Complements of plane curves with logarithmic Kodaira dimension zero

Transcription

1 J. Math. Soc. Japan Vol. 52, No. 4, 2000 Complements of plane cuves with logaithmic Kodaia dimension zeo By Hideo Kojima (Received Ap. 14, 1999) Abstact. We pove that logaithmic geometic genus of a complement of plane cuve with logaithmic Kodaia dimension zeo is equal to one. 1. Intoduction. Let B H P 2 be a educed pojective plane cuve de ned ove the complex numbe eld C. To study the cuve B, the logaithmic Kodaia dimension k P 2 B of P 2 B plays an impotant ole. Thee ae some esults fo the calculation of k P 2 B (see [18] and [20], etc.). In [12], Miyanishi and Sugie studied the stuctue of P 2 B when k P 2 B ˆ y by using the A 1 -uling theoem (cf. [11, Chapte I]). In [16] (o [15]), Tsunoda classi ed ational cuspidal cuves (i.e., ational cuves with only cusps as singulaities) B of k P 2 B ˆ1 with unique singula points by using the stuctue theoem of non-complete algebaic sufaces with k ˆ 1 due to Kawamata [6] (see also [11, Chapte II]). Recently, in [8], Kishimoto studied ational cuspidal cuves B of k P 2 B ˆ1 with two singula points. In the pesent aticle, we shall study the case k P 2 B ˆ0, mainly using the classi cation theoy of a½ne sufaces with k ˆ 0in[9]. The main esult is the following theoem. Theoem 1.1. Let B H P 2 be a educed pojective plane cuve whose complement has logaithmic Kodaia dimension zeo. Then the following assetions hold tue: (1) p g P 2 B ˆ1, whee p g P 2 B denotes the logaithmic geometic genus of P 2 B. (2) If B is not an ieducible nonsingula cubic cuve then each ieducible component of B is a ational cuve. (3) ] B (ˆ the numbe of ieducible components of B) U 3 and the equality holds if and only if P 2 B G C C, whee C ˆ C f0g. (4) If B is an ieducible ational cuve then B has unique singula point and the numbe of analytic banches of B at the singula point is equal to two. In [15], Tsunoda obtained the same esult as Theoem 1.1 when B is ieducible. As applications of Theoem 1.1, we study the fundamental goups and the topological Eule chaacteistics of the sufaces P 2 B with k P 2 B ˆ0 in Mathematical Subject Classi cation. Pimay 14J26, Seconday 14H20. Key Wods and Phases. Plane cuves, logaithmic Kodaia dimension, logaithmic geometic genus. The autho is patially suppoted by JSPS Reseach Fellowships fo Young Scientists and Gant-in-Aid fo Scienti c Reseach, the Ministy of Education, Science and Cultue.

2 794 H. Kojima By a n -cuve (n V 1) we mean a nonsingula complete ational cuve with selfintesection numbe n. A educed e ective diviso D is called an SNC-diviso (esp. an NC-diviso) if D has only simple nomal cossings (esp. nomal cossings). Let f : X 1! X 2 be a biational mophism between smooth sufaces X 1 and X 2 and let D i i ˆ 1; 2 be a diviso on X i. We denote the diect image of D 1 on X 2 (esp. the total tansfom of D 2 on X 1, the pope tansfom of D 2 on X 1 )by f D 1 (esp. f D 2, f 0 D 2 ). We efe to [5] fo the de nitions of the logaithmic Kodaia dimension k, the logaithmic geometic genus p g, the logaithmic n-genus P n n V 1 and the logaithmic iegulaity q, etc. The autho would like to expess his gatitude to Pofesso Masayoshi Miyanishi who gave the autho valuable advice and encouagement duing the pepaation of the pesent aticle. 2. Peliminaies. We ecall some basic notions in the theoy of peeling (cf. [13] and [1]). Let X; B be a pai of a nonsingula pojective suface X and an SNC-diviso B on X. We call such a pai X; B an SNC-pai. A connected cuve T consisting of ieducible components of B (a connected cuve in B, fo shot) is a twig if the dual gaph of T is a linea chain and T meets B T in a single point at one of the end components of T, the othe end of T is called the tip of T. A connected cuve R (esp. F )inbis a od (esp. fok) ifr (esp. F ) is a connected component of B and the dual gaph of R (esp. F )isa linea chain (esp. the dual gaph of the exceptional cuves of a minimal esolution of a non-cyclic quotient singulaity). A connected cuve E in B is ational (esp. admissible) if each ieducible component of E is ational (esp. if thee ae no 1 -cuves in Supp E and the intesection matix of E is negative de nite). An admissible ational twig T in B is maximal if T is not extended to an admissible ational twig with moe ieducible components of B. Let ft l g (esp. fr m g, ff n g) be the set of all admissible ational maximal twigs (esp. all admissible ational ods, all admissible ational foks), whee no ieducible components of T l 's belong to R m 's o F n 's. Then thee exists a unique decomposition of B as a sum of e ective Q-divisos B ˆ B ] Bk B such that i) Supp Bk B ˆ 6 l T l U 6 m R m U 6 n F n, ii) B ] K X Z ˆ0 fo evey ieducible component Z of Supp Bk B. We call the diviso Bk B the bak of B and say that B ] K X is poduced by the peeling of B. Definition 2.1 (cf. [13, 1.11]). An SNC-pai X; B is almost minimal if, fo evey ieducible cuve C on X, eithe B ] K X C V 0 o the intesection matix of C Bk B is not negative de nite. We have the following esult due to Miyanishi and Tsunoda [13]. Lemma 2.2 (cf. [13, Theoem 1.11]). Let X; B be an SNC-pai. Then thee exists a biational mophism m : X! W onto a nonsingula pojective suface W such that the following fou conditions (iv) ae satis ed:

