of D has exactly one positive eigen value.

Moreover, by the hypothesis in theorem 1, we have K(X, K + D) = 2. Therefore,theorem 1 follows from the Proposition, by taking A = D. Techniques involved in ourproof is very similar to the techniques of [G-S]. Proof in [G-S] heavily depends on an

tf

inequality of M. Miyaoka and the unimodularity of the adjacency matrix of the divisor at

infinity of the Z-hornology plane. In the situation of (?-homology planes and logarithmic<$-homology planes , the divisor at infinity need not be unimodular. As if to compensatefor this drawback, we now have a more generalized version of the Miyaoka-Yau typeinequality proved by R. Kobayashi for open algebraic surfaces (See [Kob]). [This wasbrought to the notice of the authors by Gurjar for which they are grateful to him.] Thisinequality plays a crucial role in our study. Below, we briefly describe the outline of theproof.

Assume that that X is not rational. Then X is either a surface of general typeor an elliptic surface. Starting with a reduced effective NC divisor D on X, we studythe contractions <J> : (X, D) —+ (Xc, Dt.) where (Xc, Dr) denotes the log-canonical model

for the pair (X, D). With the help of this study and Kobayashi's inequality, we derivean inequality involving the bark of the divisor D, the Betti numbers of X and D and(K.D) where K is the canonical divisor of X. The term (K.D) is estimated in [G-S] bystudying the contractions it : X —> X", where X" is the smooth minimal model for thefunction field of X. We reproduce this estimation for the sake of completeness of ourtreatment and for fixing up our notations. Using this estimation in the afore mentionedinequality, we get. an important auxiliary inequality involving certain integral parametersof the surface X and the divisor D. Of course, we are often interested in the case whenD = A, but sometimes we shall use the inequalities for other divisors aSso. From this stageonwards, our study is divided into two parts one, when X is a surface of general typeand two, when X is an elliptic surface. A detailed study of the inequalities gives a boundon the number of irreducible components in D and the self-intersection number for thecomponents appearing in D. Certain properties of these surfaces and hark computationsimpose restrictions on the admissible values for the parameters appearing in the auxiliaryinequality. We verify that geometric configurations predicted by this study violates theauxiliary inequality, thus proving the rationality of X and hence rationality of V.

One can say that the case when A (and hence D) is connected corresponds to the casewhen X is a smooth (P-homology plane. The additional number of connected componentsin D make life easier, and hence one can prove the following proposition with less efforts:

Proposition 2: With X and A as in Proposition 1, assume further that A is not con-nected. Then X is rational.

In the present article, we shall complete the proof of Proposition 2. The remainingcases of Proposition 1 will be taken up in two subsequent articles.

In section 2, we collect some basic definitions and results relevant to our analysis. Insection 3, we obtain an auxiliary inequality from Kobayashi's inequality. Another keyrole in our proof is played by a result due to Gurjar and Miyanishi tl it states that, a

logarithmic <?4iomology plane is strongly minimal. In particular, it does not containany contractible curve and all log exceptional curves for (X,D) are contained in D, Weshall see that this is applicable even in the slightly general situation of the Proposition 1also. Section 4 consists of certain general desprition of the geometry of A. The result inthese four sections will be used in the subsequent articles also. In section 5, the proof ofProposition 2 is completed.

2 Preliminaries

For the details in this section we refer to [F]. Let Y be a smooth projective surface. Asin [F], we call a (J-divisor F on V" pseudo-effective, if (H.F) > 0 for every ample divisor Hon V. The Zariski-Fujita decomposition of Ky + D, in ease Ky + D is pseudo-effective,is as follows:

(a) there exists H (^-effective decomposition of Ky + D. Ky + D ss P + N, where ~denotes numerical equivalence.

(b) P is numerically effective i.e., (PA) > 0 for atiy irreducible curve A.

(c) ;V = 0 or the intersection form on the irreducible components of iV is negativedefinite.

(d) {P Dt) = 0 for every irreducible component D, of N.

For a proof of the following, we refer to [M].

Theorem 2.1 Let (Y, D) be a normal completion of a quasi-projective surface Z such

that K(Z) > 0. Then KY + D is pseudo-effective and if E + N is the Zariski-Fujita

decom]x>sition of Ky + D, the following holds:

(a)K{Z) = 0 tffP~0.

(h) K(Z) = 1 iff £ / 0 and (P)2 = 0.(c) K(Z) = 2 iff (F)2 > Q.

In our study we need to estimate the value of [N)2, where N is the negative part ofa certain log-canonical divisor (Ky + D), For this, we make use of the theory of peeling

as developed by M. Miyanishi, S. Tsunoda, T. Fujita et al. (See [M-T] or [F]). For ourpurpose, the relevant definitions and results are found in sections 3 and 6 of [F]. This issummarized in section 10 of [G-S] which we reproduce below.

Let Y be a non-singular projective surface and D a reduced curve on Y, We shallassume that all the irreducible components of D are smooth rational curves and hence weshall drop the term 'rational' from Fujita's terminology. Recall that D is said to be NC(normal crossings) if all the components of D are smooth and D has at worst ordinarydouble points. By a (-n)-curve, we mean a non-singular rational curve Do with (Do)2 =

—n. We call D a MNC (minimal with normal crossings) if it is NC and blowing down of any

(-l)-curve in D disturbs the NC condition. We shall assume that D is a MNC-curve. Forany component DQ of D the branching number f}(D0) is denned by (D0.{D - DQ)) • We callDo a tip ifP(D0) = 1. A sequence F of components {Du •••, A-}, r > 1, is called a £u%ofZ?if/3(Di) = 1, p{D,) = 2and(DJ_,.DJ.) = 1 for 2 < j < r. We denote F by [IU,,...,™,.]where tu, = (A)2. We call F a maximal twig if there is a (unique) component Da of £>such that (Dr.Da) = 1 and j3{Da) > 3. Then Do is called the branching component of F.For any twig F we denote the curve D2 U ... U Dr by T\ By convention F = 0 if r = 1. Asequence F is called a club if j3(Di) = P(Dr) = I, 0(Dj) = 2, 2 < j < r - 1.

