The moduli of curves is stably rational for g\(\leq 6\) (Q1065090)

We assume that everything is defined over an algebraically closed field F. An algebraic variety X is called ''stably rational'' over F if \(X\times {\mathbb{P}}^ k\) is birational to \({\mathbb{P}}^ n\) for some k and n. Now, let \({\mathcal M}_ g(F)\) be the moduli space of curves of genus g over F. It is known that \({\mathcal M}_ g(F)\) is rational for \(g=1,2\), and that it is unirational for \(g\leq 10\). In this paper it is proved that \({\mathcal M}_ g(F)\) is stably rational over F for \(g\leq 6\).