3 Complements of plane cuves 795 (i) C :ˆ m B is an SNC-diviso. (ii) m Bk B U Bk C and m B ] K X V C ] K W. (iii) P n X B ˆP n W C fo evey intege n V 1. In paticula, k X B ˆ k W C. (iv) The pai W; C is almost minimal. We call the pai W; C as in Lemma 2.2 an almost minimal model of X; B. The following esult follows fom [1, Lemma 6.20] and [13, Theoem 2.11 (1)]. Note that a od (esp. a fok) is called a club (esp. an abnomal club) in [1]. Lemma 2.3. Let X; B be an SNC-pai with k X B V 0. Assume that any ational twig of B is admissible. If X; B is not almost minimal then thee exists a 1 -cuve E, not contained in B, such that one of the following holds: (i) E V B ˆ q. (ii) E B ˆ1 and E meets an ieducible component of Supp Bk B. (iii) E B ˆ2 and E meets two di eent connected components of B such that one of the connected components is a ational od R n of B and E meets a tip of R n. Futhe, P n X B E ˆ P n X B fo any n V 1 and hence k X B E ˆ k X B. Lemma 2.4. Let X; B be an almost minimal SNC-pai with k X B ˆ0 and p g X B ˆ1. Assume that X is ational and B is connected. Then B K 0 and B is a nonsingula elliptic cuve o a loop of nonsingula ational cuves. Poof. See [9, Poposition 1.5 (1)] o [21, Reduction theoem]. Now we ecall the constuction of a stongly minimal model of a nonsingula a½ne suface with k ˆ 0 (cf. [9, 2]). Let S ˆ Spec A be a nonsingula a½ne suface with k S ˆ0 and let X; B be an SNC-pai with X B ˆ S. We call such a pai X; B an SNC-completion of S. Note that S is ational by [9, Theoem 1.6]. Let W; C be an almost minimal model of X; B. By contacting 1 -cuves E with E C U 1 successively, we obtain a biational mophism n : W! V such that F n C > 1 fo any 1 -cuve F on V. Put D :ˆ n C and S 0 :ˆ V D. We call the suface S 0 a stongly minimal model of S. By [9, Lemmas 2.3 and 2.4 and Coollay 2.5], we have the following esult. Lemma 2.5. With the same notation and the assumptions as above, the following assetions hold: (1) S 0 is an a½ne open subset of S and S S 0 is an empty set o a disjoint union of the a½ne lines A 1. (2) D is an NC-diviso. Futhemoe, if p g S ˆ0 then D becomes an SNC-diviso and the pai V; D is almost minimal. (3) P n S 0 ˆP n S fo any n V 1. In paticula, k S 0 ˆk S ˆ0. Definition 2.6. Let S ˆ Spec A be a nonsingula a½ne suface with k S ˆ0 and let X; B be an SNC-completion of S. We call the pai X; B (esp. the suface S) tobestongly minimal if X; B is almost minimal and E B > 1 fo any 1 -cuve E on X (esp. if thee exists a stongly minimal model S 0 of S such that S ˆ S 0 ). Note

