(Fukuma Yoshiaki) 11 X n
(X, L) , . n g(X, L), △△ (X, L) . , 。。 i g i (X, L) 。 （（ , i    0 ≤ i ≤ n , 90(X, L) = Ln, 91(X, L) = g(X, L) ).
Δ i (X, L) △△ Δ- i
). i = 1 ,
," i Δ-
11
Δ-
2 , Δ-
[1]¹
, Δ(X, L)
: Δ(X, L) := n + Ln − ho(L). Δ-
.j
L
k:
2

L  k-         XつX1つ2.1 k   1 ≤ k ≤ n−1 ○つXk 1 ≤ m ≤ k m r p,q : H p (X q, Lq) → H p (X q+1, Lq+1) Lm := L|xm
: 2.1 (1) n = 1              Δ(X,L) = h¹(Ox) − h¹(L) ≥ 0 (2) L j ≤ n − 1
1 ≤
Δ(X, L) ≥  Δ(X 1, L 1) ≥ ··· ≥ Δ(X n-1, Ln-1) ≥ 0 Δ- : 2.2 ([8]) (X, L)   n (1) Δ(X, L) > dim Bs|L| 33
(2) B , L
(3) g(X,L) ≥ Δ(X,L)          , L regular 4 (a) Ln ≥ 2Δ(X, L) − 1 Bs|L| = 0 Ln ≥ 2Δ(X, L) (b) L Ln ≥ 2Δ(X, L) + 1 (c) g(X, L) = Δ(X, L) , 0 < i < n i t h²(tL) = 0 dim Bs|L| = −1 Δ- q
3
△△ X Δ( X, L) ≤ 1 (X, L) dim Bs|L| ≤ 0 2.2 (1) 2.2 (2) Ln ≥ 2Δ(X, L) − 1 g(X, L) ≥ Δ(X, L) 2.2 (3) (X, L) (X, L) L , 2.2  , Δ- 2.2 X Δ- : (I) Δ(X,L) = 0 ([2]). g(X, L) = 0 Δ( X, L) = 0 Ln = 1 (II) Δ(X, L) = 1 Δ( X, L) = 1 [6]) Del Pezzo ([3] [4] ([8]) (X, L)    Del Pezzo g(X, L) = 1 Δ(X, L) = 1 (III) Δ(X, L) = 2 ( [5]  [7] ) 3
i Δ-
Δ- Δ- 2.1 , Δ- 5 i , L 3.1 X n X L St Euler-Poincaré , t χ( L 8 t ) . t 55 4.1.1
44

5
n (3)  Δε (X, L) 44 Δ- Δ- Δ- 1 (X, L)   n 0 ≤ k ≤ n − i k: (A i ) L    (n − i)- (B i ) h 0 (Ln-i) > 0. t > 0 (C) 0 ≤ j ≤ n − 1 h³(−tL) = 0 (D k ≥ j ≤ n − i (E k ≤ j ≤ n − i Macaulay . 4.1 (X, L)
k 4.1 △△ Δ- , 1 ≤ i ≤ n i j     t X j j j Cohen- X j , L i 0 ≤ k ≤ n − i 11 , Δ- Δ-
6
(1) (1.1) 6. i    1 ≤ i ≤ n 4.1.1 ([10], [17]) (X,L)   n .         (X,L) 1 ≤ j ≤ n − i j
d im Coker(r i −1,k) k=0 Δ i (X, L) ≥ Δ i (X 1, L 1) ≥ · ·· ≥ Δ i (X n-i, Ln-i) ≥ 0 , (X, L)           1     (A i), (B i), (C)    , Δ i (X, L) ≥ L 4.1        (X, L) , : 。 0 4.1.2 ([10]) (X, L)    n 1 ≤ i ≤ n Δε(X, L) ≥ 0 i (1.2) , i Δ- ?? X ([14]) ( ) 6
4.1.1 L
77
([11])
Δ- 2 ≤ i ≤ n i .. 4.1.1 ([10]) n   3 Y     m h (Ky) = 0 m    2 ≤ m ≤ n − 1 Ky E := O(Ky)⊕n−m+1 Y S ppn—m (X, L) L Py(E) n △m(X, L) = −(n − m + 1) < 0
4.1.3 4.1.3 ([14]) (X, L)    n dim Bs|L| ≤ n − 1 Δ i (X, L) ≥ 0 i i , n ≥ 2 , i ≥ dim Bs|L| + 1 Proof.   [14, Corollary 4.1] □ ( 4.1.3 4.1.3 4.1.2 [14, Theorem 4.1] ). 4.1.3 , Ky 4.1.2 ([14])  Y            m(≥ 2) ho(Ky) = 0 .  A 1 , . . ,  A n –m      Y i hm (A i ) = 0 Bs|A ; | = 0 .  n > m . π : Py(E) → Y . i                 , E i := Ky ⊕ A1 ⊕ · · ⊕ Ai-1 ⊕ Ai+1 ⊕ ·· ⊕ An-m, D i := Py(E i) , D i ∈ |H(E) − π*A i |       DD i ∩ · ·· ∩ . Dn–m = P(Ky) L := H(E) L  .