A connected component F of D is called a fork if

(a) F has a unique component DB with /?(A>) = 3.(b) r has three maximal twigs Tl, F2, F3 with Do as the branching component.(c) F is negative definite.

(d) rf(ri)"1 + d(F2}-] + rf(r3)-' > 1 where d(-) denotes the discriminant.

Now, assume that K + D is pseudo-effective, so that by 6.13 of [F], all clubs andmaximal twigs of D are negative definite. If F is a maximal twig of D, then bark of Fdenoted Bk(T) is an element M G (Q(T) such that {N,-Di) = {(K + D).D,) for 1 < i < r.

If T is a club of D then Bk{T) is an element N2 € <?(F) such that (N2.Di) ~ ((K + D).D,),

for all Di e F. For an isolated club T = {£>,}, we have Sfc(F) = 2(-(£, ) 2 )" I£]- Porany curve D we define Bk(D) = £ Bfc(Fj) where summation runs over all maximal twigsand clubs of D.

For a fork D we define thicker bark denoted Bk*(D), as an element N € <$(D) suchthat (N.Di) = ((K + D).Dt), for every irreducible component D, of D. Finally for any con-nected component A of D which is not a fork we define Dk"{A) = Bk(A). Following [G-S],we introduce the notation bk(D) (resp. bk*[D)) for the rational number (Bk(D).Bk(D))

(resp. (Bk'(D).Bkt(D))).

We will need the following result proved in [Zar].

Lemma 2.1 Let D be a contractible Q-divisor. Let F e Q)(D) and let (F.Dj) < 0 for

any irreducible component Dj of D. Then F is effective.

Following is a result proved in [F],

Lemma 2.2 Let D be a Q-divisor and T = [Du..., Dr] a twig of D. Let Ni = E « i A €

<?(F) be the bark ofT. Then n, = rf(r)Mr), nT = d(r)'\ (Ntf = - n , anrf 0 < m < 1

/or 1 < i < r,

Proof: By definition, we have JVi 6 (?(F) and (JVJ.DJ) = ((K + D).D,) = ({K + Dj +

D - Dj).Dj) = -2 + /?(Dj) < 0. Thus by lemma 2.1 above, we see that Nt is aneffective divisor. Also, -n, is the (IJ)"1 entry of the inverse matrix of the adjacencymatrix of T. Hence we get n, = d(T)/d(r) and nr = d(V)-\ That (W,)2 = -n, is clear

from the definition of Ar1. Clearly both ni and nT are less than 1. Now, assume that

M = MaXi{n,} > 1. Take the least i such that rij = M. Since 1 < i < r, we have

0 = (A',.A) = n-,-i + n,(Dt)2 +ni+1 < M(2 + (£>,)2) < 0. This contradiction proves that

M < 1. Thus we have proved the lemma. 4

Remark 2.1 The fact that bk(T) = d{T)/d(T) implies that Wfc(F) may be obtained bythe method of continued fractions. Henceforth we shall use this fact freely.

The following lemma is useful when estimating bk(D).

Lemma 2.3 Let T denote the set of twigs of D. If any of the following is a subset of F,

thenbk(D) < -1:

{[-2], [-2]}, {[-2], [-3], [-4]}, {[-3], [-3], [-3]}, {[-3], [-3], [-4], [-6]},{[-3], [-3], [-5], [-5]}, {[-3], [-2, ^2]}, {[-2, -2], [-3, -2]},{[—n], [n x —2]} where [n x —2] denotes a twig with n vertices each of weight -2.

Proof: Note that D has to have at least three tips. Further, in case D has exactly three

tips, all three can not be maximal twigs. Then, by lemma 2.2 we see that bk{D) < —I.

Observe that bfc([—3. -2]) = \ and bk([n x -2]) = ^ ^ - 7 . *

Remark 2.2 Increase in the weight of a vertex in a maximal twig (so that all weights arestill less than -1) reduces the value of bk(D). Hence the above lemma is valid for any setof such increased weights. In the sequel, lemma 2.3 will be put to use with this additionalsense.

Lemma 2.4 (a) Let {Li,..., L^} be the tips of a graph T with the weight of Ls being Wjthen bk(T) < EjL, 1M-(b) IfT is- a fork, then bk*(T) < bk{T). Moreover, bk*(T) < - 1 .

(c) Let. T = [Ti,..., Tr] be a club with weight of Tj being wt. If r = 1,2,3 or 4, then

bk(T) = 4/ai,, (wj +u>2 — 2)/(w\W2 — 1), w2(uil +w3)/(ui1u/2ii'3 — u>i — u>3} or (wiw2w3 +

tD2t(;3«)4 — ui\ — w2 — W3 — W4 — 2 ) / ( l — wiw2 — W3W4 — W1W4 + W1H12W3W4,) (resp.).

(d) IfT is the dual graph of a minimal resolution of a quotient singularity, then bk*(T) <

, where n = d(T) is the order of the. local fundamental group of the singularity

Proof: (a) First, let k > 2. i.e., T is not a club. Let F be a twig (with tip Li). By lemma2.2 we see that bk(T) = -d{T)/d(T). Thus

d(f)

and hence bk(T) < ^ 1/tu*.

Now, let k < 2. i.e., T is a club. If T consists of exactly one component, then bfc(T) =

7

and hence (a) holds. Hence, let T = [7\, . . . ,TT] with r > 1. As in the definitions above

let Ni = T,\Ti and 7V2 = £/i,!Ti- Then N2 = BfcfT), where aa .V, has the numerical

property of Bk(T) if T1 were a maximal twig with X, as its tip. Hence aga.n by lemma 2.2

above, we see that m = -(iV|)2 > l /^i- On the other hand we have ((JV2 - Ni).T:) < 0

for every component TI and hence by lemma 2.1, iV2 > N^. In particular fii > r^ > l/u>i.

By symmetry, we get fir > l/tur. Now ife(T) = (7V2)2 = ~{/Ji + ^2) < - ( l / " ' i + l/«v).

Hence we have proved (a) of the lemma.