4 796 H. Kojima that if S is stongly minimal and p g S ˆ0 then S has a stongly minimal SNCcompletion by Lemma 2.5 (2). Lemma 2.7. Let S ˆ Spec A be a nonsingula a½ne suface with k S ˆ0 and let X; B be an SNC-completion of S such that B i B B i V 3 fo any 1 -cuve B i H B. If X; B is not stongly minimal then thee exists a 1 -cuve E, not contained in B, such that E B ˆ1 and P n X B E ˆ P n X B fo any n V 1. Poof. If X; B is almost minimal then the assetion is clea by the de nition of stongly minimality and Lemma 2.5 (3). Suppose that X; B is not almost minimal. Since k X B ˆ0 and B i B B i V 3 fo any 1 -cuve B i H B, we know that any ational twig of B is admissible by vitue of [17, Step (3) in the poof of Theoem 1.3]. Futhe, B is connected and S contains no complete cuves since S is a½ne. Hence the assetion follows fom Lemma 2.3. We state the classi cation of stongly minimal a½ne sufaces with k ˆ 0. moe details, see [9]. Fo Lemma 2.8 (cf. [9, Theoems 0.1, 4.5 and 5.4]). Let S be a stongly minimal nonsingula a½ne suface with k S ˆ0. Then we have: (1) S is one of the sufaces in Table 1, whee m S, e S and p 1 S ae espectively Table 1 Type m S q S e S p 1 S Z= Z= 2 O Z= 2 O k 4; k k V Z= k 2 O 4; Z O 2; Z O 1; 1; 1 (G C C ) Z 2 X 2Š Z= 4 H 1; 0; 1Š hy; ti= yty 1 t H 0; 0Š Z H k; kš k V Z= 4k Yf3; 3; 3g Z= 9 Yf2; 4; 4g Z= 8 Yf2; 3; 6g Z= 6

5 Complements of plane cuves 797 Figue 1 the least positive intege such that P m S S > 0, the topological Eule chaacteistic of S and the fundamental goup of S. (2) Assume futhe that p g S ˆ0 and e S U 1. Let V; D be a stongly minimal SNC-completion of S. Then the con guation of D is one of (e) in Figue 1, whee each line epesents a nonsingula ational cuve and each numbe indicates the selfintesection numbe of the coesponding cuve. Coollay 2.9. Let S be a nonsingula a½ne suface with k S ˆ0. Then the following assetions hold: (1) e S V 0 and the equality holds if and only if S is stongly minimal and of type O 1; 1; 1 o H 1; 0; 1Š. (2) Assume that e S ˆp g S ˆ0, i.e., S is of type H 1; 0; 1Š. Let V; D be an SNC-completion of S such that D i D D i V 3 fo any 1 -cuve D i H D. Then V; D is stongly minimal and the con guation of D is given as (b) in Figue 1. Poof. By Lemmas 2.5 (1), 2.7 and 2.8, the assetions ae clea. 3. Poof of Theoem 1.1, pat I. In this section, we pove Theoem 1.1 when the cuve B is educible. some lemmas to be used late. We pove Lemma 3.1. Let V be a nonsingula pojective suface with q V :ˆ h 1 V; O V ˆ0 and let D be a non-zeo educed e ective diviso on V. Then q V D V ] D V ;

6 798 H. Kojima whee V is the Picad numbe of V. V ˆ1. Futhemoe, the equality holds povided Poof. Let D ˆ Pi D i be the ieducible decomposition of D. Since q V ˆ0, we get q V D ˆdim Q Ke 0 Q D i Š!Pic V n Q i by [2, Lemma 2]. Hence q V D V ] D V : If V ˆ1 then the natual map 0 i Q D i Š!Pic V n Q is sujective. ] D 1. So q V D ˆ Lemma 3.2. Let S be a nonsingula a½ne suface with k S ˆ0. Then q S U 2. Moeove, q S ˆ2 if and only if S G C C. Poof. By [3, Theoem II], the assetions ae clea. See also [7, Theoem 2.8 and Coollay 2.9]. Now we shall pove Theoem 1.1 when B is educible. Lemma 3.3. With the same notation as in Theoem 1.1, ] B U 3 and the equality holds if and only if P 2 B G C C. In paticula, if ] B ˆ3 then p g P 2 B ˆ1. Poof. We note that p g C C ˆ1. Lemma 3.1 implies that q P 2 B ˆ ] B 1. So, by Lemma 3.2, we know that ] B U 3 and the equality holds if and only if P 2 B G C C. Among the assetions of Theoem 1.1, (3) and (1) in the case ] B V 3 ae vei ed. Next we conside the case ] B ˆ2. Lemma 3.4. With the same notation as in Theoem 1.1, assume that ] B ˆ2. Then p g P 2 B ˆ1. Poof. Put S :ˆ P 2 B. Note that p g S U 1 because k S ˆ0. Suppose to the contay that p g S ˆ0. Let B ˆ B 1 B 2 be the ieducible decomposition of B. Let m : W! P 2 be a composite of blowing-ups such that C :ˆ m 1 B becomes an SNCdiviso and that m is the shotest among such biational mophisms. Fom now on, we call such a mophism m a minimal SNC-map fo the pai P 2 ; B. Note that W C ˆ S. Since p g W C ˆp g S ˆ0, each ieducible component of C is a nonsingula ational cuve and the dual gaph of C is a tee by [11, Lemma I.2.1.3]. So B 1 and B 2 ae ational cuspidal cuves and meet in only one point P. Hence e S ˆe P 2 e B 1 e B 2 fpg ˆ ˆ 0. By Coollay 2.9 (1), S is of type H 1; 0; 1Š. Let C i i ˆ 1; 2 be the pope tansfom of B i on W. Assume that C j C C j V 3 fo any 1 -cuve C j H C. Then, by Coollay 2.9 (2), the con guation of C is given as (b) in Figue 1. Since each component of C C 1 C 2 has negative selfintesection numbe, D 4 is one of fc 1 ; C 2 g and eithe fc 1 ; C 2 g V fd 1 ; D 2 ; D 3 gˆq o fc 1 ; C 2 g V fd 5 ; D 6 ; D 7 gˆq. Then thee exists P 1 A P 2 such that D i D i 1 D i 2 ˆ