88
i i D i + π*A i ∈ | L |           . D i ∩ · · ∩ D n −m = P(K y) A , , dim Bs|L| ≤ m [9, Example 2.10 (8)]     [10, Lemma 2.12.1]        9m(X,L) = h™(Ox) Δm+1(X, L) = 0 Δm(X, L)   =    gm(X, L) − Δm+1(X, L) + (n − m)hm(Ox) − hm(L) =  (n − m + 1)h™(Ox) − h™(L). h™(Ox) = h™(Oy) = ho(Ky) = 0
=   h™(Ky ⊕ A1 ⊕ ··· ⊕ An-m) =  hm(Ky) = 1 4.1.3 △m(X,L) = −1 < 0     ,, dim Bs|L| = m dim Bs|L| ≥ m . dim Bs|L| = m
4.1.3 , dim Bs|L| ≥  i △- , Δ(X, L) > n , L
, L △- , L 4.1.3      , n i     Δε(X, L) < 0 (X, L)           , 0 ≤ i ≤ n , dim Bs|L| ≥ i κ(K x + (n − i)L) ? 4.1.1 (X, L)   n 2 < i < n  κ(K x + (n − i)L) ?.
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4.1.4 (X, L)   n (1) Δ n(X, L) < 0           , κ(X) = n
 Δ 2(X, L) < 0              ,          (X, L) (2) , κ(K x + (n − 2)L) ≥ 2 , i = 2 Δ²(X, L) ([16]). (2) 2.2 △- .. 4.1.5 ([17]) (X,L)   n                     ,i    1 ≤ i ≤ n − 1 (1) (X, L) (2)  g i (X, L) ≥  Δ i (X, L). (3)  Ln ≥  Δ i (X, L) + Δ 1 (X, L) + 2 − i. , g i (X, L) = Δ i (X, L) 4.1.6 ([17]) (X, L)    dim X = n ≥ 3 i ≤ n − 1 , i   2 ≤ (1)  (X, L) (2)  g i (X, L) ≥  Δ i (X, L). (3)  Ln ≥  Δ i (X, L) + Δ 1 (X, L) + 2 − i. k, s, t , i + 1 ≤ k, i + 1 ≤ s ≤ n − 1, t ≥ 1 Δ k (X, L) = 0, h 8 (L 8 t ) = 0 4.1.1         4.1.5           4.1.6            2.2 (3) (c) 2.2 (3) (c)             L    very ampleness i ≥ 2
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4.1.7 ([17]) (X,L)    n                       ,i    1 ≤ i ≤ n − 1 (1) (X, L) (2)  g i (X, L) ≥  Δ i (X, L). (3)  Ln ≥ Δ i (X, L) + Δ 1 (X, L) − i. .. (a)  (X, L)     regular (n − i)- 7 (b)  Δ i (X, L) =  Δ i (X 1, L 1) = · · =  Δ i (X n − i, Ln − i). (1) 2.2 (3) (a) . ,  4.1.3 i ≥ 2               2.2 (3) (b) 2.2 , Del Pezzo , Δ- Del Pezzo                (X, L)      O(K x + (n − 1)L) = O x . , Δ(X, L) = 1        g(X, L) = 1 (X, L) O(K x + (n − i)L) = O x , △ ? ([15]). , 4.1.1 (X, L)                n ≥ 3 i , 2 ≤ i ≤ n − 1 C (i, 1):  K x = − (n − i)L. C (i, 2):  Δ i (X, L) = 1           2g1(X, L) − 2 = (i − 1) L n . 2g1(X, L) − 2 = (i − 1)Ln. C (i, 3):  Δ i (X, L) > 0 2g1(X, L) − 2 = (i − 1)Ln. C(i, 4):  gi(X, L) = 1 C (i, 5):  g i (X, L) > 0 2g1(X, L) − 2 = (i − 1)Ln.ﾌﾞL   (n − i)- j ´ H 0 (X j , L j ) → H 0 (X j+1, L j+1)
