On the rationality of moduli spaces of pointed hyperelliptic curves (Q422039)

For all integers \(g\geq 1\), \(n\geq 1\) let \(\mathcal {H}_{g,n}\) be the coarse moduli space of the hyperelliptic curves of genus \(g\) with \(n\) marked points. In this paper the author proves that \(\mathcal {H}_{g,n}\) is rational if \(g\geq 3\) and \(1 \leq n \leq 2g+8\). The cases \(g=1\) (here the bound \(n\leq 2g+8 =10\) is sharp, because the Kodaira dimension of its compactification is \(0\) for \(n=11\) and \(1\) for \(n\geq 12\) (\textit{G. Bini} and \textit{C. Fontanari} [Trans. Am. Math. Soc. 358, No. 7, 3207--3217 (2006; Zbl 1105.14030)]) and \(g=2\) [\textit{G. Casnati} and \textit{C. Fontanari}, J. Lond. Math. Soc., II. Ser. 75, No. 3, 582--596 (2007; Zbl 1125.14012)] were previously known. For the proof the author constructs a variety \(V\) on which a connected algebraic group \(G\) acts and such that \(\mathcal {H}_{g,n}\) is birational to the quotient \(V/G\) (the construction is different in the case \(n\leq 3\) and in the case \(4 \leq n \leq 2g+8\)). He first proves that \(V\) is rational (easy, but the key point is to find a suitable \(V\)) and then he proves that \(V/G\) is rational (not just unirational) either by the specific geometric action of \(G\) or by specific properties of \((V,G)\) (for \(n\leq 3\), because \(V\) is a linear representation of the connected and solvable algebraic group \(G\)).