By Hiroaki Hikikata

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**Extra info for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2**

**Example text**

4, where you will see that the canonical bundle Κ is not always numerically an integral divisor in this case. In the case (ii), Κ is always numerically a divisor. 3. Suppose that S has the type (3,/, 2). Al-2 \ L-l ΑΙ ΒΙ -3 Figure 2. -2 V Canonical Bundles of Analytic Surfaces of Class VIIO 447 Then we have Κ ξ ξ -B1 2A,_i. 1, we omit it. 2. We assume that p\ > 3. Then it sufices to prove that the type must be (3,/, 2) provided Κ is numerically a divisor. 1, the canonical bundle Κ in the case η = 1 is numerically a divisor if and only if p\ = 3 , hence we may assume that η > 2.

Then A is also a nagata Ρ-ring. And in §3, along Rotthaus' idea, we prove: Theorem B . Let A be a noetherian ring with an ideal I. Suppose that 1) A is I-adically complete and A/I is quasi-excellent, 2) Resolution of Singularities holds for any excellent local domain over A. Then A is also quasi-excellent. Finally, Hironaka's Resolution of Singularities [8] and Theorem Β show: Corollary C . Let A be a noetherian ring, containing afield of characteristic 0, with an ideal I. Suppose A is I-adically complete and A/1 is quasiexcellent.

N ) . Note that pn > 2, ph q3> 3 (1 < ; < η - 1) and tjh > 1 (1 < k < η - 1). d. by the equation (0), we obtain η = 2, pi =3 and p2 — 2. Next we consider the case p\ — 2. Let S be a surface containing a GSS. 4. Let a* (resp. bk) be the coefficient Sih (resp. tjh) in -K of the curve A{h (resp. Bjh) of s elf-intersection number —pk (resp. —qk)- The coefficients in —K of the curves of self-intersection number —2 in a block between those with —pk and -pk+\ (resp. —qk and — g^+i) in C\ (resp. D\) constitute an arithmetic progression with common difference ck (resp.