Japan. J. Math. Vol. 10, No. 2, 1984
Non-complete algebraic surfaces with logarithmic Kodaira dimension -oo and with non-connected boundaries at infinity
By Masayoshi MIYANISHI and Shuichiro TSUNODA
(Communicated by Prof. M. Nagata, October 7, 1983)
Introduction.
1. Let k be an algebraically closed field of characteristic p•†0. Let X
be an algebraic variety defined over k, we say that X is affine n-ruled if X
contians a Zariski open set isomorphic to UX Ak, where U is an algebraic
variety defined over k and Ak denotes the ane n-space over k. If X is affine
1-ruled, we simply say that X is affine-ruled. On the other hand, we say that
X is affine-uniruled if there exists a dominant quasi-finite morphism: YX
such that Y is affine-ruled, when X is complete and affine-ruled then X is
ruled,; if X is a nonsingular projective ruled surface then X is affine-ruled.
However, if X is not complete, the mine-ruledness implies a more refined
structure of X involving, indeed, the data on the boundary at infinity of X.
To be more precise, we assume hereafter that X is nonsingular, we
call a triple (V, D, X) a smooth completion of X if V is a nonsingular complete variety over k, D is a reduced effective divisor on V whose irreducible com
ponents are nonsingular and whose singularities are at worst normal crossing type and X=V-D. We then call D a reduced effective divisor with simple normal crossings. We say that X is resoluble if there exists a smooth
completion of X. This is, of course, the case if char k=0 or dim X=2. The logarithmic Kodaira dimension of a nonsingular resoluble variety X, i (X) by
the notation, is defined to be -oo if the linear system n(D+KV); is empty
for every n>0 or, otherwise, the supremum of dim (D+Ky)(V) when n
ranges over all positive integers with In (D+K )~~, where KV denotes the
canonical divisor of V and D, n (D+KV), denotes the rational mapping associated with In (D+KY)J . It is known that (X) is independent of the choice of a
smooth completion (V, D, X) of X; see Iitaka [7] and Kambayashi [8] for the relevant results on the logarithmic Kodaira dimension. Note that ,(X) for
Partly supported by the Grant-in-Aid for Scientific Research, The Ministry of Edu cation, Science and Culture, Japan. 196 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA a nonsingular complete variety coincides with the usual Kodaira dimension K(X). If there exists a dominant, quasi-finite, separable morphism f: YX between nonsingular, resoluble, algebraic varieties X and Y and if i-(Y)= oo, then we have ,(X)=-oo (cf. [7]). This is an analogy of a well-known classical fact for complete varieties. We now restrict ourselves in the category of algebraic surfaces over k. As a converse of the above fact and as an analogy of the Enriques criterion of ruledness for a nonsingular complete surface, we may ask whether X is affine-ruled if X is a nonsingular surface with ii (X)=-o®, we summarize in the following theorem the results obtained up to now on this question.
THEOREM0.1. Let X be a nonsingular algebraic surface defined over k with 1(X)=oo. Then X is af fine-ruled in the following cases: (1) X is irrational; see Miyanishi-Sugie [14] and Miyanishi [10] when char k=0, and Miyanishi [11] when char k>0; (2) X is rational and connected at infinity, i.e., D is connected if (V, D, X) is a smooth completion of X; see Fujita [4], [8], [10], [14], Russell [18] and Sugie [20].
However, if X is rational and not connected at infinity, X is not always affine-ruled (cf. [10] and [20]). All examples constructed so far have an A' fiber space structure over the projective line Pk, where A;< is the affine line Alkwith one point deleted off. In a subsequent paper [15], we shall prove the following:
THEOREM0.2. Let X be a nonsingular algebraic rational surface defined over k with X)=-00. Assume that char k=0, that X is not a ffine-ruled and that, for a smooth completion (V, D, X) of X, the intersection matrix of D, i.e., the matrix ((C0. C~))1<0,~ x2+y3+z5=0 2. In order to investigate the properties of a non-complete nonsingular algebraic surface X defined over k, it is essential to consider a smooth com pletion (V, D, X) of X and look into a geometry of this triple. Hence we may as well start with a fixed triple (V, D, X). We say that (V, D, X) is a non singular triple if it is a smooth completion of X. A morphism of nonsingular triples (V1, D1, X1) *(V29 D2, X2) is a morphism f: V1V2 such that f-1(Supp D2) C Supp D1; then f (X1)cX2. Given a nonsingular triple (V, D, X), if (X)~0, i.e., In(D+KV)~c1 for some n>0, the divisor D+K~ has the so-called Zariski decomposition D+Ky=(D+Ky)++(D+Ky)-, where (D+KV)+ is a numerically (or arithmetically) effective Q-divisor, (D+KV) is an effective Q-divisor whose support has a negative-definite inter section matrix (cf. the notation and the terminology below) and ((D+Kv)+.CZ) =0 for every irreducible component Ci of the support of (D+K v)-(cf. Zariski [23], Fujita [4] and Miyanishi [10]). When k(X)•†0, Kawamata [9] has shown that there exist a nonsingular projective surface Vm, a birational morphism f: VVm and an effective Q-divisor Dm on Vm such that (D+KV)+= f*(Dm+K~m) provided char k=0. By making use of this decomposition, Kawamata succeeded in constructing a classification theory of noncomplete algebraic surfaces over an algebraically closed field of charactristic zero which is essentially parallel with the classification of complete algebraic surfaces; see Miyanishi [10] for a detailed account of Kawamata's theory. Improving on Kawamata's arguments for constructing Vm and Dm, Fujita [5] and the second author [21] have later independently found a concrete way to reach Vm and Dm as above which depends on an introduction of the notion of peeling. We shall, in the first chapter of the present article, give a full explanation of the theory of peeling. Indeed, since the peeling process does not utilize the hypothesis k (X)•†0, it can be applied as well to a nonsingular triple (V, D, X) with (X)=-00. The theory of peeling consists, roughly speaking, of the following processes: (1) Let (V, D, X) be a nonsingular triple. Write D=~2=1 Ci as a sum of irreducible components. We can then find in a canonical way rational numbers ai (0•…ai•…1; 1•…i•…r) satisfying the following condition: With D#:=~?=1aiC2 9 we have (D#+Kv•ECi•†0 for every component C1 except those called the irrelevant components of D (cf .1.4.2 below). 198 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA We set Bk (D)=D-D# and call it the bark of D. The peeling of PD is the process of finding Bk (D) and eliminating it from D. (2) If E is an irreducible curve on V such that (D#+KV• E)<0 and the intersection matrix of E+Bk (D) is negative-definite, then E is an exceptional curve of the first kind. For such an exceptional curve E, we can find a birational morphism f: VV such that, with D: =f*(D) and X:=V-D, f gives rise to a morphism of nonsingular triples (V, D+E, X)(V, D, X) and f contracts the curve E to a point. In these respects, we say that a nonsingular triple (V, D, X) (or (V, D)) is almost minimal if, every irreducible curve C on V, either (D#+KV•EC)•†0 or (D#+KV• C)<0 and the intersection matrix of C+Bk(D) is not negative definite. A nonsingular algebraic surface X is said to be almost minimal if X has an almost minimal smooth completion (V, D, X). If X is complete, X is almost minimal if and only if X is relatively minimal in the usual sense. We have the following result: THEOREM 0.4. Let V be a nonsingular projective surface defined over k and let D be a reduced effective divisor with simple normal crossings. Then there exists a birational morphism i: VV onto a nonsingular projective surface V such that, with D:=p*(D), the following conditions are satisfied: (1) D is a reduced effective divisor with simple normal crossings; (2) h0(V, n(D+Kg))=h°(V, n(D+Kr,)) for every integer n•†0; (3) ~c*Bk(D)CBk(D) and i(D#+KV)•†D#+KV; (4) (V, D) is almost minimal. With the above notations, (V, D) (rather X:=V-D) is called an almost minimal model of (V, D) (rather X: =V-D). Note that ic (X)=(X). If D#+KV is numerically effective, i.e., (D#+KV• C)•†0 for every irreducible curve C on V, then K(X)•†0 (cf. Theorem 1.12 below). Conversely, if (V, D, X ) is almost minimal, D+KV is numerically effective provided i(X)•†0 (cf. Remark 1.12). Suppose i(X)0. Then we can show that the numerically effective part (D+KV)+ in the Zariski decomposition D+Kv=(D+KV)++ (D+KV) coincides with p*(D+KV). Namely, we can take V and D# as Vm and Dm, respectively (cf. [21] and Remark 1.12 below). 3. In the second chapter of this article, we consider an almost minimal nonsingular triple (V, D, X) with k (X)=-oo. By the theory of peeling, we know that each connected component of Bk(D) is a twig, a rod or a fork which is admissible and rational (cf. 1.3 for the definitions). Thanks to Artin's theory of rational singularities, there then exist a normal projective surface V and a birational morphism f: VV such that f is the contraction of all connected components of Bk (D) to finitely many normal points on V Non-complete algebraic surfaces 199 whose singularities are at worst "quotient singularities". Moreover, there exists an integer N>0 such that, for every Weil divisor G on V, NG is linearly equivalent to a Cartier divisor. Hence we have the intersection theory for Weil divisors on V though the intersection number of two integral Weil divisors might be a rational number. Let D=f*(D#), which is an integral Weil divisor, and let KV be the canonical divisor . Then we can show that D#+Kv is numerically equivalent to f*(D+Kv) (cf. Lemma 2.5). Hence D+KV is not numerically effective. On the other hand, let NS(V)Q be the Q-vector space of all Q-divisors on V modulo numerical equivalence and let N(V)=NS(V)Q_??