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 get together of his seventieth birthday. It includes sixteen refereed articles within the parts of affine algebraic geometry, commutative algebra and similar fields, that have been the operating fields of Professor Miyanishi for nearly 50 years. Readers might be capable of finding fresh traits in those parts too. the subjects comprise 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 accordingly will offer clean examine difficulties to discover. This quantity is acceptable for graduate scholars and researchers in those components.

Readership: Graduate scholars and researchers in affine algebraic geometry.

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**Example text**

It follows that tN = 1. April 10, 2013 10:0 14 Lai Fun - 8643 - Aﬃne Algebraic Geometry - Proceedings 9in x 6in aﬃne-master I. Arzhantsev and M. Zaidenberg Since Γ0 ∩ Γ1 = {id} the map ψ|Γ0 : Γ0 → Gm is injective. So ψ(γ0 ) = ψ(γa,b (t)) has ﬁnite order dividing N . Due to claim 1 we can conclude that Γ0 = {id}. Claim 5. Γ = Ta,b . Proof of claim 5. For any γ ∈ Γ there exists t ∈ C× such that γ|C = −1 (t) ∈ Γ0 = {id} and so γ = γa,b (t) ∈ Ta,b . γa,b (t)|C. 12. 14 below the structure of the stabilizer Stab(C) for reduced (but possibly reducible) acyclic plane curves C of the remaining types (VI) and (V), respectively.

Clearly, C is not equivalent in X5, 4 to π(C1,1 ), π(Cx ), or π(Cy ). 3. Acyclic curves as orbit closures Summarizing the results of the previous subsections we arrive at the following alternative description. 8. Let X be an aﬃne toric surface over C with the acting torus T . e Furthermore, up to an automorphism of X, such a curve C is the closure of a non-closed orbit of a subtorus of T . Proof. Let C be an irreducible acyclic curve on a toric surface X. If X is smooth then this is one of the surfaces A1∗ × A1∗ , A1 × A1∗ , or A2 .

Since all such lines are T-equivalent, their images in the surface Xd,e are also equivalent under the action on Xd,e of the quotient torus T = T/Gd,e . The resulting automorphism δγ from the centralizer of the subgroup Gd,e in Aut(A2 ) rectiﬁes C and sends C 1 to a line T-equivalent to C1,1 . Consequently, the curve C on Xd,e is equivalent to π(C1, 1 ) under the T action on Xd,e and the automorphism π∗ (δγ) ∈ Aut(Xd,e ). Case 3: r > 1 and C is an acyclic curve of type (V) with smooth components C i and a non-ordinary singularity at the origin.