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4.1.4 i = 1 ,  C(1,1)    C(1,2) 4.1.1    C (1, 1)     C (1, 3), C (1, 1)     C (1, 4), C (1, 1) C (1,5) ((X, L) (g 1 (X, L) = 1 ). 4.1.8 ([15]) 4.1.1 (1)  max{2, dim Bs|L| + 2}  ≤ i  ≤ n − 1 L i        ). i 2 ≤ i ≤ n − 1 (2) i = 2 (3) i = 3 n ≥ 5 -A- 2 Δ- i △- iΔ- Δε(X, L) g i (X, L) i 2 Δ- 4.2.1 ([8]) (X, L) , g(X, L) = Δ( X, L ) = 0 0 i Δ- i . 4.2 ,  , ?
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4.2.2 (X, L) .  Δ i (X, L) = 0 g i (X, L) = 0.
? 4.2.1 (X, L)   n ? n 
4.2.1 , Δ i (X,L) ≤ 0 g i (X, L) > 0 4.2.1  ([10, Example 4.1.1]) n ≥ 4       , pn+1      n + 1 k = n + 3 . (ξο : ξι : ··· : ξn+1) . k ). G = L/kL    1 （  ρ ∈ G k (ρ) · (ξο : ξι : · · · : ξn+1) = (ξο : ρξι : · · · : ρ n+1ξn+1), ρ = exp(2πi/k). X := Y/G K x
Δ i (X, L) < 0        g i (X, L) > 0 Δ i (X, L) = 0         (X, L) ?.
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4.2.2 (X, L)    n i     Δ i (X, L) = 0            , g i (X, L) = 0 1 < i < ?. n , 4.2.1 (X, L)   n 1 < i < i  Kx+(n-i)L  nef      , κ(K x + (n − i)L) ≤ i − 1 n k i ≤ k ≤ n gk(X, L) = ∆ κ ( X , L ) . i – 1 Y, Y Proof. , H , ϕ : X → Y    , K x + (n − ( dim Y = 0 i)L = ϕ*(H) O x (K x + (n − i)L) = O x ).           [10, Lemma 1.6]         i ≤ k k
(2) κ(K x + (n − i)L) ≤ i − 1       t ≤ n − i − 1  , t h 0 (K x + tL) = 0 (3) , (2), (3), [9, Theorem 2.3]      , i + 1 ≤ k lis gk(X, L) = 0 (4) , (1), (2) 3.1 (2)        , Δn(X, L) = h n (Ox) − h n (L) = 0 i Δ- (1), (2), (4) , k Δκ (X, L) = 0

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k  .  Δ k (X, L) 0   =   Δε+1(X,L) =   g i (X, L) − Δ i (X, L) + (n − i)h i (Ox) − h i (L) =   g i (X, L) − Δ i (X, L) , g i (X, L) = Δ i (X, L) . ,         . ロ4.2.3 (X, L)   n 1 ≤ i ≤ i     0 ≤ κ(Kχ + (n − i)L) ≤ i − 1 n i ≤ k ≤ n 4.2.2 i = 1,2,n       , 4.3 △- 3 Δ-   , 3  Δε (X, L)      Δε +1 (X, L) 1 ≤ 4.3.1 ([10], [17]) (X, L)   n i ≤ n − 1 h 0 (Ln-i-1) ≥ 3 ,  Δ i (X, L) = 0            , Δ i +1 (X, L) = 0 . 4.3.1 (X, L)   n i  . 1 ≤ i ≤ ? n
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4.4 △- 4 , Δ- Δ- ? 4 i Δ- Δ- , Δ- , L , Δ- i = 2 4.4.1 ([10]) (X, L)    n (n ≥ 3)   , L Δ 2 (X, L) = 0 , (X, L) . (1) (Pn,Opn(1)). (2)  (Q", OQ" (1)). (3) scroll. (4)  Ox(Kx + (n − 1)L) = Ox, , (X, L) Del Pezzo quadric (5) (6) scroll. (7) (M, A)    (X, L)    reduction (7.1)  n = 4, (M, A) = (P 4 , O p4 (2)). (7.2)  n = 3, (M, A) = (Q³, OQ³(2)). (7.3)  n = 3, (M, A) = (P3, Op3 (3)).