_R, on which we consider a metric topology defined by some norm. Let NE(V)Q be the smallest convex cone in NS(V)Q containing all irreducible curves on V and closed under multiplication of elements of QT, which we assume to contain the zero. Define the cone NE(V)R in N(V) in a similar fashion, and let NE(V) be the closure of NE(V)R in N(V). Let L be an ample Cartier divisor on V, and set where s is a small positive number. Then we can prove the following modi fied version of the Mori theory (cf. Mori [17]): THEOREM 0.5 (Lemma 2.6 below). For every positive number; there exist possibly singular rational curves i, l•…i•…u, such that and This result implies that the side D+KY+~L<0 of the cone NE(V) is polyhedral. Hence we may think of extremal rays of NE(V) and extremal rational curves on V (cf. Mori [17]). Let l be an extremal rational curve on V which satisfies, by the definition, the conditions that (l.D+KV)<0 and that R+[l] is an extremal ray. Let l be the proper transform of l on V. Then one of the following two cases takes place (cf. Lemma 2.7): (1) The intersection matrix of l+Bk (D) is negative-semidefinite, but not negative-definite; moreover, (l2)=0. (2) The Picard number p(V) of V equals 1, and (D+KV) is ample. In the first case, the linear system |nNf*(l)| , for a sufficiently large integer n, is composed of an irreducible pencil , free from base points, whose general members are isomorphic to Pk. Therefore V is a ruled surface. Moreover, it turns out that X:=V-D is mine-ruled . In the second case, 200 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA - (D#+Kv) is numerically effective and, for an irreducible curve C on V, (D#+Kv.C)=0 if and only if C is an irreducible component of Bk (D). More over, if one writes Supp Bk(D)=U2=1 Di, then D-~2 =1 D, is connected and has at most two irreducible components which are nonsingular rational curves. In these circumstances, X:=V-D is not afne-ruled only in one of the next two cases: (2)1 Do:=D~i=1DZ is irreducible, and the connected component of D containing Do is a non-admissible rational fork with Do as the central compo nent; (2)2 Supp D=Supp Bk (D). 4. A main theme of a subsequent paper [15] is to study the case (2)1 above and to prove the following: THEOREM 0.6. Let X be a nonsingular algebraic surface defined over k with ,(X)=-co. Assume that char k is zero, that X is almost minimal and not affine-ruled and that, for a smooth completion (V, D, X) o f X, the inter section matrix of D is not negative-definite. Then the surface X possesses a structure of an A'-fiber space over Pk which has no singular fibers except three multiple fibers f=pi C2, 1•…i•…3, such that Ci is isomorphic to A and {pi, P2, P3} is, up to a permutation, one of the triplets {2, 2, n} (n•†2), {2, 3, 3}, {2, 3, 4} and {2, 3, 5}, where A is the affine line Ak with one point deleted off. Moreover, there exists a finite etale Galois covering c: XX of degree dividing 60 such that X is affine-ruled. The afore-mentioned theorem 0.2 follows immediately from this result . It is most probable that the first part of Theorem 0.6 holds in any character istic and the second part does under the hypothesis char k•‚2, 3, 5. The case (2)2 will be treated in our forthcoming paper [16]. The results treated in this article as well as the related ones were presented, partially and with a brief sketch of the proof, in Tsunoda-Miyanishi [22] and Miyanishi [12]. In com pleting this work, the authors have been constantly influenced by the works of Professors T. Fujita, S. Iitaka, Y. Kawamata and S. Mori, to whom the both authors would like to express their sincere gratitude. 5. The notations and the conventions Notations and conventions of the present article conform to the general current practice. Therefore we shall make some additional notes below. Let k be an algebraically closed field of arbitrary characteristic p•†0 which we fix as the ground field throughout this work; we assume char k=0 in the second paper [15]. The affine space and the projective space of Non-complete algebraic surfaces 201 dimension n defined over k are denoted by Ak (or An) and Pk (or Pn), respec tively. We denote by A, a k-scheme obtained from A1k by deleting one k-rational point. We denote by R+ (or Q+) the set of all non-negative real (or rational) numbers. Let V be a nonsingular projective surface defined over k. Then we use the following notations: KV: the canonical divisor of V; we denote it also by K if there is no fear of confusion. k(V): the Kodaira dimension of V. x(V, F) (or x(F)): the Euler-Poincare characteristic of a coherent sheaf F on V. h1(V, F):=dim Hi(V, F) for F as above; if D is a divisor on V, we denote H1(V, (9(D)) and hi (V, (9(D)) simply by H1(V, D) and h1(V, D), re spectively. Let D and D' be divisors on V. Then we use the following notations: (D. D'): the intersection number of D and D'. (D2) (or D2): the self intersection number of D. These notations are used extensively also for Q-divisors, i.e., finite sums of irreducible curves with coefficients in Q. D•`D': D is linearly equivalent to D'; for the algebraic equivalence and the numerical equivalence, we employ the notations D~ D' and D=D', respectively. D•†0: D is an effective divisor; D>0 if D•†0 and D!=0; thus D•†D' if and only if D-D'•†0. | D|: the complete linear system defined by D. Q: the rational mapping VPdim A defined by a linear system A provided dim A•†0. Supp (D): the underlying curve of D if D>0; more generally, if D= ~i=1aiDi with a1 e Q+ and C, irreducible, Supp (D)=Ui=1Ci. pa(D): the arithmetic genus of D, i.e., 2p a(D)-2=(D. D+Kr). By abuse of the terminology, a linear pencil whose general members are irreducible nonsingular rational curves is called an irreducible rational pencil. Let C be an irreducible curve on V and let P be a point on C. Then multp C denotes the multiplicity of C at P. If D=~i=1aiC1>0 them multP D= ~i=1ai multP Ci. For irreducible curves C and C', i(C, C'; P) denotes the local intersection multiplicity of C and C' at P. If D is reduced effective divisor on V, we denote, by abuse of the notation, V Supp (D) (the comple ment of Supp (D) in V) by V D. If D is a Q-divisor, [D] denotes the largest integral divisor with D [D]•†0. On the other hand, if D=~i =1aiC1 is an effective Q-divisor, we simply call the matrix ((C,. C~)1•…i ,j•…r) the intersection matrix of D. 202 MASAYOSHI MIYANISHI and SHUICHIRO TSUNOBA Let f: WV be a birational morphism of nonsingular projective sur faces. Then we use the following notations: f*(D): the total transform (or the inverse image as a cycle) of a divisor D on V. f'(D): the proper transform of D as above by the birational mapping f-1. f-1(C): the set-theoretical inverse image of an irreducible curve C on V. f*(A): the total transform of a linear pencil A on V by f, i.e., f *(A) is the pencil on W consisting of all f*D, where D e A. f'A: the proper transform of A as above by f-1, i,e., f'A=f*A-(the fixed part). f*D': the direct image of a divisor D' on W by f. f*A': the direct image of a linear pencil A' on W, i.e., f*A' consists of all f*D' when D' runs over all D' e A'. These notations are used extensively as well for Q-divisors and for Cartier divisors on V when f: WV is a finite morphism of nonsingular projective surfaces or a birational morphism of normal projective surfaces , of course, provided they have the right meanings. Let f: VC be a surjective morphism of a nonsingular algebraic surface onto a nonsingular curve. For a point P of C, f*(P) denotes the fiber over P, which is the complete inverse image of P by f, while f-1(P) denotes the set-theoretic inverse image of P by f, which is the underlying curve of f*(P) . Let X be a non-complete nonsingular algebraic surface defined over k. Then k(X) denotes the logarithmic Kodaira dimension of X. 6. The plan of the present article is as follows: Introduction. Chapter 1. The theory of peeling. Chapter 2. Afline-ruledness of an almost minimal algebraic surface with the logarithmic Kodaira dimension -oo. References. Chapter 1. The theory of peeling In this chapter, we develop the theory of peeling along the lines set by the pioneering works of Fujita [5] and Tsunoda [21]. A central result is Theorem 1.11 which asserts the existence of an almost minimal model . The terms "rod" and "fork" which we define below correspond to "club" and "abnormal club" , respectively, in Fujita [5]. 1.1. Let P be a weighted graph, that is, P is a one-dimensional finite simplicial complex, each vertex v being assigned an integer wv, called the weight; one-dimensional simplex is called an edge (or a segment) . For each Non-complete algebraic surfaces 203 vertex v, j3(v) is the number of edges connecting v to other vertices, called the branching number of v. Apparently, 48(v)=0 if and only if v is isolated. If j9(v)=1, the vertex v is called a tip of P. Given vertices v and v', we denote by (v, v') the number of edges connecting v and v'. Obviously, the numbers jS(v) and (v, v') depend on F, so if there is a possibility of confusion, we denote them by jS(v) and (v, v'). A subgraph I" of P with vertices {v1,•c, vr} is called a linear chain if jr(vl)=1, /3r(v1)=2 and (vi-1, v2)r=(v2, v+1)=1 for 2 If Pr(vr)=1, then such a linear chain I" is called a rod. Clearly, a rod is a connected component of P. If p(v)•†2, F' is called a twig. Furthermore, if j9(v1)3, then F' is called a maximal twig. (rod) (maximal twig) Let {v1,•c, vs} be the set of all vertices of F with the vertices suitably labelled. Let Q(F) be the s-dimensional Q-vector space consisting of all formal linear combinations of vi,•c, vs with coefficients in Q. Then Q(F) is naturally endowed with a symmetric bilinear form I defined by I (vi, vj)= (vi, vj)r if i•‚j and I (vi, vi)=wvi, The determinant of the (sXs)-matrix which has I (vi, vj) as the (i, j)-th entry is called the determinant of F and denoted by d (F). Let v be a vertex of F. We shall define two new weighted graphs from F: (1) Add one more vertex vo with weight -1 to F, connect vo and v by a single edge so that (v0, v)=1 and (v0, v')=0 for v' v, and change the weight wv of v to w, -1. Thus we obtain a new weighted graph A Then it is easy to see that d(P)=d(F). The process of obtaining f' from F is called the sprouting blowing up of F with center v. (2) Suppose (v, v')•†1 with another vertex v'. Choose an edge b con necting v and v', add one more vertex vo with weight -1 to F to subdivide the edge b by vo so that (v, v0)=(v', v0)=1 and (v0, v")=0 for v"•‚v, v', and change the weights wv, wv, to wv-1, wv,-1, respectively. Thus we obtain a new weighted graph P. Then we have d(t)=d(F). This process of obtain ing P from r is called the subdivisional blowing-up of F with center b. The inverse of any one of the above processes is called the contraction of the vertex v0. 