Table 1. morphism U P 1 dominant (MMP) 2. dim = 3 (MMP) 3. (cf. [Ii77], [Miy01]) (Table 1) 3.

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1 C T κ(t ) 1 [Projective] κ = κ =0 κ =1 κ =2 κ =3 dim = 1 P 1 elliptic others dim = 2 P 2 or ruled elliptic surface general type dim = 3 uniruled bir. to elliptic 3-fold general type Table 1 (Table 1) uniruled 1.1 (cf. [Mi-Mo86]) n T U P 1 dominant morphism U P 1 dominant T T uniruled U (n 1) (Table 1) dim = 1 dim = 2 (MMP) 2 dim = 3 (MMP) 3 - uniruled (cf. [Mi-Mo86]) 2 κ (cf. [Ii77], [Miy01]) (Table 1) 3 1 dim = 0, κ =0 κ =0 K3, Enriques, Abelian, hyper-elliptic surface 2 3 κ =0 (cf. [Fuj82], [Koj99])

2 2 : [Affine] κ = κ =0 κ =1 κ =2 dim = 1 A 1 C others dim = 2 affine ruled C -surface log general type (Table 2) Table (cf. [Miy01]) n X dim(y )=n 1 Y ϕ : X Y general fiber A 1 X affine ruled X U A 1 Zariski U dim(u) =n 1 (Table 2) dim = 1 dim = 2 (MMP) 2 κ = affine ruled - (cf. [Mi-Su80, Miy01]) κ =1 C - (cf. [Kaw79]) Abhyankar-Moh-Suzuki 1974,75 (cf. [A-M75]) 4 Gurjar-Miyanishi (cf. [Gu-Mi96]) dim 3 Zariski Abhyankar-Sathaye Jacobian dim = X κ(x) (Table 1 (Table X (1) κ(x) = X affine uniruled (2) κ(x) =2 X C -3-fold Y ϕ : X Y general fiber C X 1.1 affine uniruled uniruled ( 1.1) 1.3 X general point x X x C X C C = A 1 X affine uniruled Section Section 3 X X 3.1 Section Abhyankar-Moh original

3 Section X κ(x) (=SNC) (cf. [Ii77]) (Table 2) - [ ] Y Y W B := W Y SNC (MMP) 2 W B W W (W, B) Y = W B 1 Y [ ] (W, B) (MMP) 2 (W, B) 1 (MMP) 3 [ ] X X T D := T X SNC (MMP) 3 T D T T X = T D 2 X [ ] (T,D) (MMP) 3 (T,D) (T,D) (MMP) 3 T T 6 D D T X := T D X X T T T (i) (ii) 2 6 T (flipping curve) T

4 4 : (i) T (minimal model) K T (ii) T (Mori fiber space, Mfs) T f : T W (a) dim(w ) 2 f f O T = O W ) (b) K T f- f C T ( K T C) > 0 (c) ϱ(t /W )=ϱ(t ) ϱ(w ) 1 X X X 3 X [ ] X = T D [ ] (T,D) (MMP) 3 (T,D ) 3 (MMP) 3 [ ] T (MMP) 3 g : T T (T,D)=:(T 0,D 0 ) g (T 0 1,D 1 g ) (T 1 r 1,D r 1 ) (T gr 1 r,d r )=(T,D ), g i : T i T i+1 D i D T i X i := T i D i T i D i (1) g i : T i T i+1 E i := Exc(g i ) D i X i+1 X i g i E i D i X i+1 X i (2) g i : T i T i+1 {γ a } {γ + a } γ a D i, γ + a Di+1 X i = X i+1 {γ a }, {γ + a } T i, T i+1 X i X i+1 X i+1 X i X := T D X 2.1 T (MMP) 3 : T T X 7 Section 3 X (T,D) 2.1 (MMP) 3 X X X (T,D) (MMP)3 :(T,D) (T,D ) X