7 Complements of plane cuves 799 m 1 P 1, whee i ˆ 1 o 5. This is a contadiction. So thee exists a 1 -cuve H in Supp C such that H C H U 2. By the minimality of m, we know that H ˆ C 1 o C 2. Assume that H ˆ C 1. We claim that: Claim. C 1 C C 1 ˆ2. Poof. If C 1 C C 1 ˆ1 then k W C ˆk W C C 1 ˆ 0. Since W C C 1 ˆP 2 B 2, we have k P 2 B 2 ˆ0. In the next section, we pove that if D H P 2 be an ieducible ational cuspidal cuve then k P 2 D 0 0 (cf. Lemmas 4.1 and 4.2). So we have a contadiction. The above claim implies that thee exists a unique singula point Q A B 1 othe than P. Then, since Q is a cusp of B 1, thee exists a unique decomposition of m 1 Q as a sum of non-zeo educed e ective divisos m 1 Q ˆE F G such that the following thee conditions ae satis ed: (i) F and G ae connected. (ii) E is a unique 1 -cuve in m 1 Q and hence each component of F G has self-intesection numbe U 2. (iii) E F ˆ EG ˆ E C 1 ˆ1. The dual gaph of C is given as in Figue 2, whee we put C ~ :ˆ C C 1 E F G. We have C 1 C ˆ1. ~ Let n : W! W 0 be a sequence of contactions of 1 -cuves and subsequently contactible cuves in C, stating with the contaction of C 1, such that C 0 :ˆ n C is an SNC-diviso and that the contaction of any 1 -cuve in C 0 makes the image of D 0 lose the simple nomal cossing popety (the SNC-popety, fo shot). Then n E 2 V 0 and the weighted dual gaphs of n F and n G ae the same as those of F and G. Futhe, Ci 0 C 0 Ci 0 0 V 3 fo any 1 -cuve Ci because the dual gaph of C is a tee. Since W 0 C 0 ˆ S is of type H 1; 0; 1Š, the con guation of C 0 is given as (b) in Figue 1 by Coollay 2.9 (2). Since n E 2 V 0, n E ˆD 4 o D 5. If n E ˆD 4 then n C 1 C ˆ0 ~ and D 5 D 6 D 7 ˆ n F o n G. This is a contadiction because n F and n G contain no ieducible cuves with selfintesection numbe V 1. If n E ˆD 5 then n C ˆD ~ 1 D 4 and F and G ae ieducible 2 -cuves. This is also a contadiction because the intesection matix of E F G is then not negative de nite. The assetion (2) of Theoem 1.1 follows fom Lemma 3.5 below. Lemma 3.5 (cf. [10, Lemma 4]). Let B H P 2 be a educed cuve. Assume that k P 2 B U 1 and B contains a non-ational cuve. Then B is an ieducible nonsingula cubic cuve. Figue 2