204 MASAYOSHI MIYANISm and SHUICHIRO TSUNODA 1.2. Let V be a nonsingular projective surface defined over k and let D' be a reduced effective divisor on V with simple normal crossings. Write D =D1+•c+Du, where D's are irreducible components of D. Then we can define the dual weighted graph f'(D) of D by assigning a vertex v2 to each component A and connecting vertices vi and v1 (i•‚j) by as many edges as the points of intersection D, (1D,; the weight w1 of vZ is the self intersection number (Di). A partial sum of irreducible components of D, say D1+•c+ Dr after a suitable relabelling, is called a rational linear chain (or a rational rod, or a rational twig, resp.) if the corresponding subgraph {v1,•c, vr} of P(D) is a linear chain (or a rod, or a twig, resp.) and every component D1, 1•…i•…r, is a nonsingular rational curve. Since we consider only rational linear chains, rational rods and rational twigs in the forthcoming situations, we often drop the adjective "rational". Obviously, a sprouting (or subdivisional, resp.) blowing-up of P(D) corresponds to the blowing-up of V with center at a point on a component, say D1, which does not lie on any other components (resp. at a point of intersection of two components, say A and Dj). In the following arguments, we consider only the weighted graphs associated with divisors D of the above-mentioned kind. 1.3. Let P be a weighted graph and let L=[a1,•c, ar] be a linear chain of F, where a1 corresponds to a tip of P. For 1•…i•…r, we set Li=[a1,•c, ai-1] and Li=[ai+1, •c, ar], where L1=Lr=c and Lr+1=Lo=L. As in 1.1, d(L) denotes the determinant of L. We set d(L1)= d(Lr)=1. On the other hand, we say that a linear chain L is admissible if a1•†2 for every i. 1.3.1. LEMMA. Suppose L is admissible. Then we have: (1) d(Li-1)=a1d(L1)-d(Li+1) for 1 We express this continued fraction, as usual, as [a1,•c, ar], thus identifying it with the linear chain (cf. the assertion (3) below). (2) d(Li+1)=a1d(Li)-d(Li-1) for 2•…i•…r, d(L2)=a1, d(L1)>d(L1)>O for 1•…i•…r, and GCD (d(LT+1), d(LT))=1. Moreover we have Non-complete algebraic surfaces 205 (3) There is a one-to-one correspondence, given by between the set of linear chains and the set of rational numbers q with 0 The number d(L1)/d(L) is called the inductance of a linear chain L, denote by e(L). The proof is straightforward, so it is omitted. 1.3.2. Let L={Dl,•c, Dr} be a linear chain of nonsingular rational curves on a nonsingular projective surface V with ai=-(Di)•†2, 1•…i•…r. Write d/e=[al,•c, ar], where d and e are positive integers with d>e and GCD (d, e)=1. Suppose d is prime to p=char k. Then it is well-known (see, e.g., Brieskorn [3]) that L is identified with the set of exceptional curves of the minimal resolution of the following cyclic quotient singularity. Let G be the group of the d-th roots of the unity, and consider the following action of G on Ak=Spec k[x, y]; Let Z=A/G be the quotient variety and let P be the point of Z dominated by N the point of origin of Ak. Then the minimal resolution 2r: ZZ of the singu larity at P provides us with the exceptional set L. Note that this singularity is a rational singularity (cf. [3]). 1.3.3. DEFINITION. Let L=[al,•c, ar] be an admissible linear chain of a weighted graph F, where al corresponds to a tip of P. If L is a twig of F, we define the rational numbers pZ, 1•…i•…r, by ~i=d(Li)/d(L). By 1.3.1, we have 0 by vi=(d(Li)+d(L1))/d(L). We set di=1-pi if L is a twig and di=1-vi if L is a rod. We often use the symbol T (or R, resp.) instead of L if L is a twig (or a rod, resp.). 1.3.4. LEMMA. Suppose R=[a1,•c, ar] is an admissible rod of P. Then we have 0•…di<1, 1•…i•…r. Moreover, the following conditions are equivalent: (1) di=0 for some i, 1•…i•…r; (2) d1=0 foreveryi, 1•…i•…r; (3) ai=2 for every i, 1•…i•…r. PROOF. Since this is a purely combinatorial question, we have only to prove the above assertions in the case where R is the dual weighted graph of the exceptional set {Dl,•c, Dr} of the minimal resolution of a cyclic quotient singularity as explained in 1.3.2. So, {D1,•c, Dr} is a rational linear chain 206 MASAYOSHI MIYANISHI and SIIUICHIRO TSUNODA on a nonsingular projective surface V with ai=-(D2)•†2. By virtue of Lemma 1.3.5 below, we have for t•…j•…r. Hence we have write ~>d1D1=A-B, where A and B are effective Q-divisors free from common components. We have (A. Dj)•…(B. Dj) for every j, 1•…j•…r. Note that A and B are supported by U~_1D1 and that the intersection matrix of R is negative-definite. We have 0•…(A.B)•…(B2)•…0, whence (B2)=0. This implies B=0. Namely, d1•†0 for every 1•…i•…r. By definition, it is clear that 1>di. Suppose now dj=0 for some j. If dj+1>0 or dj-1>0, then we must have which is a contradiction. Thus the adjacent components Dj+1, Dj-1 of Dj have coefficients zero in ~>d1D1. Since R is connected, we get d1=0 for every i. This condition clearly implies ai=2 for every 1•…i•…r. Suppose the last condition holds. Then (~2=1 d1D1)2=0, which implies d1=0 for every i because the intersection matrix of is negative-definite. Q. E. D. 1.3.5. LEMMA. Let V be a nonsingular projective surface and let D be a reduced effective divisor on V with simple normal crossings. Let L={D1,•c Dr} be an admissible rational linear chain of D with a1=-(D2i), where D1 is a tip of D. Then, with the rational numbers di's defined in 1.3.3., we have for every 1•…j•…r, where D'=D-~2=1D1. PROOF. Case L is a twig. By 1.3.1 and 1.3.3, we have for j•‚1, r This implies the above equality. Non-complete algebraic surfaces 207 Case L is a rod. Again, by 1.3.1 and 1 .3.3, we find the same relations as in the preceding case, hence we obtain the stated equality . Q. E. D. 1.3.6. DEFINITION. Let V, D and L be the same as in 1 .3.5. Set Bk (L) =1(1-di)D2, where 1-d2=pi if L is a twig and 1 -d1=vi if L is a rod (see 1.3.3). By 1.3.3 and 1.3.4, Bk (L) is an effective Q-divisor, which we call the bark of an admissible rational linear chain L of D . With the notations in 1.3.5, D-Bk (L)=~Z=1 dZD1+D' is an effective Q-divisor . The process of sub tracting Bk (L) from D is called the peeling of L . Note that the peeling is defined, so far, only for an admissible rational linear chain . We shall define later in 1.5 the peeling of an admissible rational for k . 1.4. Throughout the remaining part of this chapter , we shall work in the following situation unless otherwise specified: V is a nonsingular projective surface defined over k and D is a reduced effective divisor with simple normal crossings . We set D*=D-~~ Bk (L2), where L2 ranges over all possible admissible maximal rational twigs and admissible rational rods of D . Then D* is an effective Q-divisor with Supp D* c Supp D , where Supp D Supp D* (if it is nonempty) consists of all rational rods of the type {D1 ,•c, Dr} with (D)=- 2, 1•…i•…r; the dual weighted graph is: The following is one of the central results in the theory of peeling . LEMMA. Let V, D and D* be as above . Suppose there exists an irre ducible component D0 of D such that (D*+Kv.D0)< 0 and (Do)•…-2. Then the following assertions hold true: (1) Do is a nonsingular rational curve. (2) The connected component F of D containing Do consists of non singular rational curves, and has one of the following dual weighted graphs: (Type D) where ai•†2 for 1•…i•…r. 208 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA (Type E6) (Type E7) (Type E8) Non-complete algebraic surfaces 209 PROF. Our proof consists of several steps. (I) We write a linear chain L2 as where L2 ranges over all possible maximal rational twigs and rational rods, and where D(2)1 is a tip of D. We set a(A)2=-(D() and denote by d(2), the rational number defined in 1.3.3. Then we have By hypothesis, Do U2 Supp (L2). Hence D0 is a component of D'. We have So, (D0+Kv• Do)<0, which implies pa(Do)=0. (II) We show that (~~D(2);• D0)>0. Suppose the contrary. Then we have This implies that Hence we have 0>(D-FK.D0)=(D0+K.D0)+(D-D0.D0)•†-2. Namely, (D-D0.Do)•…1. Therefore, either Do is a connected component of D or Do is a tip of D. In both cases, Do must be contained in one of L2's, which contradicts the hypothesis. Thus we have (~2D(2)2• Do)•†1. (III) Note that Do is not a component of any admissible rational rods of D. Let {T(i); 1•…i•…t} exhaust all admissible maximal twigs of D which meet Do. Then the components {D(i)1(1); 1•…i•…t} meet Do. On the other hand, let {Cj; 1•…j•…l} exhaust all irreducible components of D which meet Do and which are not contained in any admissible rational twig of D. We shall show that 1=0 and t•†3. Indeed, suppose l•†2. We then have (D*+KV•EDo)•†(Do+C1+C2+KV•ED0)•†0, which is a contradiction. Suppose 1=1. Then t•†2, for Do would, otherwise, be contained in an admissible rational twig. Note that, by definition, we have because d(T(i))>1(cf.1.3.1). Then we have 210 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA which is a contradiction. Suppose 1=0 and t•…2. Then Do would be con tained in an admissible rational rod, a contradiction. Thus we have proved the assertion. (IV) For the sake of simplicity, set 32=d(T(i)) for 1•…i•…t. The hypothe sis then implies By a straightforward computation, t=3 and {31, 32, 63} is, up to a permutation , one of the following triplets, {2, 2, n} (n•†2), {2, 3, 3}, {2, 3, 4} and {2, 3 , 5}. Now, we have the following auxiliary result. 1.4.1. LEMMA. Let T=[al, ..., ar] be an admissible twig with 3=d(T).. Then the following assertions hold. (1) 6=2 iff T=[2]. (2) 3=3 iff T=[3] or T=[2, 2]. (3) 6=4 iff T=[4] or T=[2, 2, 2]. (4) 6=5 iff T=[5], T=[3, 2], T=[2, 3] or T=[2, 2, 2, 2]. (5) 6=6 iff T=[6] or T=[2, 2, 2, 2, 2]. PROOF. The "if" part is straightforward. We prove the "only if" part. With the notations in 1.3.1, we have If 6=2, apparently r=1 and T=[2]. If 6=3, either r=1 and T=[3] or r=2' and d(T1)=2. In the latter case, T1=[2] and 6=gal-1=3. So, T=[2, 2]. Suppose 6=4. If r=1 or 3 then T=[4] or T=[2, 2, 2], respectively. If r=2 then d(T1)=2 or 3, i.e., T1=[2] or [3]. However, this is impossible because ala2=5 and a2=2 or 3. Suppose 6=5. If r=1 or 4, then T=[5] or T=[2, 2, 2, 2], respectively. Suppose r=2. Then a2=d(T,)=4, 3 or 2, and ala2=6.. Hence T=[3, 2] or [2, 3]. Suppose r=3. Then d(T1)=3 and T,=[2, 2] by the preceding cases. However, this is not the case because 6=3a1-2=5. The case 6=6 can be verified in a similar fashion. Q. E. D. (V) Note that the connected component of D containing Do consists of Do and T(i), 1•…iv3. Hence it consists of nonsingular rational curves. It is easy to write down the dual weighted graph of the connected component. We only note that, in the graph of the type (D), the twig [a1, ..., a,] has thee determinant n•†2. Q. E. D. Non-complete algebraic surfaces 211 1.4.2. REMARK and DEFINITION. Among the assumptions in the lemma, remove the assumption that (D)•…-2. Then the above proof implies that Do is a nonsingular rational curve and that Do is one of the following com ponents of D: (i) with an admissible rational twig T, which might be empty, D0 forms a rational twig; we denote it by T+D0; (ii) with admissible rational twigs T and T', which might be empty, Do forms a rational rod, T+D0+T'; (iii) Do is a component of the connected component F whose dual weighted graph is one of those listed in the above lemma; such a connected component is called a rational fork of D and the component Do is called the central component of the fork; if (Do)•…-2, the fork is said to be admissible. If (D20)•†-1, the twig T+D0, the rod T+D0+T' or the fork F is non admissible, and Do is called the irrelevant component of the twig T+D0, the rod T+D0+T' or the fork F. 1.5. Let F be a fork of D. Then the intersection matrix of F is negative-definite if and only if (D20)•…-2. This is well-known, and one may verify it by making use of the following simple criterion (cf. Satake [19]): Let A=(aij)1•…i,j•…n be an (nxn)-real symmetric matrix and let A(k)= (aij)1•…i,j•…k. Then A is negative-definite if and only if (-1)k det A">0 for 1•…k•…n. In case F is an admissible fork, write F=D0+T(1)+T(2)+T(3), as the sum of the central component Do and three admissible rational twigs T(i), 1•…i•…3. Write T(i)={D(i)1, ... , D(i)r}, 1•…i•…3. The linear chain t T(i)={D(i) r(i), ... , D(i)1} is called the transpose of T(i). We set: and where we define Bk (tT(i)) as in 1.3.6, regarding tT(i) as a twig with D(i)1(i) as a tip. Then we have the following result. LEMMA. (1) 5>1 and a>e. Hence Bk (F) is an effective Q-divisor. 