Open Access
October 2008 A rationality problem of some Cremona transformation
Akinari Hoshi, Ming-chang Kang
Proc. Japan Acad. Ser. A Math. Sci. 84(8): 133-137 (October 2008). DOI: 10.3792/pjaa.84.133


In this note we give a new approach to the rationality problem of some Cremona transformation. Let $k$ be any field, $k(x,y)$ be the rational function field of two variables over $k$. Let $\sigma$ be a $k$-automorphism of $k(x,y)$ defined by \begin{align*} &\sigma(x) = \frac{-x(3x-9y-y^{2})^{3}}{(27x+2x^{2}+9xy+2xy^{2}-y^{3})^{2}},\quad & \qquad \sigma(y) = \frac{-(3x+y^{2})(3x-9y-y^{2})}{27x+2x^{2}+9xy+2xy^{2}-y^{3}}. \end{align*} Theorem. The fixed field $k(x,y)^{\langle\sigma\rangle}$ is rational (= purely transcendental) over $k$. Embodied in a new proof of the above theorem are several general guidelines for solving the rationality problem of Cremona transformations, which may be applied elsewhere.


Download Citation

Akinari Hoshi. Ming-chang Kang. "A rationality problem of some Cremona transformation." Proc. Japan Acad. Ser. A Math. Sci. 84 (8) 133 - 137, October 2008.


Published: October 2008
First available in Project Euclid: 6 October 2008

zbMATH: 1162.14009
MathSciNet: MR2457800
Digital Object Identifier: 10.3792/pjaa.84.133

Primary: 12F20 , 13A50 , 14E07 , 14E08

Keywords: Cremona transformations , linear actions , monomial group actions , rationality problem

Rights: Copyright © 2008 The Japan Academy

Vol.84 • No. 8 • October 2008
Back to Top