Affine Algebraic Geometry: Proceedings of the Conference by Kayo Masuda, Hideo Kojima, Takashi Kishimoto

By Kayo Masuda, Hideo Kojima, Takashi Kishimoto

The current quantity grew out of a global convention on affine algebraic geometry held in Osaka, Japan in the course of 3-6 March 2011 and is devoted to Professor Masayoshi Miyanishi at the social gathering of his seventieth birthday. It comprises sixteen refereed articles within the parts of affine algebraic geometry, commutative algebra and comparable fields, that have been the operating fields of Professor Miyanishi for nearly 50 years. Readers may be capable of finding fresh developments in those components too. the subjects include either algebraic and analytic, in addition to either affine and projective, difficulties. all of the effects taken care of during this quantity are new and unique which for that reason will offer clean study difficulties to discover. This quantity is acceptable for graduate scholars and researchers in those parts.

Readership: Graduate scholars and researchers in affine algebraic geometry.

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5. + − (Nd,e , respectively) consists of all de Jonqi`eres (a) The group Nd,e transformations ϕ+ as in (7) (ϕ− as in (8), respectively) with f ∈ Ad,e (f ∈ Ad,e , respectively). ± (b) The subgroup Nd,e is the centralizer of Gd,e in the group ± Jonq (A2 ). Proof. We stick to the plus-case, the proof in the other one being similar. We have ϕ+ ◦ g ◦ (ϕ+ )−1 : (x, y) −→ y y ζ e x + f (ζ ) − ζ e f ( ), ζy β β . Hence ϕ+ ◦ g ◦ (ϕ+ )−1 ∈ Gd,e if and only if f (ζt) = ζ e f (t), if and only if f ∈ Ad,e .

3 a suitable automorphism γ ∈ Aut(A2 ) sends the reduced curve C = π∗ (C) to a union C = C 1 + . . + C r of affine lines through the origin given by equation (16) y(y − κ2 x) . . (y − κr x) = 0, where κi ∈ C× are distinct . The curve C is stable under the action on A2 of the cyclic group γGd,e γ −1 = g , where g = γgγ −1 ∈ Aut(A2 ). 9 Stab(C ) ⊆ GL(2, C), hence g ∈ GL(2, C). There exists an element δ ∈ GL(2, C) such that δg δ −1 = g is diagonal and acts via (x, y) → (ζ e x, ζy). Since no component C i of C is stable under g, the composition δγ sends each C i to a line through the origin different from a coordinate axis.

Clearly Ta,b ⊆ Stab(C). 12 Stab(C) = Ta,b . Suppose that r > 1. Consider a finite abelian group H = Stab(C) ∩ T1,0 . We claim that Stab(C) = H · Ta,b ⊆ T. Hence this is a quasitorus of rank one, as stated. Indeed, if δ ∈ Stab(C) \ Ta,b then δ(C 1 ) = C i for some i > 1. If h ∈ T1,0 is such that h(C i ) = C 1 then γ = h ◦ δ ∈ Stab(C 1 ) = Ta,b . Hence h = γ ◦ δ −1 ∈ H and so δ = h−1 ◦ γ ∈ H · Ta,b . Now the claim follows. This ends the proof. 14. Let C be an acyclic curve C of type (V) given by equation (2), where r ≥ 1 and εy + r ≥ 2.

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