212 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA (2) (F+KV• Z)=(Bk(F).Z) for every irreducible component Z of F, where the fork F is considered as a reduced effective divisor. (3) F-Bk (F) is either zero or an effective Q-divisor such that Supp (F-Bk(F))=Supp (F). F=Bk(F) if and only i f every component Z of F has (Z2)=-2. PROOF. (1) Since {d(T(1)), d(T(2)), d(T(3))} is {2, 2, n} (n•†2), {2, 3, 3}, {2, 3, 4} or {2, 3, 5} modulo a permutation, we have clearly o>1. Since the intersection matrix of F is negative-definite, we have Indeed, if T=[a1, .. . , ar], then t T=[ar, ... , al], (tT)1=t (Tr-1+1), 0•…i•…r, and hence for 1•…i•…r-1 and d((tT)r-1)=al (cf.1.3.1). (2) This is straightforwardly ascertained. (3) Suppose F Bk (F)•‚0 and write F Bk (F)=A-B, where A and B are effective Q-divisors free from common components. If B•‚0, then (B. Z) <0 for some component Z of B. Then (A-B• Z)>0 and (Z• Kv)•†0, so (F•EZ)>(Bk (F)•EZ)=(F+K.Z)•†(F•EZ), which is a contradiction. Thus F Bk (F) is effective. Since F is connected and (F Bk (F).Z)=-(Kr• Z)•…0 for every component Z of F, we have Supp (F-Bk(F))=Supp (F). It is straightforward to see that F=Bk(F) if and only if (Z2)=-2 for every component Z of F. Q. E. D. The Q-divisor Bk (F) is called the bark of an admissible rational fork F. Now going back to the divisor D on V, we consider the set {T2} of all admis sible rational maximal twigs which are not contained in the admissible rational forks, the set {R} ~of all admissible rational rods and the set {Fv} of all admissible rational forks. We set and call it the bark of D. The divisor Bk (D) is an effective Q-divisor such that : (1) D-Bk (D) is an effective Q-divisor such that Supp (D) and Supp (D Bk (D)) differ by the union of all rational rods of the form [2, 2,•c, 2] and Non-complete algebraic surfaces 213 all admissible rational forks whose components have all self-intersection numbers -2; (2) Bk (D) has the negative-definite intersection matrix; (3) (D-Bk (D)+K• Z)=0 for every irreducible component Z of T2's,, Rn's and Fr's; (4) (D-Bk (D)+V• Y)•†0 for every irreducible component Y of D except the irrelevant components of non-admissible rational twigs, non admissible rational rods and non-admissible rational forks (cf.1.4.2). We set D#=D-Bk (D) and call it the stripped form of D. 1.5.1. We note that the following result holds. LEMMA. Let ~i=1 Y, be a divisor such that Supp (~_1Yi) C Supp (Bk (D)) and that every connected component of IIY1 is an admissible rational twig, an admissible rational rod or an admissible rational fork of D. Let 4=D ~i=1Y2. Define the rational numbers al,•c, ar by the condition for 1•…j•…r. Then we have: (1) 0•…ai<1 for 1•…i•…r; (2) D#•…ƒ¢+1a1Y. PROOF. In fact, ƒ¢+aiY1 is obtained from D by the peeling of ~i=1Y1. Therefore the assertion (1) holds clearly. On the other hand, let B=~Z=1(1-a)Yi. For the assertion (2), we have only to show that Bk (D)•† B. Write Bk (D)=W+B1, where Bl is supported by {Y1,•c, Yr} and no components of W belong to {Y1,•c, Yr}. Then we have and (B1-B•Yj)=(Bk(D)-B• Y)-(W.YJ) Since the intersection matrix of Y2 is negative-definite, we conclude B1•† B. Hence Bk (D)•†B. Q. E. D. 1.6. Suppose there exists an irreducible curve E on V such that (D#+ KV• E)<0, E is not an irreducible component of D and the intersection matrix of E+Bk (D) is negative-definite. Then we have clearly (E• K) <0, i.e., E is an exceptional curve of the first kind. Let a: VV' be the contraction of E and let D'=a(D). We shall show the following: 214 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA LEMMA. D' is a reduced effective divisor on V' with simple normal cros sings. PROOF. Our proof consists of several steps. Let Z1,•c, Zn exhaust all irreducible components of D such that (Zi• E) >0. Let c be the coefficient of Zz in D#, 1•…i•…n. Then 1>ai(ZZ• E). So, 0•…ai<1 for 1•…i•…n. This implies that ZZ c Supp (Bk (D)), 1•…i•…n. We apply the following result for any non-empty set of {1,•c, n}, say {1,•c, m} after a relabelling. 1.6.1. LEMMA. Let Z1,•c, Zm be (not necessarily all) irreducible com ponents of Supp (Bk (D)) and let a, be the coefficient of Zj in D#. Define the rational numbers d1, 1•…j•…m, by the condition: for 1•…i•…m. Then we have 0•…dj•…a; for 1•…j•…m. In particular, we have 1•…j•…m, where aj=-(Z2j)•…2. PROOF. We have for 1•…i•…m. Write ~~1(o-d~)Z~=A-B with effective Q-divisors A, B free from common components. Then 0•…(A• B)•…(B2) by virtue of the above relation. Hence B=0 because Bk (D) has the negative definite intersection matrix. Thus aj•†dj for 1•…j•…m. For m=1, we have d1=1 (2/a1). Thence follows the second assertion. Q. E. D. The following result will complete the proof of Lemma 1.6. 1.6.2. LEMMA. The following assertions hold true: (1) (Zi•EE)=1, 1•…i•…n. (2) ai=2 for at most one index i. (3) (Z2•EZj)=0 for any pair (i, j) with i•‚j. (4) n•…2. PROOF. (1) we have 1>a?(Z2• E), where a2=1 (2/a1) and ai= -(Z). Hence if ai•†4 then (Zi• E)=1. Suppose a1=2 or 3 . Then 0>(Zi+2E)2=-(ai+4)+4(Zi• E)•†-7+4(Z1.E), Non-complete algebraic surfaces 215 so (Zi.E)=1. (2) Suppose a1=a2=2. Then 0>(Z1+Z2+2E)9=2(Z1.Z2)•†0, a contra diction. (3) We may assume that (i, j)=(1, 2) and a1•…a2. With the notations of 1.6.1, let m=2 and suppose (Z1.Z2)>0. Then (Z1• Z2)=1 and we have and 1>d1+d2. Hence {a1, a2}={2, 2}, {2, 3} or {2, 4}. By virtue of (2), we have {a1, a2}={2, 3} or {2, 4}. In each case, we have 0>(Z1+Z2+2E)2•†-2+2(Z1.Z2)=0, so (Z1.Z2)=0. (4) We assume a1•…a2•…•c•…an. Since 1>c, we readily see that one of the following is the case: (i) an•†2, a1=•c=an-1=2; (ii) 5•†an•†3, an-1=3, a1=•c=an-2=2. In case (i), we have n•…2 by the assertion (2) above. In case (ii), n•…3. If n=3, we have 0>(5E+3Z1+2Z2+3)2=5-a3, which is a contradiction. Thus n•…2. Q. E. D. 1.7. Let E be an exceptional curve of the first kind as in 1.6. Then E meets at most two irreducible components of Bk (D). If E meets two com ponents Z1, Z2, then the connected components of Bk (D) which contain Zi (i=1, 2), the position of Zi in the connected component and the self-intersec tion number of Zi are strongly restricted. We shall clarify these facts in the sections 1.7 and 1.3. Let f: VV be the composite of the contraction of E and the contrac tions of all possible exceptional components (i.e., irreducible curves which become exceptional curves of the first kind after a succession of several con tractions) of Bk (D) (cf. 1.6 for the notations). Let D=f*(D). Write Bk (D) =1(1-a2)D1 with ai<1, 4=D-1 D1 and D=4+1aZDZ. Suppose dim f(D3=1 for 1•…i•…r and dim f (D3=0 for r+1•…i•…s. Let Dz=f(D2), 1•… i•…r, and A=f(4). So D=4+D2. We don't know, for the moment, whether or not D is a divisor with simple normal crossings. Furthermore, since E+Bk (D) is negative-definite, D, is negative definite in the fol lowing sense: For an R-linear combination A=~i=1x1D29 we have (A2)•…0, and (A2)=0 if and only if xi=0, 1•…i•…r. Therefore we can define uniquely the rational numbers j31, 1•…i•…r, by the condition 216 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA for 1•…j•…r. We shall prove first the following: LEMMA. (1) 1>ai•†0 for 1•…i•…r. (2) D is a reduced effective divisor with simple normal crossings. (3) Each connected component of f (Supp Bk (D))=Ui_1D1 is an admis sible rational twig, an admissible rational rod or an admissible rational fork of D. (4) Let B*=~i=1(1-T)DZ. Then f*(Bk (D))•…B*•…Bk (D). PROOF. (1) Write ~ti=1j3D1=A-B with effective Q-divisors A, B free from common components. By the definition off, either pa(D3)>0 or pa(D3) =0 and (D)•…-2, for 1•…j•…r. Hence (KV.D3)•†0 and (4+KV.D3)•†0. Therefore (A-B.D3)•…0 for 1•…j•…r. Thence we have 0•…(A.B)•…(B2). Since ~i=1D1 is negative-definite, we have B=0. Namely 8•†0 for 1•…i•…r. In order to show ai 8, note that (D#+Kv)-f*(d+~i=1a2D2+Kv) is sup ported by the union of E and the components D1, r+1•…i•…s. Hence we have for 1•…j•…r, where Supp (f*(D3)-D3-aE)CUZ=r+1 Dz and a•†0. So we have This implies, by the same argument as above, that a2•†~i for 1•…i•…r. The inequality 1>ci is clear from the definition of D#. (2) We shall first show that 4 is a divisor with simple normal crossings. For this purpose, let g: VV be a birational morphism such that g factors f, i.e., f=hg with h: VV, and that g*(4) is a divisor with simple normal cros sings. Let Y be an irreducible exceptional component of g*(E+r+1D1) which should be contracted first by h, and let Y=g'(Y) be the proper trans form of Y on V. By Lemma 1.6, the contraction of E does not affect ƒ¢, and we have Y=Dj for some j, r+1•…j•…s. As in the preceding computation, we have and Non-complete algebraic surfaces 217 because pa(Y)=0, (Y2)=-1 and aj<1. Thus (g5(4). Y)•…1. This implies that either Supp g*(4) nY=~5 or Y meets only one component of g5(4) trans versally at a single point. Hence, by the contraction of Y, the direct image of g,~(4) is a divisor with simple normal crossings. Next, we shall show that ~i=1 DZ is a divisor with simple normal cros sings consisting of nonsingular rational curves. Indeed, since we know that pa(Dj)=0, i.e., Dj is a nonsingular rational curve. Suppose one of the following cases occurs: (i) Two irreducible components, say A and D2, of ~i=1Di touch each other with order of contact•†2; (ii) Three irreducible components, say D1, D2 and D3, of ~i=1 D, meet each other at one common point. In case (i), we easily see that for t•…j•…r. Hence ((/91-1)D1+(/92-1)D2.D3)•…0 for 1•…j•…r. This implies that ~(/9 -1)D, +(82 -1)D2•†0. This is a contradiction. In case (ii), we have for 1•…j•…r. Hence, by an argument similar to the above, we conclude that (j9-1)D1+ (/92-1)D2+(P3-1)D3•†0, which is a contradiction. Finally, we shall show that D=4+D, is a divisor with simple normal crossings. Otherwise, one of the following cases takes place: (iii) There are two irreducible components, say 4 and A, of 4 and DZ, respectively, such that dl and A meet each other at one point with order of contact•†2; (iv) There are three irreducible components, say ƒ¢1 of ƒ¢ and D1, D2 of ~i=1 Di, such that 2i, Dl and D2 have one common point; (v) There are three irreducible components, say v2 of 4 and D1 of D2, such that ƒ¢1, d2 and A