8 800 H. Kojima Poof. Since B contains a non-ational cuve, deg B V 3. By vitue of [4, Theoem 4], we have k P 2 B ˆk B K P 2; P 2 ˆk deg B 3 l; P 2, whee l is a line on P 2 and k B K P 2; P 2 denotes the B K P 2 -dimension of P 2 (cf. [5]). If deg B V 4 then k deg B 3 l; P 2 ˆ2. So deg B ˆ 3 and hence B is an ieducible nonsingula cubic cuve. 4. Poof of Theoem 1.1, pat II. In this section, we teat the case whee B is ieducible. All esults in this section except fo Lemma 4.3 ae stated in [15], whee thei poofs howeve ae not given. Fo the sake of completeness, we give the poofs which use the classi cation theoy of the a½ne sufaces with k ˆ 0 (cf. 2). In [14], Oevkov independently gave the poofs of Lemmas 4.1 and 4.2. Ou poofs ae almost the same as Oevkov's. Assume that p g P 2 B ˆ0 and B is ieducible. Then, by using the same agument as in the poof of Lemma 3.4, we know that B is a ational cuspidal cuve. If B is nonsingula, B is a line o a conic and k P 2 B ˆ y. So ]Sing B V 1. By [18, Theoem (II)], ]Sing B U 2. Hee we note that e P 2 B ˆe P 2 e B ˆ 3 2 ˆ 1. We shall conside the cases ]Sing B ˆ1 and ]Sing B ˆ2 sepaately. Lemma 4.1. If B H P 2 is a ational cuspidal cuve with ]Sing B ˆ1. Then k P 2 B 0 0. Poof. Suppose that k P 2 B ˆ0. Let m : W! P 2 be a minimal SNC-map fo P 2 ; B (cf. the poof of Lemma 3.4) and let C 1 be the pope tansfom of B on W. Let P be the unique singula point of B. Then thee exists a unique decomposition of m 1 P as a sum of nonzeo educed e ective divisos m 1 P ˆE F G such that the conditions (iii) fo m 1 Q as in the poof of Lemma 3.4 hold. The dual gaph of C :ˆ m 1 B ˆC 1 E F G is given as in Figue 3. Since C 1 C K W ˆ 1 < 0 and k W C ˆ0, we know that C 1 2 < 0bythe theoy of Zaiski decomposition (cf. [17, the poof of Theoem 1.3]). If C 1 2 ˆ 1 then k W C ˆk W C C 1 ˆ 0 because C 1 C C 1 ˆ1. Since C C 1 ˆ m 1 P can be contacted to a smooth point, we have k W C C 1 ˆ y, which is a contadiction. So C 1 2 U 2. Suppose that W; C is stongly minimal (cf. De nition 2.6). Then, in view of e W C ˆ1, we know that the con guation of C is given as one of (a), (c), (d) and (e) in Figue 1. Since C contains a unique 1 -cuve E, the con guation of C is eithe (a) o (e). If the case (a) occus then C contains a cuve with non-negative selfintesection numbe, which is a contadiction. If the case (e) occus then E F G ˆ D 1 D 3 D 4 D 9 since C 1 is ieducible. This is also a contadiction because Figue 3

9 Complements of plane cuves 801 the intesection matix of D 1 D 3 D 4 D 9 is then not negative de nite. Hence thee exists a 1 -cuve H, not contained in C, such that H C ˆ1 by Lemma 2.7. Let n : W! W 0 be a sequence of contactions of 1 -cuves and subsequently contactible cuves in C H, stating with the contaction of H, such that C 0 :ˆ n C is an SNC-diviso and that the contaction of any 1 -cuve in Supp C 0 makes the image of C 0 lose the SNC-popety. Then Ci 0 C 0 Ci 0 0 V 3 fo any 1 -cuve Ci H C 0 because the dual gaph of C H is a tee. Since e W 0 C 0 ˆe W C 1 ˆ 0 and p g W 0 C 0 ˆ0 by Lemma 2.7, W 0 C 0 is of type H 1; 0; 1Š and W 0 ; C 0 is stongly minimal by Coollay 2.9 (2). The con guation of C 0 is then given as (b) in Figue 1. We note that Q :ˆ n H is a unique fundamental point of n. Since each component of C has negative self-intesection numbe, Q A D 4. Then eithe n 0 D 1 D 2 D 3 o n 0 D 5 D 6 D 7 is contained in F o G. We conside the case n 0 D 1 D 2 D 3 H F o G. The case n 0 D 5 D 6 D 7 H F o G can be teated similaly. Since Q A D 4, n 0 D i i ˆ 1; 2 is a 2 -cuve and a teminal component of C. We can facto the map m ˆ m 1 m 2 : W! P 2 so that m 2 n 0 D 3 is a unique 1 - cuve in Supp m 2 E F G. Then, since n 0 D i i ˆ 1; 2 is a 2 -cuve and a teminal component of C, m 2 n 0 D i i ˆ 1; 2 emains as a 2 -cuve. This is a contadiction because the intesection matix of m 2 n 0 D 1 D 2 D 3 H m 1 1 P is then not negative de nite. Lemma 4.2. If B H P 2 be a ational cuspidal cuve with ]Sing B ˆ2 then k P 2 B V 1. Poof. By [18, Theoem (IV)], k P 2 B V 0. Suppose that k P 2 B ˆ0. Let P 1 and P 2 be two singula points of B. Let m : W! P 2 be a minimal SNC-map fo P 2 ; B. Then thee exists a unique decomposition of m 1 P i iˆ1; 2 as a sum of non-zeo educed e ective divisos m 1 P i ˆE i F i G i such that the conditions (iii) fo m 1 Q as in the poof of Lemma 3.4 hold, whee we conside espectively E, F and G as E i, F i and G i. Let C 1 be the pope tansfom of B on W and C :ˆ m 1 B ˆC 1 P 2 iˆ1 E i F i G i. The dual gaph of C is given as in Figue 4. We conside the following two cases sepaately. Case 1: C Then all 1 -cuves in C ae exhausted by E 1 and E 2 and E i C E i ˆ3 i ˆ 1; 2. If W; C is stongly minimal then it follows fom e W C ˆ1 that the con guation of C is given as one of (a), (c), (d) and (e) in Figue 4

