ON THE RATIONALITY OF MODULI SPACES OF POINTED HYPERELLIPTIC CURVES

Let Mg,n be the (coarse) moduli space of smooth, integral, projective curves of genus g ≥ 1 with n marked points defined over the complex field C. We denote by Hg,n ⊆ Mg,n the locus of points corresponding to curves carrying a $g_2^1$. It is known that Hg,n, is rational for g = 1 and n ≤ 10, for g = 2 and n ≤ 12 and for each g ≥ 3 and n = 0. We prove here that the same is true for each g ≥ 3 and 1 ≤ n ≤ 2g + 8.