(b) To prove (b), let T be a fork with the three maximal twigs Vj, F2 and P3 and let

D be the corresponding curve. Let N = Bk{T) and Nt = Bk(r,), 1 = 1,2,3. Let.

G = N - (Ni + N2 + JV3). Then we have (G.L) = 0 for every component L of £ F, and

(G.A,) = {(K + D).D0) - (fi(r,)-' + d ( r 2 ) - ' + ^ 3 ) - ' ) < 1 - 1 = 0. Hence G > 0 by

lemma 2.1 above. Now (A')2 = (A',)2 + (A'2)2 + (AT,)2 + (Of ami (G)'2 = (G.(nD0)) where

71 > 0 is the coefficient of Do in A'. Since {G.D0) < 0, it follows that (Gf < 0. Hence

bk*(D) = (A')2 < (N^2 + (Ay2 + (Ar3f < bk(D). The second part follows easily from

condition (d) in the definition of the fork. This proves (b) of the lemma.

(c) One can compute the barks directly from the definition.

(d) If T is a fork, then n > 6 and as seen above bk*{T) < - 1 . If T is an isolated tip,4

then bk"(T) = . Finally let T is a club. Then let us consider T as a maximal twig of a

larger tree with one of its leaves as the leader. Then bk*(T) < bk(T) = —-—- — —-—-.

Now, unless T — [—2, —2], by choosing the leader appropriately, we see that d(T) > 3. Of

course if T = [-2, -2] then we directly see that bk*(T) = —2, where as n = 3. Thus in3

all cases, we see that bk'(T) < . 4n

Importance of bark stems from the following crucial result due to Pujita (See Thm. ().2()of [F]).

Theorem 2.2 Let Y be a non-singular projective surface and D a MNC-curue on Y with

all its irreducible components rational. If K + D is pseudo-effective and K + D = P + N

its Zariski-Fujita decomposition, then N = Dk*(D) unless there exists a (-l)-ntrvr E not

in D satisfying one of the following:

(a) (D.E) = 0.

(b) (D.E) = 1 and E meets a component of Bk*(D).

(c) (D.E) = 2 and E meets precisely two components of D, one of which is a tip of a

dub of D.

Remark 2.3 As remarked in [G-S], even though the word precisely is not mentioned

in Fujita's theorem, it is obvious from the proof given there. We shall refer to these

conditions on (V, D) as 'Fupta's conditions. In application, when Y = X and D = A of

the Proposition, we observe that if by contracting all exceptioal curves violating Fujita's

condition, we can pass on to a surface pair (X,A) which satisfies all the conditions of the

proposition again. Therefore, without loss of generality, we can assume that the given pair

(X, A) itself satisfies Fujita's conditions. However, we note that, in carrying out these

operations, the number of connected components of A goes down by one whereas the

second bctti number goes up by one in comparison with that of the surface. (Alternatively,

indeed, using the results in [G-M], we shall see that Fujita's conditions are automatically

satisfied.) This observation is crucial, in obtaining simpler proof of the Proposition 2, as

claimed.

We will need the following important properties of D-dimensions. We refer to [I] for

definition of D-dimension and the proof of the following result.

Lemma 2.5 Let Y be a complete normal surface and let D,D\,.. ., Dr be divisors on Y.

(a) For any set of positivie integers n i ] , . . . , >nT, if n(Y, D{] > 0 for 1 < 1• < r, then

(b) If K(Y, D,) > I), then K(Y, D) < K(Y, D + A )

Now, we shall state an inequality due to R. Kobayashi. For the details we refer to [Kob].

We recall the definitions of log-canonical and log-terminal singularities of a pair (V, D).

Let (V, D.p) be a germ of a normal surface pair, i.e., (Y,p) is a germ of a normal surface

and D is a finite union of branch loci D = £ (1 — ^-)Dj where 2 < HJ; < 00 are integers

and each component D^ passes through the point p. Let ji : (Y, D, E) —* (V, D.p) be the

resolution of p such that D is a MNC curve and E is the exceptional set. Let E = LiEj

with Ej being the irreducible components of E. It is known that

pL*(KY + D) = Ky + T) + Yl ijEj

where (t} are defined by the equations (See [Mu]):

ifor each k. (1)

A germ of the normal surface pair (Y,D,p) is called log-canonical (resp. log- terminal)

singularity if n, < 1 for all j (resp. a} < 1 for all j and w; < 00 for all i). For a normal

surface pair (Y, D) with at worst log-canonical singularities, we write LCS(Y, D) for all

log-canonical singularities of (Y, D) which are not log-terminal.

Wo now recall the definition of log-minimal and log-canonical modeh for a pair (V, D),

whore Y is a projective surface with at worst log canonical singularities and D = S*(l ~~

^ ) D , (b 1 = 2 , 3 , . . .,00), a divisor on Y.

Definition: An irreducible curve E onY is a log-exceptional curve of the first kind (rcsp.

log-exceptional of the second kind) if E2 < 0 and {{KY + D).E) < 0 (resp. if E2 < 0 and

{(Ar + D).E) = OJ.

Given a smooth surface Y and a reduced effiective divisor D on V, by successive

contractions of log-exceptional curves of the first kind, we arrive at log-minimal model

(Ym, Dm). This pair is characterized by the following two properties:

(a) (Vm, Dm) is log-minimal, i.e., it contains no log-exceptional curve of the first kind;(b) there exists a bimeromorphic holomorphic mapping / : (V, D) —> (Km, Dm) such that

Dm = ft{D) and Ky + D = f*{Km + Dm) + !> ,£< , Of > 0 for all i, where Km is thecanonical divisor of Ym and £ = UEi is the exceptional set of / .

By contracting ali log-exceptional curves of the second kind in the log-minimal model(Ym, Dm) we arrive at log-canonical model (YC,DC). This pair is characterized by thefollowing two properties:

(a) (Yc, Dc) is log-canonical, i.e., it contains no log-exceptional curve of the first or secondkind;

(b) there exists a bimeromorphic holomorphic mapping g : (Ym, Dm) —> (Yc, Dc) suchthat Dc = g,(Dm) and g*(Kc + Dc) = Km + Dm where Kc denotes the canonical divisorof Yc.