10 802 H. Kojima Figue 5 Figue 1. This is, howeve, a contadiction. So thee exists a 1 -cuve H, not contained in C, such that H C ˆ1 by Lemma 2.7. Let n : W! W 0 be a sequence of contactions of 1 -cuves and subsequently contactible cuves in C H, stating with the contaction of H, such that C 0 :ˆ n C is an SNC-diviso and that the contaction of any 1 -cuve in Supp C 0 makes the image of C 0 lose the SNCpopety. We know that Q :ˆ n H is a unique fundamental point of n and the con guation of C 0 is given as (b) in Figue 1 by the same agument as in the poof of Lemma 4.1. Since H C ˆ1, we may assume that H E 1 F 1 G 1 ˆ0. Then n E 1 2 V 1 and the dual gaphs of n F 1 and n G 1 ae the same as those of F 1 and G 1. So n E 1 ˆD 4 o D 5. Since each component of n F 1 and n G 1 has self-intesection numbe U 2, n E 1 ˆD 5. Hence F 1 and G 1 ae 2 -cuves. This contadicts that the intesection matix of E 1 F 1 G 1 is negative de nite. Case 2: C 1 2 ˆ 1. Let f : W! W 0 be the contaction of C 1 and put C 0 :ˆ f C. The dual gaph of C 0 is given as in Figue 5, whee the dual gaphs of Fi 0 :ˆ f F i and Gi 0 :ˆ f G i i ˆ 1; 2 ae the same as those of F i and G i. The diviso C 0 contains no 1 -cuves. If W 0 ; C 0 is stongly minimal then the con guation of C 0 must be (a) in Figue 1. Then k ˆ 0 in Figue 1 (a) and F 1, F 2, G 1 and G 2 ae 2 -cuves. This is a contadiction because the intesection matix of E i F i G i i ˆ 1; 2 is then not negative de nite. So thee exists a 1 -cuve H 0, not contained in C 0, such that H 0 C 0 ˆ1 by Lemma 2.7. Let n : W 0! W 00 be a sequence of contactions of 1 -cuves and subsequently contactible cuves in C 0 H 0, stating with the contaction of H 0, such that C 00 :ˆ n C 0 is an SNC-diviso and that the contaction of any 1 -cuve in Supp C 00 makes the image of C 00 lose the SNCpopety. By the same agument as in the poof of Lemma 4.1, we know that Q :ˆ n H 0 is a unique fundamental point of n and the con guation of C 00 is given as (b) in Figue 1. We may assume that H 0 C 0 ˆ H 0 E2 0 G 2 0 ˆ1. Then n E2 0 2 V 0, n E1 0 C 00 n E1 0 ˆ 3 and the dual gaphs of n F1 0 G1 0 ae the same as those of F 1 and G 1. So n E1 0 ˆD 5 and F 1 and G 1 ae 2 -cuves. This is a contadiction. The poof of (1) of Theoem 1.1 is thus completed by Lemmas 3.3, 3.4, 4.1 and 4.2. Poof of (4) of Theoem 1.1. Let B H P 2 be an ieducible ational cuve with k P 2 B ˆ0. Then, by Lemmas 4.1 and 4.2 and [18, Theoems (II) and (III)], B has a unique singula point, say P, and P is not a cusp. We denote the numbe of analytic banches of B at P by P B. Then P B ˆ2 follows fom Lemma 4.3 below.

