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$Contact Geometry of second order I - Dept. Math, Hokkaido Univ ...$

y them. Further, we have T3 = (a + b + c)T2F in the Chow ring of w. We put p=a+b+c+3 (1.1) and assume that p > 4. Let s be an integer, and let Q be a general member of the linear system I2T +sFl. We remark that Q is irreducible and has only rational double points of type Al if Choose general. k fibers Fl, - - - , Fk of T. We assume that Q and Fi meet transversally, a,nd that the intersection Q; = Q n F; is an irreducible conic (in Fi E p2) for each i. We let v :
ProoJ. TI I) holds, the11 Bs l4T - (p - 5 + k)FI = 6. We assume that 2) holds. We let Xo, XI and Xa be sections on W of O(T - aF), O(T - bF) and O(T - cF), respectively, such that they form a system of homogeneous coordinates on each fiber of n- : W -+ I?'. If 46 > p- 5 + k, then )4T - (p - 5 + k)FI has no base point outside the rational curve B defined by XI = X2 = 0. On the other hand, the equation of & call be written as where q,, is a homogeneous form on P' 01 degree (2 - i - j)a + ib + jc+ s. Thus, if s = -2a, then we can assume that, 400 is a nonzero consta,nt. Then Q does not meet B. Thus 2) is also sufficient to imply 2) of Leinrna 1. q. e. d. Lemma 3: Suppose that I LI contains an irreducible nonsing-ular member S. Then S is a minimal surface satisfjring: I) The canonical map ofS is a birational molphism. 2) p,(S) =p, q(S) = 0 and ci(S) = 3p,(S)-7+k, wl~ereh, q alrd c: denote the geometric genus, the irregularit-y and the Chel-n number of S, respecljvely. Proof. For any complex manifold M, we denote by I
Page 1 and 2: Proceedings of the Meeting and the
Page 3 and 4: Tadashi Ashihga(Z E ) (1)A Resoluti
Page 5 and 6: of P, and let p, be the geometric g
Page 7 and 8: (2) s s s, + - - - a ST In- 1 571 1
Page 9 and 10: nonsingular rational curves N' and
Page 11 and 12: 0. Let o : -+ V be the minimal reso
Page 13 and 14: 3 An inequality for a singularity E
Page 15 and 16: [lo] H. Laufer: On normal twedimens
Page 17 and 18: some sence. So it seems that the si
Page 19 and 20: RPn-,O(ICx + S) = 0 for p - > 1, we
Page 21 and 22: 1.6 Let S be a surlace defined by (
Page 23 and 24: and gi by r respectively. Put r = 7
Page 25 and 26: 0 d OE~L: (Dl + C hi) 4 OC(yl+)Lc)
Page 27 and 28: By the above argument, the number O
Page 29 and 30: Corollary 2.5 pg(~)=pg(S)-k, q(S)=g
Page 31 and 32: y3 t Mk, let C'$ be the curve on W
Page 33 and 34: where St = pP1(t) (t # 0) is a srno
Page 35 and 36: Fig. 1
Page 37: Appendix to T. Ashikaga's paper KAZ
Page 41 and 42: of Lemma 2, we can cover the region
Page 43 and 44: quadric hppersurface Q . Precisely
Page 45 and 46: 5.1 General properties of a smooth
Page 47 and 48: Prcof. he ha\-e the equality: I< =
Page 49 and 50: decomposition by th? finiten~ss of
Page 51 and 52: each pair ( ul ,u,, in 1- > I ( 1 L
Page 53 and 54: - 1 * (3n the other hand for such a
Page 55 and 56: 5.2 The property of the tangent bun
Page 57 and 58: Hence we ha\-e a diagram where G(Q)
Page 59 and 60: Remark 2.6.3. Cncler the exact seql
Page 61 and 62: ~qhere I is a lin3bundls on R and E
Page 63 and 64: exact sequence 2.5 is locally free
Page 65 and 66: Proposition 1.2 (2) yields the desi
Page 67 and 68: f(c(q)) with some point P (= f(ci))
Page 69 and 70: 3ext ~orc.llar~- 3.4.3 and Proposit
Page 71 and 72: isomorphism: ( = I : T - nncl i'( r
Page 73 and 74: At last Ice ha-~e come to the final
Page 75 and 76: Remarl< 3.22. Let 1 be a hyperplane
Page 77 and 78: asstme that R has the isolated sin2
Page 79 and 80: * is GS, so is r" T._ , hence so is
Page 81 and 82: M . - At last we hare come to the f
Page 83 and 84: Reference R.Hartshorne, Ample vecto
Page 85 and 86: to be reduced, irreducible or effec
Page 87 and 88: we mean that Y is a principal fiber
Page 89 and 90:
Definition (1.1,) We write Z = LDta
Page 91 and 92:
If we put L := e($), we have L E pi
Page 93 and 94:
Condition(A1. Via the natural isomo
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( sufficiency We chosse integers a'
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1 1 1 Since -1 E H (X, 0) z H HX, @
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with a finite order. Hence for a su
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proposition does not necessarily ho
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enjoys the following conditions. (1
Page 105 and 106:
Proc. Japan Acad., 50 (19741, 533 -
Page 107 and 108:
51. Mukai - Umemura's construction.
Page 109 and 110:
00 3 m Lemma 2. (1)([2])- V5 - H5 C
Page 111 and 112:
Lemma 5. (a) O(-l) (33 O(-l) Proof.
Page 113 and 114:
Lemma 11. BslkI21 = R2 Proof. Since
Page 115 and 116:
Lemma 17. On the other hand, by a d
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curve, by Mori [6], W is smooth. Th
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Algebraic Surfaces of General Type
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- exceptional divisor of a, respect
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2 Surfaces of type 111. In this sec
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3 Quadrics through canonical surfac
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M7e turn our attention to So. By (2
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4) a non-conic normal Del Pezzo thr
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in H0(31
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of o satisfying SU~~(E) = Supp(E) a
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Ho(o*Ii - ii)) is bijective for i >
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we get (a - 4)(P + 1) = 26 - 4. Rec
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and F2 I C are the pull-back of a p
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U2. Thus 3 is the normalization of
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7.5 Conversely, we start from St de
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8.3 We consider the case ii) of 8.1
Page 147 and 148:
We first extract inforination from
Page 149 and 150:
Leinina 9.3 Let S* be a.s in Lemnza
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In order to calculate x(Ost), we us
Page 153 and 154:
Thus, in general, the canonical deg
Page 155 and 156:
We first consider the case a+ c 5 3
Page 157 and 158:
Deformations of a Complex Manifold
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Theorem 2. Let N be a smooth strong
Page 161 and 162:
1. Formulation of deformations of a
Page 163 and 164:
2. Proof of Theorem 1 * We fix a,
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By (2.3) and (2.4) , q(t1 satisfies
Page 167 and 168:
~?oof. We may assume that R2ko c .
Page 169 and 170:
3. Proof of Theorem 2 Let rr:5 + (T
Page 171 and 172:
Let Eq be a subbundle of T~@A~('T")
Page 173 and 174:
Proof. Let B"ltB1 CF 4 3 , we have
Page 175 and 176:
Since $~($"(sl +$;+l (s1 1 =-i~($"(
Page 177 and 178:
References [All Akahori,T.: The new
Page 179 and 180:
CohomoBogicd Criterion sf Naamerica
Page 181 and 182:
On Finite Galois Covering Germs Mak
Page 183 and 184:
Theorem 1 (Grauert-Remmert[2]). If
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sense : There exist a lot of finite
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y the following commutative diagram
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(see Namba[6]). Remark. (1) f(0) is
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a-decomposition may have infinitely
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e an n-dimensional projective compl
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Lemma (1 .I). (1) I im or(B + &AIg
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. Therefore we have E + a (D + 6A)
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Proof. 1 Let {Dni (n E a) be a sequ
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Lemma - (1.11). Let Qw(X) 3 X be th
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Proof. Let Z be the intersection of
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Remark (2.4). We can calculate the
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Therefore $@OC is an ample vector b
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Q n*n.O ( 2 airi )/torsion is inver
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Proof. Let- u2 : X2 4 X1 be the blo
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original f , then the assumption of
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Proof. If G is semi-stable, we have
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References. [Cl S. D. Cutkosky, Zar
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Rational curves on Weierstrass mode
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normal bundles OP1(-l) @ op~(-l) ),
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Theorem ([6,(3.4)1) Let X be a smoo
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la" 227b2 not identically zero. The
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?j 2. Preliminaries + Let S and S b
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For example, if we assume that ever
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Taking the discriminant D(F), we wr
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+ Moreover if we assume that deg q
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+ (effective divisors disjoint from
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3 3. Rational curves 1. In this par
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C 0 C' := the reduced subscheme of
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+ We shall denote by C (resp. C the
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.where pl (resp. P2) is the first(r
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elliptic fibration obtained has onl
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1 1 the Euler number e(W) of W is e
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dim C = 1, H2(QplC) = 0 . We shall
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which has no intersections with CI
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the following diagram: K 3 K o = W
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Appendix. the calculation of H 2(W
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C) Let F' be a singular fibre of a.
Page 262:
10. Schoen, C.: On fibre products o
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