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,~<r if D=~i-1 C, is a decomposition of D into irreducible components, is not negative-definite. Then X is an A* fiber space over Pik. Moreover, X is affine-uniruled. Namely, there exists a. dominant quasi-finite morphism: UoX A -±X, where Uo is an a ffine curve. The case where the intersection matrix of D is negative-definite will be treated in our forthcoming paper [16]. As an application of this result, we can prove the following: THEOREM0.3 (Miyanishi [12], [13]). Let X be an affine normal surface defined over the field of complex numbers C. Assume that X has AC as a finite covering, i.e., there exists a finite surjective morphism 8: A2X. Then X is isomorphic to the quotient variety AC/G with respect to a small finite subgroup G of GL(2, C). Moreover, if the coordinate ring P(X, &~) of X is factorial, X is Non-complete algebraic surfaces 197 isomorphic to either the affine plane Ac or a hypersurface in A=Spec C[x, y, z] defined by 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.