By Hiroaki Hijikata

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**Additional resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1**

**Example text**

From the ex act sequence 0 = ^^Β,(ί^φ)-^^|Β^(Βφ)-> ^Β^{Αφ Θ Gφ) ^ H'^A^V) —> ^ φ Β we have ί^φβφ(Βφ) = O, which contradicts depth Β φ = 2. Now the proof is completed. References [1] [2] Y. Aoyama, On the depth and the projective dimension of the canonical module, Japan. J. Math. ), 6 (1980), 61-66. Y. Aoyama, Some basic results on canonical modules, J. Math. , 23 (1983), 85-94. 34 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Y. ΑοΥΑΜΑ and S. G O T O Y. Aoyama and S. Goto, Some special cases of a conjecture of Sharp, J.

4) Any pair of coplanar tangents to S belongs to the three tangents of some plane W{s,t), s :t ePi. Proof Pairs of points χ G 5 with coplanar tangents are described in 5 χ 5 by the condition that a 4 χ 4-determinant vanishes. The variety of coplanar pairs of tangents therefore has no discrete components. Projecting 5 to P i from some tangent Τ = T^(S), χ G 5 , we obtain a covering 5 P i of degree < 4. Its number of branch points is < 6. On Rational Plane Sextics with Six Tritangents 49 If X = χ(Χ,μ) with Δ ( λ , μ ) φ O there are three distinct planes W{si,ti) contaming Γ .

3 . The proof of the theorem is by induction on the number of variables X i , . . , X n . , Xn-i]' Under this assumption we prove the following lemma: L e m m a : Let σ : R[X,Y] -> ( X - ( X i , . . ,Γλγ)) be the homomorphism defined by σ{Υ{) = y¿ G m¿[[X]], and let / i , . ,Ν. ,m. 1) Because of condition (i), we get, after performing a Weierstrass trans formation if necessary: g{X,y) = ä{Xn)e, where ε G R[[X]Y is invertible and ä ( X n ) = Σ > = ί polynomial with coefficients äj G R[[Xi,...