11 Lemma 4.3. Let D be an ieducible ational cuve on a nonsingula pojective ational suface V with k V D ˆ0. Let s D be the numbe of singula points on D which ae not cusps. Then s D U 1 and if s D ˆ1 then P D ˆ2, whee P is the singula point on D which is not a cusp. Poof. Assume that s D V 1. Let f : ~V! V be a minimal SNC-map fo V; D and let ~D :ˆ f 1 D. Then ~D contains loops of nonsingula ational cuves. So p g ~V ~D ˆp g V D ˆ1. Let W; C be an almost minimal model of ~V; ~D. Lemma 2.4 implies that C is a loop of nonsingula ational cuves. The dual gaph of ~D then contains only one loop by the constuction of almost minimal models (cf. [13], etc.). Hence the assetions hold. The poof of Theoem 1.1 is thus completed. 5. p 1 P 2 B and e P 2 B. In this section, we study the fundamental goups p 1 P 2 B and the topological Eule chaacteistics e P 2 B of the sufaces P 2 B with k P 2 B ˆ0 by using Theoem 1.1. Poposition 5.1. Let B H P 2 be a educed cuve with k P 2 B ˆ0. Then p 1 P 2 B is abelian. In paticula, if B is ieducible then p 1 P 2 B ˆZ= deg B Z. Poof. Put S :ˆ P 2 B. Let S 0 be a stongly minimal model of S. Then, by Theoem 1.1 (1) and Lemma 2.8 (1), p 1 S 0 is an abelian goup. So p 1 S is abelian since S 0 is a Zaiski open subset of S. If B is ieducible then H 1 S; Z G Z= deg B Z by the duality. Poposition 5.2. Let B H P 2 be a educed cuve with k P 2 B ˆ0. Then e P 2 3; if B is a nonsingula cubic cuve B ˆ 3 ] B ; othewise: Poof. By Theoem 1.1, the assetion holds unless ] B ˆ2. So we conside the case ] B ˆ2. Put S :ˆ P 2 B. Assume that S is stongly minimal. Since q S ˆ] B 1 ˆ 1 by Lemma 3.1, S is of type O 4; 1 o O 2; 2 (cf. Table 1). If the latte case occus then S G P 1 P 1 C 1 C 2, whee C i i ˆ 1; 2 is a cuve of bidegee 1; 1 and C 1 C 2 is an SNCdiviso (cf. [9, Theoem 3.1]). This is a contadiction because Pic S is then not a nite goup. Hence we know that e S ˆ1. Assume that S is not stongly minimal. Let S 0 be a stongly minimal model of S. Since S S 0 consists of disjoint a½ne lines A 1 V 1 by Lemma 2.5 (1), we have e S ˆe S 0. Put B 0 :ˆ P 2 S 0. Then B 0 is puely of codimension one. Since ] B 0 ˆ] B ˆ 2, we have S 0 G C C and ˆ 1 by Theoem 1.1 (3). Hence e S ˆe S 0 ˆ The case B is ieducible. Complements of plane cuves 803 Let B H P 2 be an ieducible cuve with k P 2 B ˆ0. Thoughout this section, we assume that B is not a nonsingula cubic cuve. Theoem 1.1 (4) implies that B is

12 804 H. Kojima a ational cuve with unique singula point P and P B ˆ2. Let B 1 and B 2 be two analytic banches of B at P. Then we have the following thee cases: Case (I): P is a smooth point of B 1 and B 2. Case (II): P is a smooth point of eithe B 1 o B 2, but not fo both. Case (III): P is a singula point of B 1 and B 2. We call the cuve B to be of type (I) (esp. (II), (III)) if the case (I) (esp. (II), (III)) occus. We conside the case (I). Poposition 6.1. Suppose that B is of type (I). Then B is pojectively equivalent to one of the cuves de ned by the following polynomials, whee X; Y; Z denotes the system of homogeneous coodinates in P 2 and d ˆ deg B. d de ning equation 3 XYZ X 3 Y 3 4 YZ X 2 2 tx 2 Y 2 XY 3, t A C f0g 5 YZ X 2 YZ 2 X 2 Z ty 2 Z tx 2 Y 2XY 2 Y 5, t A C f0g Convesely, if C t is a cuve whose de ning equation is one of the above list with deg C t ˆ 4 o 5 then k P 2 C t ˆ0. Moeove, C t and C s ae pojectively equivalent if and only if t 3 ˆ s 3, i.e., t 3 is the pojective invaiant. Poof. Since B 1 and B 2 ae smooth at P, the multiplicity of B at P is equal to two. So the assetions follow fom [20, Popositions 1 and 3] (o [19]). We give examples of the cases (II) and (III). We denote by F a, M a and l a Hizebuch suface of degee a, the minimal section of F a and a geneal be of the uling on F a, espectively. Example 1. Let C 0, C 1 and C 2 be thee ieducible cuves on F a a V 3 such that C M a a 1 l (the epesents the linea equivalence of divisos), C 1 ˆ M a, C l and C 0 C 1 C 2 is an SNC-diviso. See Figue 6-(i). Let m : V! F a be the composite of a 1 -times blowing-ups such that the con guation of C 0 :ˆ m 1 C 0 C 1 C 2 is shown as in Figue 6-(ii), whee Ci 0 i ˆ 0; 1; 2 is the pope Figue 6

