On the degrees of irrationality of hyperelliptic surfaces (Q1924883)

Let \(L\) be a field, which is a finitely generated extension of a ground field \(k\), and assume that \(\text{tr deg}_k L=n\). We denote by \(d_r(L)\) the degree of irrationality of \(L\) over \(k\), which is defined to be the number \[ \begin{aligned} \min\{m |\;& m=[L: k(x_1,\dots, x_n)], \text{ where }x_1,\dots,x_n \text{ are }\\ & \text{algebraically independent elements of }L.\}\end{aligned} \] We call the field \(k(x_1,\dots, x_n)\), which defines the value \(d_r(L)\), a maximal rational subfield of \(L\). For an algebraic variety \(V\) defined over \(k\), we define the degree of irrationality of \(V\) to be \(d_r(k(V))\), where \(k(V)\) is the rational function field of \(V\). Clearly it is a birational invariant of algebraic varieties, it is the minimal degree of a dominant rational map from \(V\) to the projective \(n\)-space. We assume that \(k=\mathbb{C}\) hereafter. In this note we announce the results for \(d_r(S)\) of hyperelliptic surfaces \(S\). Details will appear elsewhere.