It is known that log-canonical model of (Y, D) is unique if K(Y \ D) = 2 (See Fact 2,page 350 of [Kob]). Observe that, all the exceptional curves of / need not be characterisedby property in the above definition, on the surface Y itself, but could satisfy this propertyafter successive contractions.

In all our study we shall deal with only those divisors D such that K(X \ D) = 2.Hence we always have unique log-canonical model. The following result is proved byR. Kobayashi (See theorem 1 and theorem 2 of [Kob]).

Theorem 2.3 Let (Y, D) be a normal surface, pair with K(Y \ D) — 2. Suppose {Y, D)

has at worst log-canonical singularities. Let {Yc, Dc) be the log-canonical model for (Y, D)

where. Dc = Y.,{\-{,)DV is the image of Dc. Define YQ ~ Yc.\(\JWi^Dr,i)\LCS(Yc.,Dc)

and D°4 := DC|I n Ya. Then

cf < 3 jc(y&) E \r(p)\(2)

where, e( —) denotes the topological euler number, d{ is the number of singularities of

{YC,DC) lying over £>J?, and \V(p)\ is the order of the local fundamental group F(p) of a

log-terminal singular point p of (Xc, Dc).

In the application of this theorem, we always have a situation in which Y = X issimply connected, D is an (integral) reduced effective divisor, i.e., wt; = oo for all i. Thus

9

any singularity which lies in the image Dc of D is going to be LCS. Moreover, since we

begin with log projective surfaces, all singularities outside Dc are going to be log terminal.

Let {ph . .. ,ps) be the number of such singularities, and let yt denote the order of the

local fundamental group at p{. Then (2) reduces to:

(K c + DJ2 < 3 f (3)

Let us introduce the notation (h(~) denote the ith betti number of a topological space,and let 3, = bi(X),bi = bi(D). Since X is simply connected, we have, ft = ft = 0. Also,Ho = /?4 = 1. Hence, e(X) - 2 + P2. In order, to relate e(X), e(X^) and e(D), let us makethe following technical definition:

Definition: We say (V, D) is log-content if all the log-exceptional curves for the pair(V, B) are contained in B.

In what follows, we shall assume that (X, D) is log-content, are also quotient singulari-ties. (Since Y is assumed to irregularity q = 0, and hence by lemma4 and bi(D) = b\{Dc).

However, we shall number of D with that of Dr, we will

.saying that (V, D) is log-content.

Let now (V, D) be log-content. Let / be the number of connected components of Dc.

Then s + I = (jn. (This follows because, tacitly we are assuming that D is a MNC curve.)Let c be the number of irreducible components in the exceptional set. Then it follows thatb-i(Y,) = b-2(Y)-c.bl(Dc) = bub2(Dc) = bt{D)-c. Therefore, e(Yc\Dc) = e(Yc)-e(Dc) =

1 + {,% - c) + 1 - (( - bi + b2 - c) = 2 + (02 - b2) +bi-l. T h u s we may rewri te (3) as

[Kt + Dcf < 3(2+ {02 - b,) + h - &o + £,-I 1=1 Ti

(4)

We will need the following important result due to Sakai relating the Zariski-Fujitadecomposition and the log- minimal model (See [Sa]).

Lemma 2.6 Let f : (Y, D) —* (Ym,Dm) be the bimeromorphic holomorphic mapping

conducting the log-exceptional curves of the first kind. Then P = f"(Km + Dm) where

KY + D = P + N is the Zariski-Fujita decomposition of the log-canonical divisor Ky + D.

As a consequence of this lemma, we see that

10

Thus, if K + D = P + JV is the Zariski-Fujita decomposition of K + D, we may rewrite

0 < (P)2 = (Kc + Dcf < 3 12 + (02 - b?) + 6, - 60 + £ - (5)

Next, we compute (K + D)2. where D is any divisor on X with normal crossings

and all its components are rational. Infact, if D is connected and simply connected,

then it follows that, {(K + D).D)) = - 2 . FFrom this, we derive that if h(D) = (, then

K + D.D = —2 + 21. Taking sum over all the connected components, we have, in general,

K + D.D — 2(&i - bn). Therefore, using Nother's formula, and the simply connectedness

of X we have,

(K + Df = ft"2 + {K.D) + ((K + D).D) = 10 - & + 2(f>, - bQ) + (K.D). (G)

Using (K + D)2 = P2 + N2 also in the inequality (5), we get

Q < 10 - li-i + 2(6, - M + KI3 - {Nf < 3 (2 +

Introducing the notation, (N)2 + 3 ^ = ^ = V(D)> w e fic*-i

- i/ < 4ft - 3&2 + 6, - b0 - K.D - 4 (7)

3 Auxiliary inequality

As seen in the previous section, we have been lead to the problem of estimating K.D.

This is done in this section, more or less exactly as in [G-Sj, which we shall recall here,for the sake of completeness, and for fixing up the notation.

;,Prom now onwards we shall assume that X is smooth simply connected protectivesurface which is not rational and D is a MNC divisor on X satisfying Fujita's conditionsand such that (X, D) is log-content. Our aim here is to derive a number of auxiliaryinequalities, resulting from Kobayashi's inequality.

As an consequence of non-rationality of X we have the following result.

Lemma 3.1 (a) Any two (-l)-curves on X art; disjoint.

(b) For any irreducible component C of D. we have (C)2 < 0.(c) There, is at least one branching curve in D.

We will use this lemma tacitly throughout our study.Let n : X —> X" be a composition of contractions of —1 curves, where X" is the

smooth minimal model X" for X. Let D" := 7r(D). Let £• be the exceptional set for 7r.Write 7r = 4>m ° 4>m-\ •• • ° 4>\ where each < j is a contraction of the (-1) curve E}. Let•ip0 = Idx and V'j = <pj ° ' ' ' ° 4>i f°r j - l- By lemma 3.1(a), we see that, any two (-1)

11

curves in A' are disjoint. Hence we can arrange 0jS in such away that if TTJ = rj>n[ o- • -o^i,then(a) 0' : = JTi(O) has all the components still smooth and(b) for each j > n,, (Ej.C) > 2 for at. least, one component C of i/,'j_,(.D).