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(7.4) n = 3, M F p²- , (F, A|F) ≒ (P², OP² (2)) (X, L) reduction 4.1 (1) X (resp. Y) H) X (resp. Y) simple blowing up
(1.1) 1 blowing up E π-exceptional effective (1.2)  L = π*(H) − E reduced divisor (2) X (resp. M) (resp. M ) reduction
(2.1) μ simple blowing up simple blowing up (2.2) (M,A) , Δ 2 (X, L) = 1 4.4.2 ([10], [12]) (X, L)
Δ 2 (X, L) = 1 ). (1) (M, A)

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(3) (M,A) M → S f*(K s + H) quadric , S . f : H     K M +(n-2)A = (S, H) . (3.1)  S S  C 2 = 1 (resp. p ） . (3.2)  S 4.4.1 (X, L) , L Δ i (X, L) (X, L) 4.5 △-
5
5 i Δ- Δ- 4.6 △-
6

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(1975),
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[2] T. Fujita,  On the structure of polarized varieties with △-genera zero, J. Fac. Sci. Univ. of Tokyo 22 (1975), 103–115. [3] T. Fujita, On the structure of polarized manifolds with total defi- ciency one, I, J. Math. Soc. Japan 32 (1980), 709–725. [4] T. Fujita, On the structure of polarized manifolds with total defi- ciency one, II, J. Math. Soc. Japan 33 (1981), 415–434. [5] T. Fujita,  Polarized manifolds of ∆ -genus two, part I  J. Math. Soc. Japan 36 (1984), 709–730. [6] T. Fujita, On the structure of polarized manifolds with total defi- ciency one, III, J. Math. Soc. Japan 36 (1984), 75–89. [7] T. Fujita, Polarized manifolds of degree three and △ -genus two, J. Math. Soc. Japan 41 (1989), 311–331. [8] T. Fujita,  Classification Theories of Polarized Varieties,  London Math. Soc. Lecture Note Ser. 155, Cambridge University Press, (1990). [9] Y. Fukuma,  On the sectional geometric genus of quasi-polarized varieties, I, Comm. Algebra 32 (2004), 1069–1100. Δ- , Δ- [1] 27 4 , 316–326. [10] Y. Fukuma,  A generalization of the ∆ -genus of quasi-polarized varieties, J. Math. Soc. Japan 57 (2005), 1003–1044. [11] Y. Fukuma, On invariants of polynomial functions, Japan. J. Math.  31 (2005), 345–378. “On the sectional geometric genus of [12] Y. Fukuma,  Addendum: quasi-polarized varieties, I", Comm. Algebra 36 (2008), 3250–3252. [13] 61 4 (2009), , 395–417. [14] Y. Fukuma, A note on non-negativity of the ith △ -genus of quasi- polarized varieties, Kyushu J. Math. 64 (2010), 17–34. [15] Y. Fukuma, A numerical characterization of polarized manifolds (X, L) with K x = −(n − i)L by the ith sectional geometric genus and the ith △-genus, preprint, arXiv:1005.4722. [16] Y. Fukuma,  A lower bound for the second △ -genus of polarized manifolds, in preparation. [17] Y. Fukuma, Remarks on the ith △-genus of quasi-polarized vari- eties, in preparation. [18] , 1977. Yoshiaki Fukuma Department of Mathematics Faculty of Science Kochi University Akebono-cho, Kochi 780-8520 Japan E-mail: fukuma@kochi-u.ac.jp
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