A rationality problem of some Cremona transformation

Abstract

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 $$\begin{array}{rlr}& \sigma (x)=\frac{-x(3x-9y-y^2{)}^3}{(27x+2x^2+9xy+2x{y}^2-y^3{)}^2},& \sigma (y)=\frac{-(3x+y^2)(3x-9y-y^2)}{27x+2x^2+9xy+2x{y}^2-y^3}.\end{array}$$ Theorem. The fixed field $k(x,y{)}^{⟨\sigma ⟩}$ 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.