Let A"' = TTI(X), 7r2 := <t>n o • •• o 4>7n+\ '• X' —> X" and let £j be the exceptional set forTfi, i = 1,2. Write m = Tii + n2. Clearly ^(fj) = TJJ and bj(£) = rn = n,! -I- n2. Weintroduce some more notations. The integer /3(E3) := (i/'J_](D) — Ej).Ej is the branchingnumber of Ej w.r.t, i/'j-i(-"}-/3(Ci) := (V'j-ifC1;)) for any component C, of £ where j issuch thai. V'j-i(C{) is a (-1) curve.Rt := U{Lj € £i | /3(Lj) = •:}, i > 2.S := &i n Z/, n := b2(Ri),

Kj := rt! - 62(£i n D), and cr := n2 - T.t:^s(E'2 + 2)-

Now, let D = {Dr\, D' = {D^} and Z)" = {£>"}. Let {P,.;} be aii the singular pointsof D" including the infinitely near ones - and let the multiplicities at these be mti{> 2).We define

For 1 < j < TI|, let. now <jij contract ipJ_i(L]) where L3 C £•, n Ru for some i > . Thenclearly.

r ( ( ^ { D r ) ) S + 2 ) _ ic D

Here we take (Vvi(£>r))2 = 0 (resp (4>j{Dr))

2 = 0 ) if ipj-i(Dr) (resp. ^j(Dr) ) is a point.By the adjunction formula we have

-(K.D) = £(D? + 2) = 2)

and

2) = 2),

Now by repeated application of (S) for j = 1,... ,nj and the fact that n, = Ei>2rii

obtain

-(K.D) = -(!<'.D') - 52 lTi + b2{£i H D)

and hence,

On the other hand, by genus formula, for each t, we have,

';? + 2) + (D'l.K") = j : m « K ( - 1)

12

and since D't is the smooth model of D",

f 2 = <£?)' + 2 -

Recalling the definition of r and A we get

? + 2) = - T - 2n2 - A.

Since

we have,

Let us put

-(K'.D1) = + 2) = )2 + 2) + 2),

-(#.£>) = -A - T - a - e, -

Then we have,- (K.D) = -$- m.

Substituting this in (7) and noting that (}'{ := bj[X") = ft - m, we get,

— u < f1" - 4 — 8 + {62 — b2) + by — b0.

(9)

(10)

(11)

When we combine this with the inequaities obtained in the previous section, we will

get bound on 0. Since each term in the expression for 6 is non negative, in turn, we get

bound on each of these terras and some inter-relation. The idea is to show that, these

relations are not compatible and hence to arrive at a contradiction. This part of the proof

is quite a detailed case by case study. The following qualitative observations taken from

[G-S], aid us in this task considerably.

Lemma 3.2 (a) T.. each (-1)-curve E\ in S there exist D[ C D' such that (E[.D[) > 2

and some point r, € E[C\D\ such that either b^-ny1 fa)) > 2 and-n^lfa) contains a curve,

Li £ R-j U /?4 or 7r["'(XJ) = [Li] for some (-l)-curve Lj which is not m D. In particular,

if p:=number of (-1) curves in S, then p < rij.

(b) Ifa — ti then £? = S and S is a disjoint union of (-l)-curves and hence n2 = p < n,.

(c) Ifa^l then either

(i) £2 \ D' = {£J} and S is a disjoint union of (-1) curves or

(it) £2 = S consists of a disjoint union of (-1) curves and {E\,E'2} with (Ex)"2 =

- 1 , (E2)'2 = - 2 and (E[.E<2) = 1.

13

Proof: (a) Let E[ be any (-l)-curve in S. By definition of £2, there exists D[ C D1 with

(D[.E[) > 1. Since D is a system of divisors with normal crossings and E\ C D', it follows

that for all x 6 D[ n £J such that (£>•.£,')* > 2 we have the said property for irf '(XJ). If

(DJ.E;)^ = 1 for all x £ DJ P £J then D| U E[ is not simply connected. Since D is simply

connected, it follows that for some x € D'{ n EJ we have (DJ.£J)r > 2 and so we are done.

(h) By definition we have,

^2).

Since bi(S} < JIJ and ( £ ' ) 2 < - 1 for each B C 5, a = 0 implies that (E'f = - 1 for all

£* € 5 and 62(5) = n j . Hence t-2 = S and £2 is a disjoint union of (-l)-curves.

(c) If (7 = 1 then either b2(S) = rii — 1 or &2(S) = n^. In the former case, we further

have {E')2 = —1 for all £" e S as before. In the latter case £2 = S and all except one

curve, say E\. in 5 arc {-l)-curves and (JEJ) 2 = —2. The rest of the claim of the lemma

is obvious. A

We shall now prove:

L e m m a 3 .3 With (X, A) as in the Proposition, (X, A) is log content.

Proof : : In fact, in order to apply the results of [G-M], viz., lemma 4 and 6 directly, the

only trouble is that X \ A is not necessarily affine. However, what is needed instead is

that. A' \ A does not contain any complete curve and this follows from the hypothesis that

the irreducible components of A generate the Pic(X)®Q. If [X, A) were not. log-content,

then there will be a (-1) curve E on X not contained in A and such that E meets at most

two distinct connected components of A and that too transversally in single points. This

already violates Fujita's condition and hence the lemma. 4k

As an immediate corollary, we deduce the following results:

L e m m a 3.4 Let (A*, A) be as in lemma 1.1. Then

/H,(A) = /^.2) Components of A form a Q basis for Pic(X) 04 Q.3} The intersection form on A has exactly one positive etgen value.

4) The connected component of A that supports the positive eigen value has non-linear

dual graph (if X is not rational).

5) (X, A) satisfies Fujita 's condition.

6) A has at most two connected components.

Proof: Assume b2{A) > 0'2'. By the above lemma, we can apply inequality (5),

0 < P2 < 3 {2 + (h - h + ^ ~ b0 + J2~) <

14

T

Now, recall that 7, > 2 for all i and 0 < s < d0 — 1. Hence, we obtain,

U < o^l — UQ + \pu — L)/ZJ,

which is absurd. Therefore 1) follows, The the above inequality, now yields.

