Rationality of some projective linear actions (Q1570370)

Let \(K\) be a field, \(V\) a finite-dimensional vector space over \(K\), \(K[V]\) a coordinate ring of \(V\) over \(K\), and \(K(V)\) the function field of \(V\) over \(K\). Specifically, if \(x_1,x_2,\dots,x_n\) is a basis of \(V^*\), the dual space of \(V\), then \(K[V]= K[x_1,\dots,x_n]\), the polynomial ring of \(n\) variables over \(K\), and \(K(V)= K(x_1,\dots, x_n)\), the quotient field of \(K[x_1,\dots,x_n]\). Let \(P(V)\) denote the projective space associated with \(V\). Then define \(K(P(V))\) to be the subfield \(K(x_1/x_n,\dots,x_{n-1}/x_n)\) of \(K(V)\). Let \(G\) be a subgroup of \(GL(V)\). In this paper, the authors study under what conditions the fixed fields \(K(V)^G\) and \(K(P(V))^G\) are rational over \(K\). The authors do not restrict themselves to the assumptions that \(G\) is finite or that \(K\) is algebraically closed. The authors show the following result. Suppose that \(G=\langle\sigma\rangle\) and \(\dim_K V\leq 3\). Then both \(K(V)^G\) and \(K(P(V))^G\) are rational over \(K\). In addition, if \(G\) acts diagonally on \(V\oplus\cdots\oplus V\), then both \(K(V\oplus\cdots\oplus V)^G\) and \(K(P(V)\times\cdots\times P(V))^G\) are rational over \(K\). The authors also show the following result. If \(G=\langle \sigma\rangle\) acts on \(K(x_1,\dots,x_n,y_1,\dots,y_m)\) by \(\sigma(x_j)=(a_1x_j+b_1)/(c_1x_j+d_1)\) and \(\sigma(y_k)=(a_2y_k+b_2)/(c_2y_k+d_2)\), \(j=1,\dots,n\), \(k=1,\dots,m\), \(a_id_i-b_ic_i\neq 0\), \(a_i,b_i,c_i,d_i\in K\), \(i=1,2\), then \(K(x_1,\dots,x_n,y_1,\dots,y_m)^G\) is rational over \(K\).