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.
References
M. Hajja and M. Kang, Finite group actions on rational function fields, J. Algebra 149 (1992), no. 1, 139–154. MR1165204 10.1016/0021-8693(92)90009-BM. Hajja and M. Kang, Finite group actions on rational function fields, J. Algebra 149 (1992), no. 1, 139–154. MR1165204 10.1016/0021-8693(92)90009-B
M. Hajja and M. Kang, Three-dimensional purely monomial actions, J. Algebra 170 (1994), no. 3, 805–860. MR1305266 10.1006/jabr.1994.1366M. Hajja and M. Kang, Three-dimensional purely monomial actions, J. Algebra 170 (1994), no. 3, 805–860. MR1305266 10.1006/jabr.1994.1366
A. Hoshi and K. Miyake, Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 3, 21–26. MR2317305 10.3792/pjaa.83.21 euclid.pja/1176126885
A. Hoshi and K. Miyake, Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 3, 21–26. MR2317305 10.3792/pjaa.83.21 euclid.pja/1176126885
A. Hoshi and Y. Rikuna, Rationality problem of three-dimensional purely monomial group actions: the last case, Math. Comp. 77 (2008), no. 263, 1823–1829. MR2398796 10.1090/S0025-5718-08-02069-3A. Hoshi and Y. Rikuna, Rationality problem of three-dimensional purely monomial group actions: the last case, Math. Comp. 77 (2008), no. 263, 1823–1829. MR2398796 10.1090/S0025-5718-08-02069-3
I. Ya. Kolpakov-Miroshnichenko and Yu. G. Prokhorov, Rationality of fields of invariants of some four-dimensional linear groups, and an equivariant construction connected with the Segre cubic, Mat. Sb. 182 (1991), no. 10, 1430–1445; translation in Math. USSR-Sb. 74 (1993), no. 1, 169–183. MR1135933I. Ya. Kolpakov-Miroshnichenko and Yu. G. Prokhorov, Rationality of fields of invariants of some four-dimensional linear groups, and an equivariant construction connected with the Segre cubic, Mat. Sb. 182 (1991), no. 10, 1430–1445; translation in Math. USSR-Sb. 74 (1993), no. 1, 169–183. MR1135933
R. Swan, Noether's problem in Galois theory, in Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), edited by B. Srinivasan and J. Sally, 21–40, Springer-Verlag, New York, 1983. MR713790R. Swan, Noether's problem in Galois theory, in Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), edited by B. Srinivasan and J. Sally, 21–40, Springer-Verlag, New York, 1983. MR713790
O. Zariski, On Castelnuovo's criterion of rationality $p_{a}=P_{2}=0$ of an algebraic surface, Illinois J. Math. 2 (1958), 303–315. MR99990 euclid.ijm/1255454536
O. Zariski, On Castelnuovo's criterion of rationality $p_{a}=P_{2}=0$ of an algebraic surface, Illinois J. Math. 2 (1958), 303–315. MR99990 euclid.ijm/1255454536