0 < P 2 < 3 J 2 + ;32-£>2 + &i- '> 0 + 5 2 - i < 3 | 2 - ( > [ ) + ] r - (13)

Hence 0 < 3(2 - be + (bu - l)/2) which implies that b0 < 2. This proves 6). Statements2), 3) 4) are all straight forward consequences of 1).

If (X, A) did not satisfy Fujita's condition, let there be a (-1) curve E as in the theorem2.2. Then contracting all such curves we still obtain a smooth surface pair (X. D) whichsatisfies the conditions of the Proposition 1, and b?{Di) > bi{X) cont.rdict.ing 1). Thisproves 5) 6

Corollary On a logarithmic (ty-homology plant V of Kodaira dimension 2, we ran have

at most one singularity.

Finally, for D = A, since bi = 0, ti2 = Ai, etc., from (11), we get,

0 < -v(A) < ,;%' -4-0-bo.

Since, the terms on the rhs are integers, it follows that,

0 < $ ' - 5 - 0 - bB.

(14)

(15)

Lemma 3.5 // equality is reached in (15), then X docs not contain any -1 curve E winch

intersects A in exactly two points transversely -

Proof: Since we have shown that {X,A) satisfies Fujita's condition, it follows that, such

a curve E will have to intersect the same connected component of A viz.. the one at

infinity. Hence, if we take D = A + E, then ba(D) = bo{A),b^D) - 1, and b2{D) = >%•

Moreover, it is easily seen that k(X, K + D) — 2, and {X, D) is log-content. Therefore,

we can appply (7) to (X, D) to obtain:

~u(D) < 4/32 - 4 - 3b2(D)

On the other hand,

K.D = A'.A + K.E =

- bo(D) - K.D.

Substituting this is the above inequality, we get ~u(D) < 0. But, since the connected

component of A at infinity should have at least three tips, the corresponding connected

component of D should have at least one tip. Hence -v{D) > 0. This contradiction

proves the lemma. 4

15

4 Some general results in the two cases

In this section we shall recall some basic facts about a smooth surface of general type,and also tmstablish some results when X happens to be an elliptic fibre space. Most ofthese can lie found in [G-S] but the proofs given there are not always valid any more inour situation.

Theorem 4.1 Let Y be a minimal surface of general type. Then the following holds:

(a) (A'y)2 > 0 and hence the second betti number 0(Y) < 9.(b) For any irreducible curve D on Y, we have (KyD) > 0 and (Ky.D) — 0 iff D is a

(•2}-ciirui'.

{(•) All the (-2)-curves on Y can be contracted to finitely many rational double point

singularities.

(d) The number of (-2)-curves is at moat equal to (p(Y) - 1), where p(Y) denotes the

Pi card number ofY.

A proof of (c) may be found in [Sa]. For the o ther resul ts we refer t o section 2, chap te r

VII of [B-P-V].

L e m m a 4 .1 / / £ (f. A, then X > 2.

P r o o f : For. amongs t t he curves contracted by TT, there is a t least one curve which is no t

in A a n d hence 6 2 (A") > /3J. On the other h a n d , a t mos t fj" — 1 curves on X" can b e —2

curves. Hence the lemma. 4k

In the rest of th i s section, we shall assume t h a t X is an elliptic: fibre space. So, let

<l> : X —> C be an elliptic fibration and <j>" : X" —* C be the corresponding min ima l ell iptic

t ibrat ion. Clearly C = Pl. Since X" is a minimal elliptic surface, we have, /?." = 10 a n d

hence (14) may be wr i t ten as

- i ^ < 6 - 0 - 6 o (16)

Since l b s is an integer and —v is positive ra t ional number , we see t h a t

6 < 5 - i (17)

Recall that an irreducible curve not contained in a fibre of ij> or (resp ij>") is called avertical component-otherwise, it is calledan horizontal component. Note that if A has nohorizontal components, then every component of A is in a fibre and hence the intersectionform of the adjacency matrix of A has no positive eigen value contradicting lemma 3.4.Hence A has at least one horizontal component, i.e., A > 1. We will need the followingresult about multiplicity of the multiple fibres, as a consequence of non-rationality of X,

The proof is exactly as in [G-S], and we reproduce it below for the sake of completeness.

16

Lemma 4.2 There are exactly two multiple fibres of ip" , viz., r«iP," and m^P^. Also

{mi,m2} = {2,3} or {2,5}. / / (K".H") < 2 for some horizontal component, then

{m,,m2} = {2,3}.

Proof: Let {n^P"^ be the multiple fibres of <j>". By the simply connectivity of X" itfollows that r < 2 and m\ and m2 are coprime. Since ps = <y = 0, we have the canonicalbundle formula

iy ' l ( m i - i ) „ . . . f p " l ( m r - l )

Thus if r < 1, it follows that \nK"\ ~ 0 for all n, and hence \nK\ = 0 for all n. But thenX is rational, a contradiction to our basic assumption. Hence r = 2.

Now, if P" denotes a general fibre of <£", we have the linear equivalence

P" ,

" - PS.and hence

K" ~ (m2 - l)Pj - P" ~ (m, -

Thus A'" is numerically equivalent to the ($-divisor mP" where ?ri = {m\m-i - rri\ —

mi)lm\m-2. Now let if" be a horizontal component in D". Then 1 < (K".H") =m(P".H"). Since rn, and m2 are coprime it follows that mirn2 — niy — rn2 divides(K".H"). On the other hand by (17), we have A < 4 and hence (K".H") < 4. Hence{mi,m-i} = {2,3} or {2,5}, and if {K".H") <2 for some H", then {mi,m2} = {2,3}, asclaimed. A

Now, we prove that A does not contain any fibre of <fi. First note that this is equivalentto say that A" does not contain any fibre of tj>". Also, the independence of the irreduciblecomponents of A implies that A cannot contain two or more fibres. In fhe sequel, we letP denote the fibre of <f> contained in A and P" the image of P under TV. Also, let H (resp.H") be a horizontal component in A (resp. A").

