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Families of hypersurfaces with noncancellation property


Author: Kayo Masuda
Journal: Proc. Amer. Math. Soc. 145 (2017), 1439-1452
MSC (2010): Primary 14R20; Secondary 13N15, 14R25
DOI: https://doi.org/10.1090/proc/13489
Published electronically: December 27, 2016
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Abstract: Let $ k$ be a field of characteristic zero and $ R$ a factorial affine $ k$-domain. Let $ B$ be an affine $ R$-domain. In an earlier work of the author, the criteria are given in terms of locally nilpotent derivations for $ B$ to be $ R$-isomorphic to the residue ring of the form $ R[X,Y,Z]/(XY-\varphi (Z))$ for some $ \varphi (Z)\in R[Z]\setminus R$. We give a criterion for $ B$ to be $ R$-isomorphic to $ R[X,Y,Z]/(X^mY-F(X,Z))$ where $ m \ge 1$ and $ F(X,Z)\in R[X,Z]$ is such that $ F(0,Z)\in R[Z]\setminus R$ when $ B$ is factorial. We also show that for $ m \ge 1$, the hypersurfaces defined by $ x^my-F(x,z)=0$ have noncancellation property under some conditions on $ F(X,Z)\in R[X,Z]$.


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Additional Information

Kayo Masuda
Affiliation: Department of Mathematical Sciences, Faculty of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan
Email: kayo@kwansei.ac.jp

DOI: https://doi.org/10.1090/proc/13489
Keywords: Affine fibration, locally nilpotent derivation, Cancellation Problem
Received by editor(s): August 26, 2015
Received by editor(s) in revised form: May 29, 2016
Published electronically: December 27, 2016
Additional Notes: This research was supported by Grant-in-Aid for Scientific Research (C) 22540059, JSPS
Communicated by: Lev Borisov
Article copyright: © Copyright 2016 American Mathematical Society