have one common point. In case (iii) or (v), we have for 1•…j•…r. 218 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA Hence we have (j3-1)D1•†0, which is a contradiction. In case (iv), we have for 1•…j•…r. Hence we have (j31)D1+(J32-1)D2•†0, which is a contradiction. (3) The assertion is verified by virtue of the following: 1.7.1. LEMMA. Let V be a nonsingular projective surface and let D be a reduced effective divisor with simple normal crossings. Suppose we have for 1•…j•…r, where D=4+~z=1 Di and jai is a rational number with 0 Furthermore, suppose that the intersection matrix of ~jZ=1 Di is negative-definite and that (Dc•…-2 for 1•…i•…r. Then every connected component o f Di is an admissible rational twig, an admissible rational rod or an admissible rational fork of D. PROOF. Note, first of all, that pa(D3=0, 1•…i•…r (cf. the above proof). Let {T2}, {R} and {F} vbe all possible admissible rational twigs, admissible rational rods and admissible rational forks of D, respectively, such that the supports of T2's, Ru's and F's are contained in Uz=1 Di. We take T2's to be maximal in the sense that Supp (T2) C U~=1 Di. Let and write D*=4+4'+~z_1 riDi, where Di, 1•…i•…t, ranges over all irreduci ble components of ~i=1 Di which are contained in the above T2's, Re's and Fr's, and where ri is a rational number with or, <1. Furthermore we have for 1•…j•…t. Suppose 4'O and (4+4'+J1r1D1+K.Y)<0 for an irreducible com ponent Y of 4'. Let C1, ..., C1 be all irreducible components of 4+4' Y which meet Y, and let T(i), 1•…i•…q, be all admissible rational twigs in 1 Di which meet Y. Write T (i)={D(i)1, ... , D(j)r}, where D(i)1 is the tip of T(i). Then we have Therefore 1=0 or 1. Suppose l=0. Then we have Non-complete algebraic surfaces 219 Hence q•…3, and Y+q=1T (j) is an admissible rational rod (if q•…2) or an admissible rational fork (if q=3) contained in Ui _1 Di (cf. the proof of Lemma 1.4). This is a contradiction. Suppose 1=1 . Then we have Since 1-(1/d(T(j )))•†1/2, we conclude that q•…1 . Then Y+~q=1T(j) is an admissible rational twig contained in Ui =1D.. This is a contradiction. Therefore we must have for every component Y of ƒ¢' provided ƒ¢'•‚0. Now consider ~jz=1~~DZ-(4'+~2=1 r2D3, which is a Q divisor supported by Ui_1D.. We have Hence we conclude that =1j2D2•†4'+~i =1TD. This implies that d'=0 because ji<1 for 1•…i•…r. It is now easy to see that ii=Yi for 1•…i•…r . Q. E. D. (4) The last assertion holds clearly in virtue of the assertion (1) and Lemma 1.5.1. This completes the proof of Lemma 1 .7. 1.8. Let V, D and E be the same as in 1.6 and 1 .7. If E meets only one irreducible component Z of Bk (D) then (Z. E)=1 and E does not meet any other components of D; Z can be arbitrary and we have nothing more to say . So, we consider the case where E meets two irreducible components Y , Z of Bk (D), Y and Z then belonging to distinct connected components A , B of Bk (D), respectively (cf. Lemma 1.7). By virtue of the same result, we have only to consider the following cases, (changing the roles of A and B if neces sary): (1) A and B are linear chains, (2) A is a fork and B is a linear chain. Note that if A is a linear chain in Bk (D), A may be either a twig or a rod in D. For an irreducible component Y of Bk (D), JSD(Y) (or PBk (D) (Y) , resp.) is the number of irreducible components of D (or Bk (D), resp .), other than Y, which meet Y. For the sake of simplicity, we denote PA(Y) and J3Bk (D) (Y) by p(Y) and ~3(Y), respectively. In the subsequent computation, which is simple in princple but fairly lengthy and hence we don't complete, the key is the following: 220 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA 1.8.1. LEMMA. Let Yi be a divisor such that Supp (~v=1Y1) Supp (Bk (D)). Let 4 be the reduced effective divisor which is the sum of all irreducible components of D not in Supp (Bk (D)). Define the rational numbers al,•c, ar by the condition for 1•…j•…r. Then we have: (1) 0•…ai•…<1 for 1•…j•…r. (2) ai•†ai, where ai is the coefficient of Yi in D#. PROOF. Note that every connected component of ~i=1YZ is a twig, a rod or a fork of the divisor 4+~2=1Yi and it is admissible and rational. Therefore the assertion (1) is clear from 1.3.5 and 1.5. In order to prove the assertion (2), note that for 1•…j•…r. Therefore we have for 1•…j•…r. Since the intersection matrix of ~i=1Y, is negative definite, we conclude that ai•†ai for 1•…i•…r. Q. E. D. 1.8.2. LEMMA Let a=-(Y2) and b=-(Z2). Then the pair {a, b} is, up to a permutation, one of the following: {2, m} (m•†3), {3, 3}, {3, 4} and {3, 5}. PROOF. Since (D#+KV• E)<0, we have by 1.6.1, Hence the pair {a, b} is one of the above ones. In case {a, b}={2, m}, we have m•†3 because E+Y+Z is negative-definite. Q. E. D. 1.8.3. LEMMA. Suppose ~(Y)•†2. Then the following assertions hold: (1) {a, b}={m, 2} (m3) or {3, 3}. (2) The connected component B of Bk (D) containing Z is a rod with Z as a tip. If (Y)=3 then B=[2, 2,•c, 2] and the set A U E U B shrinks down to an admissible rational fork, where A is the connected component of Bk (D) containing Y. Non-complete algebraic surfaces 221 (3) Suppose (Y)=2. Then A U E U B shrinks to an admissible rational fork unless B=[2, 2,•c, 2]. Let Yi (i=1, 2) be the adjacent component of Y in A and let a2=-(Y). If {a, b}={3, 3} then {a,, a2}={2, 2} unless A={Y1 , Y, Y2}, B={Z}, (Y)=-2 and (Y)=-3. PROOF. (1) As in the assertion (3), let Y, (i=1, 2) be two of the adjacent components of Y in A and let a2=-(Y). Considering Y1+Y+Y2 and Z as admissible rational rods, we have by 1.8.1 (or rather 1.6.1), This inequality is rewritten as Hence we have (a-2)(b-2)•…1, i,e., a=2 or b=2 or {a, b}={3 , 3}. Suppose a=2. Then we have This implies that b=3, 4, 5 or 6 and, for each value of b , the pair {a1, a2} is determined, up to a permutation, as follows: If b=3 then {a1, a2}={2, n} (n•†2), {3, 3}, {3, 4} or {3, 5}; If b=4 then {a1, a2}={2, 2}, {2, 3}, {2, 4} or {2 , 5}; If b=5 or 6 then {a1, a2}={2, 2} or {2,3}. It is now straightforward to check, case by case , that Y1+Y+Y2+E+Z is not negative-definite. Hence a•‚2. In case {a, b}={3, 3}, the pair {a1 , a2}= {2, 2} or {2, 3} up to a permutation. (2) By virtue of Lemma 1.7, A+E+B shrinks to a twig, a rod or a fork after the contraction of all possible exceptional components. Then B must be a rod with Z as a tip. If (Y)=3 then A is a fork and all components of E+B should be contractible. Thus B=[2 , 2,•c, 2]. (3) The first assertion holds clearly. We shall prove the second asser tion. So, suppose {a, b}={3, 3} and {a1, a2}={2 , 3} (cf. the assertion (1)). If B•‚{Z}, let Z, be the adjacent component of Z in B and let b1=-(Z) . Then we obtain whence b1•…1, a contradiction. Thus B={Z}. Then it is easy to show that A U E U B has one of the following weighted graphs: 222 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA It is now easy to show that (D#+KV• E)<0 only in the first case. Thus, except this case, {a1, a2}={2, 2}. 1.8.4. LEMMA. Suppose (Y)=2, {a, b}={3, 3} and {a1, a2}={2, 2} (cf.. 1.8.3). Then the configuration of A+E+B is one of the following: (1) (2) The configuration obtained from the configuration o f nonsingular rational curves with self intersection number -2 whose weighted dual graph is the Eynkin diagram of type En (n>4), E6, E7 or E8, by blowing up any one of three points on the central component in which the central component meets three twigs. PROOF. Let L={Z1,•c, Zn} be a rational linear chain such that B= {Z, Z1,•c, Zn}. Let L1={Z2,•c, Zn}. Then we have: where we set d(L)=1 and d(L1)=0 if L=~. Hence we have d(L)=d(L1)+1, which implies L=[2, 2,•c, 2]. On the other hand, since A+E+B shrinks to an admissible rational fork, let M be a linear chain such that Y2+M is a maximal twig meeting Y. Then we have Non-complete algebraic surfaces 223 where we set d(M)=1 and d(M1)=0 if M=ƒÓ. Hence we have n(4d(M)-3d(M1)-4)<6-2(d(M)-d(M1))•…4. If n=0 then either d(M)=d(M1)+1 (i.e., M=[2,...,2]) or d(M)=d(M1)+2 (i.e., M=[2,...,2,3]). If n=1 then M=ƒÓ, [2], [2,2] or [2,2,2]. If n=2 then M=ƒÓ or [2]. If n•†3 then M=ƒÓ. Finally we note that if n•†1 we may assume {Y1} is another maximal twig of A meeting Y. Now, making use of the above observation and the condition that A+E+B shrinks to a fork, we can readily list up all possibilities and thus verify the stated assertion. Q.E.D. 1.8.5. REMARK. (1) In case ƒÀ(Y)=2, {a,b}={m,2}(m•†3) and B is a rod not of the type [2,2,...,2], A•¾E•¾B shrinks to a fork. Therefore, going the other way, i.e., blowing up several points on the central component which are infinitely near to a point in which the central component meets one of three twigs, we get back A•¾E•¾B, E being the last exceptional curve. However, the condition (D#+KV¥E)<0 reflects in a subtle way on the length of the rod B as shown by the following: 1.8.5.1. EXAMPLE. Suppose A•¾E•¾B shrinks to a fork whose dual graph is: Tnen A•¾E•¾B has one of the following weighted dual graphs: (2) Now suppose ƒÀ(Y)•…1. We may assume ƒÀ(Z)•…1. Then Y and Z are the tips of rational linear chains. Furthermore, either A or B, say B, is a linear chain. If A is a fork or a twig, then B is a rod. We consider the case where A is a fork and B is a rod. Then A•¾E•¾B shrinks to a fork F, 224 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA and A•¾E•¾B is regained from F by the blowing-ups as explained above. We exhibit it by the following: 1.8.5.2. EXAMPLE. Suppose the fork F has the same weighted dual graph as in 1.8.5.1. Then A•¾E•¾B has one of the following weighted dual graphs: (3) The remaining case, i.e., the case where A and B are rational linear chains, can be treated in a similar fashion. Further computations and details are left to the readers. 1.9. Let V and D be as above. Let E now be an exceptional component of D, i.e., E•¼Supp (D), pa(E)=0 and (E2)=-1, such that PD(E)•…2, (D#+ Kv¥E)<0 and the intersection matrix of E+Bk (D) is negative-definite. E is then called a superfluous exceptional component of D. By virtue of Remark 1.4.2, such a component E appears only in one of the following situations: (i) E is an isolated component, (ii) there exists an admissible rational twig T such that T is a con nected component of Bk (D) and T+E is a twig of D, where T might be empty, (iii) there exist admissible rational twigs T and T' such that they are connected components of Bk (D) and T+E+T' is a rod of D, where one (or both) of T and T' might be empty. As in 1.7, let f:VV be the contraction of all possible contractible com ponents of E+Bk (D) including E, i.e., irreducible components of E+Bk (D) which become exceptional curves of the first kind after a succession of several contractions. Let D=f*D. Then it is clear that D is a reduced effective divisor with simple normal crossings. Define Di(1•…i•…r), d, the rational numbers ƒÀi(1•…i•…r) and the effective Q-divisor B*=rƒ°i=1(1-ƒÀi)Di exactly in the same way as in 1.7. Then we have f*Bk(D)•…B*. By virtue of Lemma 1.5.1, we have: LEMMA. f*Bk(D)•…B*•…Bk (D). The divisor, D, of course, might have superfluous exceptional com ponents. We perform the contractions of the above kind as long as there are Non-complete algebraic surfaces 225 superfluous exceptional components. If there are no superfluous exceptional components in D one of the following cases takes place for an irreducible curve C on V: (i) (D#+Kv¥C)>0, (ii) C is an exceptional curve of the first kind such that C_??