Lemma 4.3 There are. at least two horizontal components in A. In particular, A > 2 and

{rnum2} = {2,3}.

Proof: Assume that there is only one and denote it by H. Let H be an /i-fold section

for 0. Then as observed above we know that h is an integral multiple of m ^ ^ . Let

Si, 1 < i < k, be the simply connected fibres and T3, 1 < j < I be the fibres of with

bi(Tj) — 1. Since A is simply connected it follows that no Tj is completely contained in

A. Hence,

62(A) < £>(£ , ) + 2>2p)) - 1) + 1. (18)

17

On the other hand, by the well known addition formula for euler characteristic, we have,

2 + b-2(D) = e(X) = (19)

Putting these two together, we get fe + I < 3, Since irreducible components of A arelinearly independent, it follows that no two fibres of cf> are completely contained in A.Therefore, denoting the unique fibre contained in A by S\, we have,

1) + - 1 ) +1. (20)

Combining this with (19) yields, 2k + I < 4. Of course, we have k > 1, / > 0.

Now suppose k = 1. Then / > 1, for otherwise, e(X) = ^(Si), which is absurd.Let. now / = 1. Let C be the closure of T, \ A. Let A = 7\ U # = S U C, say, where, B

is the union of all the components of A not in C. Then e{A) = e(B) + e(C) — ?;, where, r)

is the number of points in B fl C- Now,

= C(X) = 2 + i2(A) = = 1 + e,(S\)

Hence, b2(C} = 1 + 62(r,) - b2{B) = 1 + fc^TO - e(Ti) + 1 = 2.It may happen that Ti is one of the two multiple fibres Pj. Assume for a while that

this is not so. Then observe that H may intersect T\ in atmost /; points, and hence,^i(^) 5! h- Hence, we have, e(A) > 1 — h + b2(A). Since, B is simply connected and hasat most two connected components, we have,

r.(C) - T) = e(A) - e.{B) > \ - h + b2(A) - (2 + b2(B)) = - 1 - h + b2(C).

Therefore, K(T, \ A) = e(C) - 7j > 1 - h. On the other hand, if P, denote the multiplefibres, then // intersects them in at most /i/m, distinct points, and hence it follows thate(P, \ A) = e(Pi \ H) > -ft/rrij. Consider the restricted fibration, <t>* • U —* (T. Amongstthe singular fibres of this, we have, T] \ A, Pj \ A, £ = 1,2, and possiblly some others.Thus 1 = e(U) >-h + (l~h + h) + (-h/rni + h) + (-h/in2 + h) > 2. This is absurd.

Now, if T] happens to be one of the multiple fibres, say 7\ = P], then H will intersectTt in at. most, /i/m, distinct points, and hence, we get e(C) — 17 > 1 — hJTH\. Hence,

1 = e(U) > -h + (1 - h/mx + h) + (-h/m2 + h) > 2,

which is absurd.Let us now consider the case, fc = 1, i = 2. Then arguing as above we first observe

that, exactly one cotnponent of T; will be outside A. If these are denoted by Cj then asbefore, it. follows that, e(T, \ A) = e(C, \ A) > — 1 — at + 1 = ai} where, a, is the numberof distinct points in which H may intersect 7). Once again, if none of the Tl is a multiplefibre, then we have, a; < h, and hence,

18

7

:*&.- .*::

Hence, equality holds everywhere. This, in particular implies that, h = 6, and <t>'

has no other singular fibres. Hence, H may have singularities only on Si. Further, since,S\ U H is contained in A which is simply connected, it follows that, Si n H is single pointand of course should be counted with multiplicity h = 6. Now consider the 6-fold covering4> : H —* P' • We have proved that this has exactly three ramification points, Xi with onepoint on xu two on i2 and three on X3, It easily follows that this is a normal (Galois)covering. This means that we have a group of order 6 with the following presentation:

(x, y,z;xR = yJ = z2 = xyz = 1).

But it is well-known that such a group is actually of infinite order. This eliminates thecase when k = 1, / = 2.

Finally let k = 2. Then I ~ 0. If S^ is the second fibre, then arguements similar to theabove cases show that only one component say C of S2

is n()t contained in A. Moreover, Cwill intersect A in at most /i+l distinct, points, ands hence, e(C\ A) > 2 —(ft + 1) = 1 - / i .For the fibration 4>* we thus have at least three singular fibres and hence,

1 = c(U) > -h +{l-h + h) + {-h/mi + h) + {-h/m2 + h) > 2.

This eliminates the last case also.Therefore, there must be at least, two horizontal components in A. A

5 The singular case

We shall complete the proof of the Prposition 2, in this section. As promised earlier,we shall deal with the singular case first and see that it is easier to handle. By theassumption, we have now &o = 2, and JJ = 1. From 15, we have,

6 < 02 ~ 7.

We shall make two subsections here to deal with the two major cases.

(21)

5.1 The general type case

In this subsection, we shall consider the case when X is a surface of general type. Then

the above inequality reduces to 0 < 2. Also, we know that A > 1. Clearly r; = 0,1 > 4.

While estimating u we should remember that A has tow connected components, one

corresponding to the resolution of the single quotient singularity.

19

1 • #*-*•-

Suppose A = 1. Then by lemma 4.1 it follows that £3A. In particular, eL = 0. Assumethat £ is 11011 empty, i.e., X is not minimal. Then, it is necessary, that r3 = 1, and0 + T - 0. Therefore, IT may consist of at most three contractions. It is not difficult tosee that —v>\, which is a contradiction.

On the other hand, if X = X" is minimal, then j3" = 8 or 9. In any case, the weightset for the dual graph of A consists of one -3 curve and all other -2 curves. Thus it is notdifficult to see again that — v > 2, again a contradiction.

If A = 2 then of course, ei = a = r = 7-3 = 0 and hence X = X" is minimal. Alsothe weight set. then consists of {—4, - 2 . . . , —2} or {-3, - 3 , - 2 . . . . -2} . In either case,we can easily sec that —v > 1 leading to a contradiction.

This show that X cannot be of general type.