13 Complements of plane cuves 805 Figue 7 tansfom of C i. Then we obtain the biational mophism n : V! P 2 which is the contaction of the cuve C 0 C0 0 E a 1 in the ode C2 0; E 3;...; E a ; C1 0. Put B :ˆ n C0 0. We know that deg B ˆ a 1 V 4. We have k V C 0 ˆk F a C 0 C 1 C 2 ˆ 0 because C 0 C 1 C 2 K 0. Since E a 1 is a 1 -cuve and E a 1 C 0 E a 1 ˆ1, k V C 0 E a 1 ˆ k V C 0. So k P 2 B ˆk V 0 C 0 E a 1 ˆ 0. The cuve B is of type (II). Example 2. Let b, s and t be thee integes such that b V 5, s; t V 2 and s t ˆ b 1. Let C 0 ;...; C 3 be fou ieducible cuves on F b such that C M b bl, C C l, C 3 ˆ M b and C 0 C 3 is an SNC-diviso. See Figue 7-(i). Let m : V! F b be the composite of s t -times blowing-ups such that the con guation of C 0 :ˆ m 1 C 0 C 3 is shown as in Figue 7-(ii), whee Ci 0 i ˆ 0;...; 3 is the pope tansfom of C i. Then we obtain the biational mophism n : V! P 2 which is the contaction of the cuve C 0 C0 0 E s F t in the ode C1 0; E 1;...; E s 1 ; C2 0; F 1;...; F t 1 ; C3 0. Put B :ˆ n C 0 0. We know that deg B ˆ b V 5. Since C 0 C 3 K 0, k P 2 B ˆ0 (cf. Example 1). The cuve B is of type (III). By the above examples, we have the following esult. Poposition 6.2. Fo any intege n V 4 (esp. V5), thee exists an ieducible ational cuve B H P 2 of degee n such that k P 2 B ˆ0 and B is of type (II) (esp. (III)). Refeences [ 1 ] T. Fujita, On the topology of non-complete algebaic sufaces, J. Fac. Sci. Univ. Tokyo, 29 (1982), 503±566. [ 2 ] S. Iitaka, On logaithmic K3 sufaces, Osaka J. Math., 16 (1979), 675±705. [ 3 ] S. Iitaka, A numeical citeion of quasi-abelian sufaces, Nagoya Math. J., 73 (1979), 99±115. [ 4 ] S. Iitaka, The vitual singulaity theoem and the logaithmic bigenus theoem, ToÃhoku Math. J., 32 (1980), 337±351. [ 5 ] S. Iitaka, Algebaic Geomety, Spinge GTM76. [ 6 ] Y. Kawamata, On the classi cation of non-complete algebaic sufaces, Poc. Copenhagen Summe meeting in Algebaic Geomety, Lectue Notes in Mathematics, No. 732, 215±232, Belin-Heidebeg- New Yok, Spinge, [ 7 ] Y. Kawamata, Chaacteization of abelian vaieties, Compositio Math., 43 (1981), 253±276. [ 8 ] T. Kishimoto, Pojective plane cuves whose complements have k ˆ 1, Thesis fo Maste's degee, Osaka Univ., [ 9 ] H. Kojima, Open ational sufaces with logaithmic Kodaia dimension zeo, Intenat. J. Math., 10 (1999), 619±642.

14 806 H. Kojima [10] H. Kojima, On Veys' conjectue, Indag. Math., 10 (1999), 537±538. [11] M. Miyanishi, Non-complete algebaic sufaces, Lectue Notes in Mathematics, No. 857, Belin- Heidebeg-New Yok, Spinge, [12] M. Miyanishi and T. Sugie, On a pojective plane cuve whose complement has logaithmic Kodaia dimension y, Osaka J. Math., 18 (1981), 1±11. [13] M. Miyanishi and S. Tsunoda, Non-complete algebaic sufaces with logaithmic Kodaia dimension y and with non-connected boundaies at in nity, Japan. J. Math., 10 (1984), 195±242. [14] S. Yu. Oevkov, On ational cuspidal cuves I, shap estimate fo degee via multiplicities, pepint. [15] S. Tsunoda, The complements of pojective plane cuves, RIMS-KoÃkyuÃoku, 446 (1981), 48±56. [16] S. Tsunoda, The stuctue of open algebaic sufaces and its application to plane cuves, Poc. Japan Acad., 57 (1981), 230±232. [17] S. Tsunoda, Stuctue of open algebaic sufaces, I, J. Math. Kyoto Univ., 23 (1983), 95±125. [18] I. Wakabayashi, On the logaithmic Kodaia dimension of the complement of a cuve in P 2, Poc. Japan Acad., 54 (1978), 157±162. [19] H. Yoshihaa, Some poblems on plane ational cuves, Poc. Kinosaki Symp. Algebaic Geomety, Kinosaki, 1978, 80±117 (in Japanese). [20] H. Yoshihaa, On plane ational cuves, Poc. Japan Acad., 55 (1979), 152±155. [21] D.-Q. Zhang, On Iitaka sufaces, Osaka J. Math., 24 (1987), 417±460. Hideo Kojima Depatment of Mathematics Gaduate School of Science Osaka Univesity Toyonaka, Osaka , Japan smv088kh@mail.goo.ne.jp