_Supp(D), (D#+Kv¥C)<0 and the intersection matrix of C+Bk (D) is negative-definite, (iii) (D#+Kv¥C)<0 and C+Bk (D) is not negative-definite. 1.10. For a Q-divisor Z=nƒ°i=1 aiCi, Ci being an irreducible component, we set [Z]=nƒ°i=1[ai]Ci, where [ai] is the largest integer not exceeding ai. The divisor [Z] is called the integral part of Z. Clearly we have Z•†[Z]. Now we prove the following: LEMMA. Let V, D be the same as above. Then the following assertions hold: (1) h0(V,n(D+Kr))=h0(V,[n(D#+Kr)]) for every integer n•†0. (2) Let f:VV be the contraction as in 1.7 or 1.9. Then we have h0(V, [n(D#+Kr)])=h0(V,[n(D#+Kr)]) for every integer n•†0. PROOF. (cf. [10; Chap. II. 1.81). (1) Fix an integer n•†0. Set Y=n(D#+Kr)-[n(D#+Kr)]=nD#-[nD#] Z=nBk(D) and W=Y+Z. Since W=n(D+Kr)-[n(D#+Kr)], W is an integral divisor. We may assume | n(D+KV)|•‚ƒÓ. Otherwise, |[n(D#+KV)]|=ƒÓ and the equality holds. Let A•¸| n(D+KV)|. Then we have only to show A•†W. Choose a positive integer n0 such that [n0(D#+Kr)]=n0(D#+KV) and [n0Bk(D)]=n0Bk(D). Then n0A•†n0Z because (n0A-n0Z¥C)=(nn0(D#+KV)¥C)=0 for every component C of Bk (D) and the intersection matrix of Bk (D) is negative-definite. Write A=A'+F, where A'•†0, F•†0, Supp (F) Supp (Bk (D)) and A' and F have no common components. Then F•†Z. Now choose an arbitrary component C of Y and let ƒ¿,ƒÀ,ƒÁ be the coefficients of C in Y,Z,F, respectively. Then 0<ƒ¿<1, ƒ¿+ƒÀ•¸Z and ƒÁ•¸Z. Since ƒÁ>ƒÀ and 0<ƒ¿<1, we have ƒÁ•†ƒ¿+ƒÀ. Therefore A•†Y+Z=W. (2) It is easy to see that h0(V,n(D+KV))=h0(V,n(D+Kr)) for every integer n•†0, because f*(D+KV)=D+Kv. Therefore we have h0(V,[n(D#+K)])=h0(V,[n(D#+KV)]) 226 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA by virtue of the assertion (1). Q.E.D. 1.11. Summarizing the above results, we obtain the following: THEOREM. Let V be a nonsingular projective surface defined over k and let D be a reduced effective divisor with simple normal crossings. Then there exists a birational morphism ƒÊ:VV onto a nonsingular projective surface V such that, with D=ƒÊ*(D), the following conditions are satisfied: (1) h0(V,n(D+K))=h0(V,n(D+Kr)) for every integer n•†0; (2) ƒÊ*Bk(D)•…Bk(D) and ƒÊ*(D#+KV)•†D#+KV; (3) For every irreducible curve C on V, we have either (D#+KV¥C)•†0 or (D#+KV¥C)<0 and the intersection matrix of C+Bk (D) is not negative definite. PROOF. A birational morphism ƒÊ:VV is obtained as a composite of the following operations: (1) If there is a superfluous exceptional component E, contract E and all possible exceptional components of E+Bk (D) (cf.1.9). Repeat this oper ations as long as there are superfluous exceptional components. (2) If there are no superfluous exceptional components in D then peel the barks of all maximal twigs, rods and forks of D which are admissible and rational. (3) Find an exceptional curve E of the first kind such that E¢ Supp (D), (D#+KV¥E)<0 and the intersection matrix of E+Bk (D) is negative-definite. If there is none, then we are done. If there is one, contract E and all pos sible exceptional components of Bk (D) (cf.1.7). Repeat this operation as long as there are exceptional curves of the first kind like E above. Now repeat the operations (1), (2) and (3) all over again. The condition (3) follows from the above construction, and the other conditions follow from Lemmas 1.7, 1.9 and 1.10. Q.E.D. A pair (V,D) is called almost minimal if, for every irreducible curve C on V, either (D#+KV¥C)•†0 or (D#+KV¥C)<0 and the intersection matrix of C+Bk (D) is not negative-definite. The pair (V,D) constructed in the above theorem is called an almost minimal model of (V,D); see also Tsunoda [21; p. 105]. 1.12. THEOREM. Let V be a nonsingular projective surface defined over k and let D be a reduced effective divisor with simple normal crossings. Sup pose D#+Kv is numerically effective, i. e., (D#+Kv¥C)•†0 for every irreducible curve C on V. Then k(V-D)•†0. PROOF, we have only to show that |n(D+Kv)|•‚ƒÓ for a sufficiently large integer n>0. If k(V)•†0, i.e., |nKV|•‚ƒÓ for some integer n>0, we have Non-complete algebraic surfaces 227 | n(D+KV)|•‚ƒÓ for the same n. So, we may assume k(V)=-•‡, i.e., V is either a rational surface or an irrational ruled surface. We first consider the case V is rational. Choose a sufficiently large integer n such that [n(D#+Kr)] = n(D#+Kr). Then n(D#+KV) is a pseudo-ample divisor (cf. Hartshorne [6]). Hence (D#+KV)2•†0. On the other hand, H2(V,n(D#+KV))_??_H0(V,KV - n(D#+Kr))=(0); indeed, if n(D#+KV) is numerically trivial, i. e., (n(D#+Kr) ¥ C)=0 for every irreducible curve C, then n(D#+KV)•`0 because V is rational, whence H2(V,n(D#+KV))=(0); votherwise, (D#+KV¥A)>0 for an ample divisor A, and so (KV-n(D#+KV)¥A)<0 for a sufficiently large integer n, which implies H2(V,n(D#+KV))=(0). Now the Riemann-Roch theorem provides us with Hence |n(D#+K)|•‚ƒÓ. Suppose next that V is an irrational ruled surface. Then V is endowed with a P1-fibration f:VC over a nonsingular curve C of positive genus. Let l be a general fiber off. Then (D#+Kv¥l)=(D#¥1)2 •† 0. Thus D contains a horizontal component, say ƒ¡, i.e., (ƒ¡¥l)>0, which appears in D# with coefficient 1. Suppose h0(V,[n(D#+KV)])=0 for every n>0. Since D#+Kv•†ƒ¡+Kv, we then have h0(V,n(ƒ¡+KV))=0 for every n>0, i.e., k(V-ƒ¡)=-•‡. By virtue of [10; Chap. I, 2.3] and [11; Th. 1], we know that ƒ¡ is a cross-section of f, i.e., (ƒ¡¥l)=1. Since (D#¥l)•†2, there are at least two cross-sections, say ƒ¡1 and ƒ¡2, appearing in D as horizontal com ponents. We shall then show that k(V,ƒ¡1+ƒ¡2+KV)•†0, hence k(V,D+KV)•† 0. Note that ƒ¡1+ƒ¡2 is a reduced divisor with simple normal crossings. If V is not relatively minimal, take an exceptional curve E of the first kind. If (E¥ƒ¡1+ƒ¡2)•…1, we may contract E without loss of generality (cf. [10; pp. 24-25] for a related argument). Since E is contained in a fiber F of f, we have (ƒ¡1¥E)•…1 (i=1,2). Suppoes (ƒ¡1¥E)=(ƒ¡2¥E)=1. Then there exists another exceptional curve E' of the first kind in the same fiber F such that (ƒ¡1¥E')=0 (i=1,2) (cf. [10; Chap. I. 4.4.1]). Then we contract E' instead of E. Therefore we may assume that V is relatively minimal. Then, since (ƒ¡1+ƒ¡2+Kv¥l)=0, we have ƒ¡1+ƒ¡2+KV•`f*•¢ with a divisor •¢ on C. Since Pa(ƒ¡1)>0 and (ƒ¡1¥ƒ¡2)•†0, we have (ƒ¡1+ƒ¡2+Kv¥ƒ¡1)=deg •¢•†0, where deg d =0 if and only if Pa(ƒ¡1)=1 and (ƒ¡1¥ƒ¡2)=0. If deg •¢>0, we have apparently k(V,ƒ¡1+ƒ¡2+Kr)>0. Suppose deg •¢=0. Then ƒ¡1 is a nonsingular elliptic curve, so (ƒ¡1+Kv)|„s1•`0. Thus f*•¢|„s1•`(ƒ¡1+ƒ¡2+Kv)|„s1•`0. Namely ƒ¡1+ƒ¡2 KV•`0 and K(V,ƒ¡1+ƒ¡2+KV)=0. This completes the proof of the theorem. REMARK. With the same notations in the above theorem, assume, con versely, that k(V-D)•†0. Let (V,D) be an almost minimal model of (V,D). Then, by virtue of [21; Th. 1.3], we know that D#+KV is numerically effective, 228 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA D#+KV=ƒÊ*(D#+KV) and D#+Kv is the numerically effective part (D+KV)+ in the Zariski decomposition D+KV=(D+KV)++(D+KV)- (see also [10; Chap. II]). In particular, D#+KV is numerically effective. Indeed, (V,D-Bk(D)) is relatively minimal model of (V,D) in the sense of Kawamata [8] (cf. [21; Prop. 1.5]). Chapter 2. Affine-ruledness of an almost minimal algebraic surface with the logarithmic Kodaira dimension -•‡ 2.1. A nonsingular algebraic surface X defined over k is said to be almost minimal if there exists a smooth completion (V,D,X) such that the pair (V,D) is almost minimal. In general, we have the following: LEMMA. Let X be a nonsingular algebraic surface defined over k. Then there exist a Zariski open set U of X and a proper birational morphism ƒÊ:U X where X is an almost minimal, nonsingular, algebraic surface with k(X) =K(X). If X is affine-ruled, so is X. PROOF. Let (V,D,X) be a smooth completion of X. Then there exists a birational morphism ƒÊ:VV onto a nonsingular projective surface V such that, with D:=ƒÊ*D which is a reduced effective divisor with simple normal crossings, (V,D) is almost minimal (cf. Th. 1.11). Let X:=V-D and let U:=ƒÊ-1(X). Then X is almost minimal and ƒÊ:UX is a required proper birational morphism. Q.E.D. Throughout this chapter, we consider a pair (V,D) consisting of a non singular projective surface V and a reduced effective divisor D with simple normal crossings such that D#+Kv is not numerically effective. 2.2. Since D#+KV is not numerically effective, (D#+KV¥C)<0 for some irreducible curve C on V. If C_??_Supp (D) then (Kr¥C)<0. Suppose C•¼ Supp (D). Then C_??_Supp Bk (D), so [D#]•†C and (C+KV¥C)<-(D#-C¥C) •… 0. This implies that pa(C)=0 and that C is, in fact, an irrelevant com ponent of a twig, a rod or a fork which is rational and non-admissible. Hence (C2)•†-1 and (KV¥C)•…-1. Therefore (KV¥C)<0 always, and KV is not numerically effective. Then, according to the Mori theory [17], we have where NE (V) is the closed effective cone of curves on V, ƒÃ is a small positive number, NEƒÃ(V)={ZƒÃNE(V);(Z¥Kv)•†-ƒÃ(Z¥L)}, L being a (fixed) ample Non-complete algebraic surfaces 229 divisor, and li(1•…i•…s) is an extremal rational curve such that (li¥KV)< -ƒÃ(li¥L) . We shall first modify the Mori theory to match our situation by replacing KV by D#+KV (cf. [11], [12] and [22]). 2.2.1. LEMMA. Let D=rƒ°i=1 Di(Di: an irreducible component) and let D*=rƒ°i=1 aiDi with nonnegative rational numbers ƒ¿i. Set Then we have where e, L and li's are the same as specified above. PROOF. Let S=sƒ°i=1 R+[li]+rƒ°j=1 R+[Di]+NEƒÃ(D*, V). We have only to show NE(V)_??_S. So, take Z•¸NE(V) and let S(Z) be the set of all ex pressions of Z (modulo numerical equivalence): where ai, ƒ¿j •¸ R+, 1•…i•…s, l•…j•…r, and W•¸NE(V). Of course, S(Z) is not empty. Note that, for any expression of Z as above, we have Note that S(Z) is a closed subset of NE(V). Thus we can choose an element sƒ°i=1 aili+rƒ°j=1 ƒ¿iDj+ W of S(Z) such that takes the largest value. Then we claim that W•¸NEƒÃ(D*,V), i.e., (W¥D*+ KV)•†-ƒÃ(W¥L). So, suppose the contrary. We consider the following two cases separately. Case (W¥D*)<0. Since W•¸NE(V), we may write W=limn•‡ Wn, where {Wn} is a sequence of effective Q-divisors. Write Wn=Wn'+rƒ°j=1 x(n)j Dj, where x(n)j•†0, 1•…j•…r, and Wn' is an effective Q-divisor with Dj_??