5 Section Section [ ] X T D := T X SNC D ( ) X (T,D) ( ) ( ) : H := D smooth member 3.1 X X ( ) (T,D) ( ) (1) (5) ( ) (T,D)=:(T 0,D 0 ) g (T 0 1,D 1 g ) (T 1 s 1,D s 1 ) (T gs 1 s,d s ) gs (T s+1,d s+1 ) gs+1 (T s+2,d s+2 ) (T r 1,D r 1 ) gr 1 (T r,d r )=:(T,D ) T i Q- D i D T i X i := T i D i D i g i : T i T i+1 (1) 0 i<s g i : T i T i+1 g i g i D i g i D i+1 (2) s i<r g i : T i T i+1 E i := Exc(g i ) p i+1 := g i (E i ) T i+1 g i p i+1 T i+1 (weighted blow-up) b i N wts = (1, 1,b i ) 1 b s b s 1 b r 1 (3) 0 i s X i = X (4) s i<r X i = X i+1 X i X i+1 (b i,k i )- half-point attachment k i N 1 k i b i half-point attachment 3.1 X i+1 X i X i X i+1 = C (ki 1) A 1 C (ki 1) A 1 (k i 1)- (5) (T,D ) κ(x) (i) κ(x) = T (ii) κ(x) 0 K T + D κ(x) =κ(t ; K T + D ) 3.1 (half-point attachment) [Miy01, Definition] 3.1 Z V Z Zariski B := V Z p B V f : V E V p wts = (1, 1,b) E = P(1, 1,b) b N B f B := f 1B E

6 6 : B E = k j=1 m jl j l j E = P(1, 1,b) ruling m j N k j=1 m j = b Z := V B Z (b, k)- half-point attachment Z Z Zariski Z Z = C (k 1) A ( ) T D X := T D X 3.1 κ(x) = ( ) π : T W D D X κ(x) = X ( ) (T,D) (1) π : T W 3.1 (5) D T D [C 2 - ] π : T = P(E) W W P 1 - E := π O T (D ) W 2 D O(1) [D 2- ] π : T W W (quadratic bundle) general fiber (F, D F ) = (P 1 P 1, O P1 P1(1, 1)) π G G = Q 2 0 v (T,v) o (xy + z 2 + t l =0) C 4 :(x, y, z, t), l 1 [D 3 - ] π : T = P(E) W W P 2 - E := π O T (D ) W 3 π F (F, D F ) = (P 2, O P 2(1)) [D 3- ] π : T W W P 2 - general fiber F (F, D F ) = (P 2, O P 2(2)) π G G = S 4 v hyper-quotient singularity (T,v) o (xy+z 2 +t l =0) C 4 :(x, y, z, t)/z 2 (1, 1, 1, 0), l 1 S 4 P 5 4 [Q- ] T ϱ(t )=1 Q- (T,D ) (i) (P(1, 1, 2, 3), O(6)); (ii) (T 6 P(1, 1, 2, 3,a), {x 4 =0} T 6 ) a {3, 4, 5}; (iii) (T 6 P(1, 1, 2, 2, 3), {x 3 =0} T 6 ); (iv) (T 6 P(1, 1, 1, 2, 3), {x 0 =0} T 6 ); (v) (P(1, 1, 1, 2), O(c)) c {2, 4}; (vi) (T 4 P(1, 1, 1, 1, 2), {x 0 =0} T 4 ); (vii) (T 4 P(1, 1, 1, 2,a), {x 4 =0} T 4 ) a {2, 3}; (viii) (P 3, O P 3(c)) c {1, 2, 3}, (Q 3, O Q (c)) c {1, 2}; (ix) (T 3 P(1, 1, 1, 1, 2), {x 4 =0} T 3 ); (x) (T 3 P 4, O(1)); (xi) (T 2,2 P 5, O(1)); (xii) (V 5, O(1)) V 5 P 6 P 4 Gr(2, 5) P 9 general

7 7 (2) X T D D X = T D half-point attachment 3.1 half-point attachment (1) π : T W [C 2 - ] X = X X X (1, 1)- half-point attachment X X A 2 [D 2- ] X = X X X (1, 1)- half-point attachment X X A 2 [D 3 - ] X = X [D 3 - ] X = X X X (b, k)- half-point attachment 1 b 2, 1 k b [Q- ] (v), c =2 (viii), c =1 X = X (iv) (vi) (viii) (P 3, O P 3(2)) (x) (xi) (xii) X = X X X (1, 1)- half-point attachment X X A 2 (iii) (vii), a =2 (viii) (Q 3, O Q (2)) (ix) X = X X X (b, k)- half-point attachment 1 b 2, 1 k b (ii), a =3 (vii), a =3 (viii) (P 3, O P 3(3)) X = X X X (b, k)- half-point attachment 1 b 3, 1 k b (ii), a =4 (v), c =4 X = X X X (b, k)- half-point attachment 1 b 4, 1 k b (ii), a =5 X = X X X (b, k)- half-point attachment 1 b 5, 1 k b (i) X = X X X (b, k)- half-point attachment 1 b 6, 1 k b κ(x) =2 κ =1 C - Table X κ(x) =2 X ( ) (T,D) X C - ϕ : X Y Y Section , 3.3 RIMS RIMS (cf. [Ki03]) 10 [ 3.1 ] Mella (cf. [Me02]) Mella [Me02] uniruled T Q- S T T (MMP) 3 S [Me02] 11 Mella 10 11

