Algebraic Geometry and Commutative Algebra. In Honor of by Hiroaki Hikikata

By Hiroaki Hikikata

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Additional resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2

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1. d. Remark. Nakamura studied a surface of class VIIo satisfying the condition (e) (cf. Nakamura[9 bis]), and the results of this section might be derived from [9 bis]. §5. In this section, we review a classification and the properties of surfaces containing global spherical shells (GSS). Let 5 be a surface contaning a GSS. For the definition of GSS, we refer to Kato[4]. It is easy to see that S is a surface of class V I I . We assume that S has no exceptional curves of the first kind and that the second Betti number b2{S) is positive.

6). 18. And ht q* > 0 if q* = q*nHB* with 7 G Γ ( η ) . 7) A(y) is a finite set. 9. But we shall prove these lemmas after giving some more notation and definition. 4. 1) c ^ f l ^ ^ q ^ q ^ n . 5. 1) (1 < k < i 7 ) . (cf. 1)]). 7. 6. and Q = QDB,Qe Ass(Cyk/yCyk)}. With notation as above, we have A7(y) C A'7(y) = ( J m 67 A'[m}(y) for any 7 € Γ ( η ) . Proof. The first inclusion is obvious by definition. So, we show the second B^ a reduced P-homomorphism, we equality. , tm} might be empty (cf. 1)).

Note that ωγχ = O y x ( - X ) i >2 Ri - ^ 2 ) , where A2 is an effective divisor on Y\ supported by R2 4- · · · + Rk- Then we have an exact sequence similar to ( * ) with Y and φ replaced by Y\ and φ\. 2 Note that H (0Yl) Ξ H°{0Yl{-Σΐ>2 ~ ^ 2 ) ) * = 0. Let Dx be the part of -Ks = aC + X) aiCi such that (Di)red = Ri. The virtual genus of Di is π(Ό1) 1 2 = \{D\ + D1K) + 1 = -{D\ - D ) + 1 = 1. 4, pg(x\) Therefore we have 1 H°{R UtOs) 439 = hPföipi + Os) is not less than π(ϋχ) = 1. =C and 1 H {OYl) = 0.

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