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Picard group of moduli of hyperelliptic curves. (English) Zbl 1132.14011
Call \(H_g\) the coarse moduli space of hyperelliptic curves of genus \(g\), defined over an algebraically closed field of characteristic \(\neq 2\). Call \({\mathcal H}_g\) the moduli stack of hyperelliptic curves. The authors compute the divisor class group \(\mathrm{Cl}(H_g)\) of the coarse moduli space, for most values of the characteristic. As a consequence, they prove that \(\mathrm{Pic}(H_g\)) is trivial, and exclude the existence of a universal family over the open set parametrizing curves with no extra automorphisms. The authors compare then the group \(\mathrm{Cl}(H_g)\) with the Picard group of \({\mathcal H}_g\), already computed by A. Arsie and A. Vistoli [Compos. Math. 140, No. 3, 647–666 (2004; Zbl 1169.14301)], and produce an explicit set of generators for \(\mathrm{Pic}({\mathcal H}_g)\).

MSC:
14D22 Fine and coarse moduli spaces
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