_Supp Wn', 1•…j•…r. Since {x(n)j} is bounded from above by (W¥L), replacing {Wn} by a suitable subsequence if necessary, we may assume that xj:=limn•‡x(n)j, 1•…j •…r, and W':=limn•‡ Wn' exist. Namely, we have W=W'+rƒ°j=1 xjDj, where xj•†0, 1•…j•…r, W'•¸NE(V) and (W'¥D*)•†0. On the other hand, since 230 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA we have rƒ°j=1 xj(Dj¥D*)<0. In particular, rƒ°j=1 xj>0. Furthermore, we have for which This is a contradiction.Case. Since we have Write Then sƒ°i yi=1>0, and we have This is again a contradiction. Thus W•¸NE(D*,V) and so Z•¸S. Q.E.D. 2.2.2. COROLLARY. Suppose D#+Kv is not numerically effective. Then, for every small positive number s, we have where Ci is a nonsingular rational curve surch that 0>(Ci¥D#+Ky)•†-3 for 1•…i•…m, and NEƒÃ(D#,V) is defined as in Lemma 2.2.1 with D* replaced by D#. Furthermore, if (V,D) is almost minimal, we may assume that every Ci satis fies one of the following conditions: In case Ci_??_Supp D, Ci is an irrelevant component of a rational twig, a rational rod or a rational fork; In case Ci_??_Supp D, Ci is an exceptional curve of the first kind or a fiber of a P1-bundle (when V is a P1-bundle over a curve), or a line on P2(when V_??_ P2) PROOF. Apply Lemma 2.2.1 to D*=D#, and pick all curves C2, 1•…i•…m, from li's and Dj's such that (D#+KV¥C)<-ƒÃ(L¥Ci)<0. If Ci_??_Supp D then Ci is one of li's, so pa(Ci)=0 and (D#+KV•Ci)•†(D#¥Ci)-3•†-3 (cf. Mori 117]). If Ci•¼Supp D then Ci is an irrelevant component of a rational twig, a rational rod or a rational fork (cf.1.4.2). Hence pa(Cj)=0 and (D#+Kv•Ci) =(Ci+KV¥Ci)+(D#-Ci¥Ci)•†-2. The second assertion is now clear from the Mori theory. Q.E.D. Non-complete algebraic surfaces 231 2.3. We assume that (V,D) is almost minimal and D#+KV is not nu merically effective. We then recall that the intersection matrix of Bk (D) is negative-definite, that Bk (D) contains no exceptional curves of the first kind and that every connected component ƒ¡ of Bk (D) is an admissible rational twig, an admissible rational rod or an admissible rational fork. The dual weighted graph of ƒ¡ has one of the following forms: (1) where ai•†2, 1•…i•…r, if ƒ¡ is a twig or a rod; (2) where bij•†2, 1•…i•…3, 1•…j•…ri, and b•†2, if ƒ¡ is a fork. In case (1), define positive integers n, q with GCD (n,q)=1 by the continued fraction n/q= [al,...,ar] (cf.1.3.1 for the notation). In case (2), define positive integers ni, qi (1•…i•…3) with GCD (ni,qi)=1 by ni/qi=[bi1,...,biri]; then {n1,n2,n3}= {2,2,n} (n•†2), {2,3,3}, {2,3,4} or {2,3,5} up to a permutation. It is known (cf. Brieskorn [3]) that, if char k=0, ƒ¡ is identified with the set of exceptional curves of a minimal resolution of a quotient singularity (hence a rational singularity); a cyclic quotient singularity in case (1), and a quotient singu larity (A2k/G,0) with respect to a small finite subgroup G of GL(2,k) in case (2). Even if char k=p>0, we shall see, in 2.4 below, that ƒ¡ contracts down to a rational singular point and that ƒ¡ is identified with the set of exceptional curves of a minimal resolution of this singularity. Thus, even in case char k > 0, we call the singularity a "quotient singularity". 2.4. Now we have to recall several basic facts on rational singularities (cf. Artin [1], [2]). Let ƒ¡=•¾si=1 Ci be a set of nonsingular irreducible curves on a nonsingular projective surface V. For any effective divisor Z= sƒ°i=1 ƒ¿iCi with ƒ¿i•¸Z+, we set pa(Z)=1/2(Z¥Z+KV)+1, the arithmetic genus of Z. On the other hand, let Q be a singular point of a normal projective surface V and let f:VV be a minimal resolution of all singularities on V if it exists. Let ƒ¡=•¾si=1 Ci be the set of exceptional curves on V over the 232 MASAYOSHI MIYANISHI and SHUICHIRO TSTJNODA point Q. We say that Q has rational singularity if pa(Z)•…0 for every effective divisor Z with Supp (Z)•¼ƒ¡ and Z•‚0. It is clear that if Q has rational singu larity, Ci is a nonsingular rational curve for 1•…i•…r. LEMMA (Actin [1; Th. 2.3 and Cor. 2.6]). (1) Let ƒ¡=•¾si=1 Ci be a set of nonsingular irreducible curves on a nonsingular projective surface V. Then the following conditions are equivalent: (a) ƒ¡ is contractible to a rational singular point, and if f:VV is the contraction off then x(V,_??_v)=x(V,_??_v). (b) The intersection matrix of ƒ¡ is negative-definite, and pa(Z)•…0 for every effective divisor Z with Supp (Z)•¼ƒ¡ and Z•‚0. Moreover, V is a normal projective surface. (2) Let f:VV be the contraction as in the above condition (a). Then there exists an integer N>0 such that, for every Weil divisor D on V, ND is linearly equivalent to a Cartier divisor. PROOF. We give some comments on the assertion (2). For any divisor D on V, we can define the direct image f*D exactly as in the case where f is a morphism of nonsingular surfaces. Moreover, if D•`D' (linear equivalence) on V then f*D•`f*D' Now, given a Weil divisor, D, we can easily find a divisor D on V such that f*D=D. Find then the rational numbers ƒ¿1,...,ƒ¿3 by the condition Then dƒ¿i•¸Z, 1•…i•…s, where d is the determinant of the matrix (-(Ci¥Cj))1•…1,j•…s. Let Dl:=d(D+sƒ°i=1 ƒ¿iCi). Then D1 is an integral divisor such that (Dl¥Cj)=0 for 1•…j•…s. By virtue of [ibid; Cor. 2.6], there exists a divisor D2 on V such that D1•`D2 and Supp (D2)•¿ƒ¡=ƒÓ. Then dD=f*D1•` f*D2 and f*D2 is apparently a Cartier divisor on V. Thus we can take N=d. Q.E.D. If ƒ¡ is contractible, let Z0 be the fundamental cycle of ƒ¡ (or the normal singular point obtained by the contraction of ƒ¡). The singular point is rational if and only if pa(Z0)=0 (Actin [2; Th. 3]). Now taking ƒ¡ to be a connected component of Bk (D) (with the notations in 2.3) and applying the above lemma, we see readily that ƒ¡ contracts to a rational singular point. If one uses Brieskorn's result [3], pa(Z0)=0 for the fundamental cycle Z0 of ƒ¡ provided char k=0. However, Z0 is determined in a combinatorial way, hence independently of whether or not char k=0. Thus ƒ¡ contracts to a rational singular point in case char k>0, as well. 2.5. Let (V,D) be a pair as set in the paragraph 2.3. Let f:VV be Non-complete algebraic surfaces 233 the contraction of all connected components of Bk (D). Then V is a normal projective surface carrying at worst "quotient singular points". Further more, there exists an integer N>0 such that, for every Weil divisor G on V,. NG is linearly equivalent to a Cartier divisor on V. We assume also [ND#] =ND#. We note that the intersection multiplicity (A¥G) of a Weil divisor G and a Cartier divisor A on V is well-defined. Hence, for Weil divisors G and H on V, the intersection multiplicity (G¥H) is defined to be (1/N)(G¥NH). Therefore, we have the intersection theory for Veil divisors on V, though the multiplicities are, in general, rational numbers. We set D:=f*(D0); D is an integral (Weil) divisor, On the other hand, let KV be the canonical divisor on V which is defined as follows: Let V0 be the smooth part of V, i.e., V0:=V-Sing V. Then KV is the unique Well divisor on V such that KV•¿V0=Kv0, which is determined up to linear equivalence. Let G be a Well divisor on V. We define a numerical equivalence class f*(G) in N(V)Q:=NS(V)_??_ZQ by f*(G):=(1/N) f*(NG), where NS(V) is the Neron-Severi group of V and where f*(NG) is the inverse image of a Cartier divisor NG in the usual sense. Evidently, f*(G) is independent of the choice of an integer N. Now we shall prove the following: LEMMA. (1) Kv•`f*(KV). (2) D#+KV•ßf*(D+KV) in N(V)Q. PROOF. Let V0:=f-1(V0). Then f*(K)•¿V0=(f0)*(Kv•¿V0)•`KV•¿V0, where f0:=f|V0. Hence f*(K)•`KV. This proves the assertion (1). To verify the assertion (2), note that N(D#+K)-Nf*(D+KV) is an integral divisor and linearly equivalent to a divisor B with Supp B•¿Supp Bk (D)=ƒÓ. Thus we have only to show that, for an irreducible curve G or V, we have N(D#+KV¥f*G)=N(f*(D+KV)¥f*G). This can be shown as follows: N2(f*(D+Kv)¥f*G)=(f*(N(D+Kr))¥f*NG)=(N(D+KV)¥NG) =(f*N(D#+KV)¥NG)=(N(D#+KY)¥f*NG)=N2(D#+KV¥f*G). Q.E.D. 2.6. Let (V,D) be the pair as set in the paragraph 2.3. Note that V is a normal projective surface and the numerical equivalence of Weil divisors on V is well-defined because the intersection theory is available for such divisors. So, let NS(V)Q be the Q-vector space of all Q-divisors, i.e., formal fi nite sums of irreducible curves on V with coefficients in Q, modulo numerical equivalence. Set N(V):=NS(V)Q_??_Q R, whose dimension equals the Picard number p(V) of V. We consider any norm on N(V) so that it defines the 234 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA metric topology on N(V). Let NE(V)Q be the smallest convex cone in NS(V)Q containing all irreducible curves on V and closed under multipli cation by elements of Q+. The cone NE(V)R in N(V) is defined in a similar fashion, and let NE(V) be the closure of NE(V)R with respect to the metric topology on N(V). Let L be an ample (Cartier) divisor on V. On the other hand , write Supp Bk (D)=•¾ri=1 Di, Di being an irreducible component. Let L be an ample divisor on V such that L=f*L. Set L-f*L=-W, where Supp W_??_ Supp Bk (D). Since the intersection matrix of Bk (D) is negative-definite , W is an effective divisor such that (W¥Di)<0 for 1•…i•…r . We fix these divisors L and L in the subsequent arguments. For a small positive number E , we set NEƒÃ(D,V)={Z•¸NE(V),(Z¥D+Kv)•†-ƒÃ(Z¥L)}. With these notations, we have the following: LEMMA. For every positive number ƒÃ, there exists (possibly singular) rational curves li, 1•…i•…u, such that and 0>(D+Kv•li)•†-3, 1•…i•…u. PROOF. For every integer n>0, we set where L=f*(L)-W, the ample divisor as chosen above. By virtue of Cor. 2.2.2, there exist nonsingular rational curves C(n)j, 1•…j•…u(n), such that (i) NE(V)=u(n)ƒ°j=1 R+[C(n)j]+NE1/n(D#,V), (ii) -(1/n)(C(n)j¥L)>(C(n)j¥D#+KV)•†-3, 1•…j•…u(n), whence C(n)j_??_ Supp Bk (D). We may assume that: (iii) if n'>n, we have u(n')•†u(n), C(n')j=C(n)j for 1•…j•…u(n), and (C(n')j¥D#+KV)•†-(1/n)(C(n')j¥L) for u(n)+1•…j•…u(n'). Let l(n)j:=f*(C(n)j). Then, since (l(n')j¥L)=(C(n')j¥L)+(C(n')j¥W)•† (C(n')j¥L), the condition (iii) implies: (iv) (l(n')j¥D+KV)•†-(1/n)(l(n')j¥L), whenever n'>n and u(n)+1•…j•… u(n'). The condition (iv) implies, in particular, that almost all elements of {l(n)j; n>0, 1•…j•…u(n)}, except finitely many elements, say l1,...,lu, belong to NE(D,V), where the number is fixed throughout the proof. Let Non-complete algebraic surfaces 235 Then we have only to prove that S=NE(V), because the condition (ii) above entails (li¥D+KV)•†-3, 1•…i•…u. For this purpose, it suffices to show that every irreducible curve C on V with (C¥D+KV)<0 belongs to S. Let C be the proper transform of C on V. Since (C¥D#+KV)=(C¥D+KV)<0, C is ex pressed as where Zn is written as (cf, the proof of Lemma 2.2.1). Furthermore, let Ci be the proper transform of li on V, and write where Ci_??