5.2 The singular elliptic case

Here we have, 0 < '.i but also A > 2. As in the general type case, we must remember herethat A has two connected components and one of them corresponds to the resolution ofthe quotient singularity.

Let us consider the case A = 3 first. It follows that, X = X" is minimal and the weightis either { - 3 , - 3 , - 3 , - 2 , . . . , - 2 } or { - 4 , - 3 , - 2 , . . . , - 2 } . Remember that A = A" nowconsists of two connected components one of which has to have at least three tips. Henceit easily follows that u > 1 which is contradiction.

Let now A = 2. Then a + r + r3 = 0. If r3 = 1 then u + T = 0 and hence, it followsthat, 7T may be a contraction of at most three curves, as discussed below: Let Lo € R3

and let Li,i — 1, 2,3 be the components of A = A' which meet Lo-

a) If may happen that (L,)2 < - 3 and hence after contracting La we have arrived at A".Since no fibre of c> is completely contained in A, it follows that not. all of the LJJ are -2curves, say (Lj)2 = - 3 . Depending on whether one of the other (Li)2, the weight set ofA will now he equal to either

{-4, -4 , -X - 2 , . . . , -2 , -1} or {-4, - 3 , - 3 , - 2 , . . . , - 2 , - 1} .b) It. may happen that, say, (I1)2 = —2 and (Ls)

2 < —4, i = 2, 3. Then we have tocontract both Lo and then L\ to arrive at X". Observe that L\ is now necessarily a tipin A. Again it. folows that not both Li and L3 could be -2 curves. So, the weight set isnow, either

{ - 5 , - 5 , - 2 . . . , - 2 , - 2 } or {-5, -4 , - 3 , - 2 , . . . , - 2 , -1} . c) We may have (L,)2 = - 2and {L-i)1 = 3. Then, it follows that L\ is a tip. We claim that L2 is also a tip. Let us say,that, D,, 1 < i < s be the other components of A, which L2 intersects. So, after contractingLo, L, and L-2 successively, we arrive at the minimal surface X". ( For otherwise, it sholudbe that image of one of the £\ is now a -1 curve and if that is the case then r > 0.) Again,

20

since L\ is a rational curve with a single ordinary cuspidal double point singularity, itfollows that L" cannot be a fibre and hence has to be a horizontal component. Therefore,we must have (L3)

2 = - 1 . We have, (K.L3) = 5, K.L2 = 1, K.Li = 0,K.LB = - 1 . Also,we hace K.D, = K" .D'[ + 1 for each 1 < i < s, and we have K.D} = K".D] otherwise.Therefore, it follows that K.A = 5 + 1 - 1 + (A - 1) + s = 6 4- s. On the other hand,K.D = # + m = 3 = 3 = 6. Hence, s = 0. It follows that Li is also a tip and the weightset for A is {-7, - 3 , - 3 , - 2 , . . . , - 2 , - 1} .

In any of these cases, it is fairly easy to see that v > 1, which leads to a contradiction.

Therefore, we consider now the case: A = 2, r^ = 0. Suppose now that et = 1. Thenof course r3 = 0 and hence the lone component E of £i \ A has to intersect A in twotranseversal points. This contradicts the lemma 3.5. Therefore et = 0.

Suppose that a = 1. Then it follows that A = A', £-2 — {E} which has to intersectDelta in at least three points and also it should intersect a single component say D\ withmultiplicity two, i.o., E.D\ = 2. Let DU2 < i < s be the other components which E

intersects. Then since r = 0 it follows that E.Dl = 1, 2 < i < s. Then A". A = K".D" +

1, 2 < i < s. Also, K.Di = K".D" + 2, and K.D, = K".D" for all other components ofA. Therefore K.A = 2 + s + A = 4 + s. On the other hand, K.A = 0 + m = 3 + l = 4.Hence, we obtain s = 0 which is absurd, hence (7 = 0.

Thus we have shown that when A = 2, it must be that ei = r3 = a = 0. It followsthat r = 0 and A' = X" is minimal. Clearly the weight set is {-3, - 3 , - 2 . . . , - 2} . Wemust show that —v > 2 to arrive at a contradiction.

If the connected component A^ has four tips then this is easy to see. So, we havoto consider only the case when A,*, has precisely four tips. Observe that A,*, cannot, beconsisting of only -2 curves, for then it will be contained inside a fibre and hence cannothave a positive eigen value. Let L be the branching component. Then it follows that( i ) 2 = -2 (otherwise AK becomes negative definite!). Let 1 , i = 1,2,3 be the threebranches. Then, at most one of F, can be an isolated component, for if two of them areso, then it is easily seen that Aoo is negative definite.

So let F] be an isolated component. Remember that A^ can have at most, components.It follows that other F; should each have three components, and if one of them has threethen the third one should have four components, again to avoid negative definiteness.This leaves with exactly one component in S, which is a -2 or a -3 curve. From thisdescription, it is now easy to see that —v > 2.

Now all F; have at least two components. Let Fi have two components. Suppose F2

also has two components. Then it follows that Fg should have at least three components.Assume that F3 does have three components. Then we are Jeft with two componets in S.

If both these are -2 curves then it follows that — v > 2. So let 5 = [-2, —3]. Then at mosttwo of Fj can have a -3 curve. Hence we easily see that -v > 4/5 + 2/3 + 2/3 + 1/3 > 2.

21

If F3 has four components, then also, it is easy to see that -v > 2. Finally consider the

case when both T2 and T3 have three components. This leaves S to have one component.

One can check in a routine way that, in these cases also, -v > 2.

Thus we have eliminated the case A = 2. But then A = 3 and once again we must haveX = X" a minimal surface with the weight set for A equal tp either {-4, - 3 , - 2 , . . . , -2}°r { - 3 , —3, —3, —2,..., -2} . In either case, we have only to show that ~v > 1, whichseen fairly easily. This eliminates the case A = 3 also.

This completes the proof of the Proposition 2.

Acknowledgments

One of the authors (A.R.S.) would like to acknowledge the International Centre for The-

oretical Physics, Trieste, for hospitality during his visit as an Associate and where this

paper was prepared. C.R.P. is supported by the National Board for Higher Mathematics.

22

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