Proceedings of the Japan Academy, Series A, Mathematical Sciences

A rationality problem of some Cremona transformation

Akinari Hoshi and Ming-chang Kang

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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.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 8 (2008), 133-137.

First available in Project Euclid: 6 October 2008

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Primary: 14E07: Birational automorphisms, Cremona group and generalizations 14E08: Rationality questions [See also 14M20] 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 12F20: Transcendental extensions

Rationality problem Cremona transformations linear actions monomial group actions


Hoshi, Akinari; Kang, Ming-chang. A rationality problem of some Cremona transformation. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 8, 133--137. doi:10.3792/pjaa.84.133.

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