_Supp Jn. Hence we have and Note that {x(n)i} and {a(n)i} are the sequences bounded from above by (C¥L). Therefore they have convergent (infinite) subsequences. Therefore {Jn+Z'n} has a convergent subsequence as well. To simplify the notations, we assume that {x(n)i}, {a(n)i} and {Jn+Z'n} are convergent. Then {f*Jn+f*Zn} is conver gent, and limn•‡ (f*Jn+f*Z'n)=f*(limn•‡ (Jn+Zn)). Moreover, we have we shall show that limn•‡ (f*Jn+f*Z'n)•¸NEƒÃ(D,Y). Indeed, by the defini tion of Jn, we have (f*Jn¥D+KV)•†-ƒÃ(f*Jn¥L). Take n so large that nƒÃ>1, and let Wi=limn•‡ rƒ°i-1 a(n)i Di. Then we have: 236 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA where andis very close to (W1¥W) if n>>0. Therefore we have so we obtain Namely, limn•‡. (f*Jn+f*Zn)•¸NEƒÃ(D,V), and NE(V)=S. Q.E.D. 2.7. Since NE(V) is polyhedral on the side D+Kv+ƒÃL<0, we can define an extremal rational curve l as follows: a half line R=R+[Z] in NE(V) is an extremal ray if (i) (D+KV¥Z)<0 and (ii) Z1 and Z2 in NE(V) satisfy Z1, Z2 E R if Z1+Z2•¸R; a rational curve l on V is an extremal rational curve if 0 >(D+KV¥l)•†-3 and R+[l] is an extremal ray. As an analogy of the Mori theory, we shall prove the following: LEMMA. Let the notations and the assumptions be the same as above. Let l be an extremal rational curve on V and let l be the proper transform of 1 on V. Then one of the following two cases takes place: (1) The intersection matrix of l+Bk (D) is negative-semidefinite, but not negative-definite. Moreover, (l2)=0. (2) The Picard number p(V) equals 1, and -(D+KV) is ample. PROOF. As in Mori [17], there exists a numerically effective Q-divisor lion V such that H•Û•¿NE(V)=R+[l]; where H•Û={Z•¸NE(V); (H¥Z)=0}. Let H:=f*(H). Then N is numerically effective on V. We consider the fol lowing three cases separately. (1)(H2)>0; (2) (H2)=0 and H is not numeri cally trivial (H_??_0); (3) H is numerically trivial (H•ß0). In case (1), it is readily ascertained, by the lodge index theorem, that the intersection matrix of l+Bk (D) is negative-definite. Since (l¥D+KV)<0, the existence of l contradicts the hypothesis that the pair (V,D) is almost minimal. Therefore, the case (1) cannot occur. Non-complete algebraic surfaces 237 In case (2), by virtue of the Hodge index theorem, the intersection matrix of l+Bk (D) is negative-semidefinite. Furthermore, by the same reasoning as in the case (1) above, the intersection matrix of l+Bk (D) cannot be negative-definite. Suppose (l2)<0. Note that mH-l is numerically effective on V for a sufficiently large integer m (cf. Mori [17; Lemma 3.7]). Then f*(mH-l) is numerically effective on V. Hence (mH-1)2=(f*(mH-l))2•†0 and (mH-1)2=m2(H2)+(l2)•†0, so (H2)=(H2)>0, a contradiction. Therefore, (l2)=0. In case (3), we conclude that N(V)=R[l]. Hence p(V)=1, and l is ample. Since (D+Kv¥l)<0, -(D+KV) is ample. Q.E.D. 2.8. LEMMA. Let the notations and the assumptions be the same as in Lemma 2.7. Suppose the first case occurs. Then the following assertions hold: (1) For a sufficiently large integer n, the linear system |nNf*(l)| is com vosed of an irreducible pencil, free from base points, whose general members are isomorphic to P1(cf, the paragraph 2.5 for the definition of the integer N). Therefore V is a ruled surface. (2) V-D is affine-ruled, so i(V-D)=-•‡. PROOF. (1) Define rational numbers ƒ¿l,...,ƒ¿r by the condition where Supp Bk (D)=•¾ri=1 Di. Since l_??_Supp Bk (D), we have ƒ¿i•†0 for 1•…i•… r. Set Z=l+rƒ°i=1 ƒ¿iDi. Then (Z2)•…0 by the hypothesis, and (Z2)=0, indeed; for, otherwise, l+Bk (D) would be negative-definite. By the defini tion, we have (l¥rƒ°i=1 ƒ¿iDi)=-(rƒ°i=1 ƒ¿iDi)2. Then (Z2)=0 implies (l2)= (rƒ°i=1 ƒ¿iDi)2. So, (Z¥l)=0, and (l2)=0 if and only if ƒ¿i=0, 1•…i•…r. On the other hand, Nf*(l):=f*(Nl) is a Cartier divisor such that (f*(Nl))2=N2(l2) =0 and f*f*(Nl)=Nl. Hence f*(Nl)-N1 is supported by •¾ri =1 Di. It is easy to see that f*(Nl)=NZ. So, we denote the Q-divisor Z by f*(l). By Lemma 2.5, we have (f*(l)¥Kv)=(l¥K)<-(l¥D)•…0 because (l2)=0 implies (l¥D)•†0. Noting that H2(V,nNf*(l))_??_H0(V,KV-nNf*(l))=(0) for a sufficiently large integer n, by the Riemann-Rosh theorem, we have Note that f*(l) is connected. Indeed, if l meets an irreducible component Di of Bk (D), then Supp f*(l) contains the connected component of Bk (D) con taining Di. This remark and the above Riemann-Rock inequality implies 238 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA that |nNf*(l)| is composed of an irreducible pencil •È. Let ƒ¡ be a general member of •È, and let C=ƒ¡red; if char k=0 then C is, of course, reduced, while, in case char k=p>0,ƒ¡ might be of the form paC with C=Fred and a>0. The curve C is an irreducible reduced curve such that (C2)=0 and (C¥KV)<0. Hence pa(C)=0 and (C¥KV)=-2. Therefore, taking n to be sufficiently large, we know that ƒ¡ is reduced and hence |nNf*(l)| is composed of an irre ducible pencil whose general members are nonsingular rational curves. Thus V is a ruled surface. (2) Let C be as above and let C=f*(C). Since (C¥Di)=0, 1•…i•…r, the curve C is a nonsingular rational curve which does not pass through any singular point of V. Hence (C2)=0. (C¥K)=-2 and (C¥D)•…1, because (C¥D+K•…)<0 implies (C¥D)<-(C¥KV)=2. Therefore we know that C does not meet any irreducible component of Bk (D) and that C meets at most one irreducible component (then transversally in a single point) of Supp (D) Supp Bk (D). This implies that V-D is affine-ruled. Q.E.D. 2.9. Consider the case (2) of Lemma 2.7. Then we have the following result: LEMMA. Let the notations and the assumptions be the same as in Lemma 2.7. Suppose the second case occurs. Then the following assertions hold (1) -(D#+KV) is numerically effective and, for an irreducible curve C on V, (D#+KV¥C)=0 if and only if C is an irreducible component o f Bk (D). Moreover, k(V-D)=-•‡. (2) Let Supp Bk (D)=•¾ri=1 Di. Then D-rƒ°i=1 Di is connected and has at most two irreducible components. Each irreducible component is a nonsingular rational curve, and if D-rƒ°i=1 Di has two irreducible components Y1,Y2 then (Y1¥Y2)=1. (3) If D-rƒ°i=1 Di has two irreducible components Y1,Y2 then the con nected component of D containing Y1,Y2 has the following dual weighted graph (R1) where Ti(i=1,2) is an admissible maximal rational twig which might be empty. (4) Suppose Y:=D-rƒ°i=1 Di is irreducible. Then the connected com ponent of D containing Y is a non-admissible rational fork or has the following dual weighted graph: Non-complete algebraic surfaces 239 (R2) where Ti(i=1,2) is an admissible maximal rational twig which might be empty. PROOF. (1) Since D#+KV•ßf*(D+KV) and -(D+KV) is ample , the assertion (1) holds clearly. (2) Let D':=D-rƒ°i=1 Di and D#-D'=rƒ°i=1 ƒ¿iDi. Let D'=1 sƒ°j=•¢j be the decomposition of D' into connected components . Let •¢j be the connected component of D containing •¢j. Then it is easy to see that d •¢i•‚•¢j whenever •¢i•‚•¢j and that f*(D)=sƒ°j=1 f*(•¢j) is the decomposition of f*(D) into con nected components. Since p(V)=1, we obtain s=1 . Namely, D' is connected. Now suppose D' has more than two irreducible components . Then D' has an irreducible component Y such that (D'-Y¥Y)•†2 , and hence we have which is a contradiction. Thus, D' has at most two irreducible components . If D' has two irreducible components Y1 and Y2, then where (Y2¥Y1)>0. Thence we get pa(Y1)=0 , (Y1¥Y2)=1 and (rƒ°i=1 ƒ¿iDi•Y1)<1. Similarly, pa(Y2)=0 and (rƒ°i =1 ƒ¿iDi¥Y2)<1. If D'(=Y) is irreducible, we have Thence we get pa(Y)=0 and (rƒ°i =1 ƒ¿iDi¥Y)<2. (3) when D'=Y1+Y2, we have (rƒ°i=1 ƒ¿iDi¥Yj)<1(j=1,2). Note that any connected component of Bk(D) which meets Y1(or Y2) is an admissible maximal rational twig T and the irreducible component of T meeting Y 1(or Y 2) has the coefficient 1-1/d(T) in rƒ°i=1 ƒ¿iDi, where d(T) is the determinant of T (cf.1.3). It is then easy to ascertain that there is at most one twig meeting Y1 (or Y2) and that the graph of the connected component •¢ of D containing Y1 and Y2 is the one given in the statement . (4) The same argument as in (3) above works also in this case. We only note that there are at most three admissible maximal rational twigs of D meeting Y and that if there are three, the connected component •¢ of D con taining Y is a non-admissible rational fork . Q.E.D. 240 MASAYOSHI MIYANISHI and SHUICHIRO TSUNODA 2.10. LEMMA. In the situation of Lemma 2.9, suppose that the connected component of D containing D-rƒ°i=1 Di has the dual graph o f the type (R1) or (R2) in the same lemma. Then V-D is affine-ruled. Therefore, if k(V-D) =-•‡ and V-D is not affine-ruled , then either D=rƒ°i=1 Di, i.e., Supp D= Supp Bk (D) or D0:=D-rƒ°i=1 Di is irreducible and the connected component of D containing D0 is a non-admissible rational fork for which D0 is the central component. PROOF. Suppose that the connected component •¢ of D containing D-rƒ°i =1 Di(•‚ƒÓ) has the dual graph of the type (R1) or (R2). Then d is a non admissible rational rod. If (Y21)•†0 (or (Y22)•†0, or (Y2)•†0) then we are done by [10; Chap. I, Cor. 2.4.3]; the quoted result holds in an arbitrary charac teristic case. Otherwise, (Y21)=(Y22)=-1 (or (Y2)=-1). After contracting Y1 in the case (R1), and after contracting Y and several (successively) adjacent components in the rod in the case (R2), we find an irreducible component with non-negative self intersection number. This is possible because the inter section matrix of •¢ is not negative definite. Thus we are done by the above quoted result. Q.B.D. 2.11. We shall summarize Theorem 1.12 and a part of the results ob tained in this chapter in the following: THEOREM. Let V be a nonsingular projective surface and let D be a reduced effective divisor with simple normal crossings. Suppose the pair (V,D) is almost minimal. Then we have: (1) k(V-D)•†0 if and only if D#+KV is numerically effective. (2) Suppose D#+KV is not numerically effective. Let f:VV be the con traction of all connected components of Bk (D), V being a normal projective surface carrying at worst "quotient singular points", and let D=f*(D#). Then V-D is affine-ruled possibly except the case: (i) p(V)=1, (ii) -(D+KV) is ample, and (iii) Supp (D)=Supp Bk (D) or there exists a unique irreducible component D0 of D not in Supp Bk (D) such that the connected component of D containing D0 is a non-admissible rational fork for which D0 is the central component. 2.12. Let (V,D) be a pair as in Theorem 2.11. The pair (V,D) (or rather V) is called a logarithmic del Pezzo surface of rank one if p(V)=1 and -(D+KV) is ample. It is said to have the non-contractible (or contractible, resp.) boundary if Supp (D)•‚Supp Bk (D) (or Supp (D)=Supp Bk (D), resp.). Let V be anew a normal projective surface possessing at worst "quotient singular points". 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