8 8 : (T 0, H 0 ):=(T,H) H 0 + K T κ(x) = t 0 := sup{µ Q H 0 + µk T 0 } 0 t 0 < 1 (extremal ray) NE (T 0 ) {H 0 + t 0 K T 0} R 0 = R + [l 0 ] R 0 (contraction) g 0 : T 0 T 1 T 0 g 0 g κ(x) 0 g 0 g 0 : T 0 T 1 t 0 (1) t 0 =0 g 0 D 0 g 0 (2) t 0 > 0 g 0 E 0 p 1 := g 0 (E 0 ) T 1 g 0 : T 0 T 1 p 1 T 1 wts = (1, 1,b 0 )(b 0 N) 12 (D 0 l 0 )=1 l 0 E 0 = P(1, 1,b 0 ) ruling H 0 D 0,S 0 g 0 H 1 D 1,S 1 S 0 H 0 ( ) smooth member (1) H 1 S 1 H 1 smooth member (2) κ(x) =κ(t 1 ; K T 1 + H 1 ) (T 0, H 0 ) (T 1, H 1 ) (MMP) X (MMP) 3 Mella [Me02] [Ka01] (2) 13 [ ] ( ) 1.1 (2) ( ) (1) κ(x) = affine uniruled 1.3 ( ) 3.2 X X 12 T 0 g 0 b 0 =1 g 0 13 wts = (1,a,b), a, b N, gcd(a, b) =1 smooth member S 0 H 0 wts = (1, 1,b)

9 9 half-point attachment 3.1 X affine uniruled X affine uniruled X π : T W D C 2, D 2, D 3, D 3- X A 1 T ϱ(t )=1 Q- 3.2, (2) X = T D affine uniruled T Q- X A 1 κ = Zariski [Ki04] 15 References [A-M75] S.S. Abhyankar and T.T. Moh, Embeddings of the line in the plane, J. reine angew. Math.,276 (1975), pp [Fuj82] T. Fujita, On the topology of non-complete algebraic surfaces, J. Fac. Sci. Univ. Tokyo, 29 (1982), pp [Gu-Mi96] R.V. Gurjar and M. Miyanishi, On contractible curves in the complex affine plane, Tohoku Math. J. 48 (1996), pp [Ii77] S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, Complex Analysis and Algebraic Geometry (W. L. Baily, Jr. and T. Shioda, eds.), Iwanami Shoten, Tokyo, 1977, pp [Ka01] M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math., 145 (2001), no. 1, pp [Kaw79] Y. Kawamata, On the classification of noncomplete algebraic surfaces (Copenhagen, 1978), Lecture Notes in Math., 732, Springer, Berlin, 1979, pp [Ki03] T. Kishimoto, On the logarithmic Kodaira dimension of affine threefolds, RIMS preprint series, RIMS (2003); available at http : // u.ac.jp/home page/preprint/list/2003.html. [Ki04] T. Kishimoto, On the compactification of contractible affine threefolds and the Zariski Cancellation Problem, to appear in Math. Zeit.; available (for subscribers) at the home page in Math. Zeit. [Koj99] H. Kojima, Open rational surfaces with logarithmic Kodaira dimension zero, Internat. J. Math., 10 (1999), pp [Me02] M. Mella, -Minimal models of uniruled 3-folds, Math. Zeit., 242 (2002), pp [Miy01] M. Miyanishi, Open algebraic surfaces, CRM Monograph Series 12, AMS Providence, RI, [Mi-Su80] M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ., 20 (1980), pp [Mi-Mo86] Y. Miyaoka and S. Mori, Numerical criterion for uniruledness, Ann. of Math., 124 (1986), pp

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