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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
##### Keywords:
Hyperelliptic curves
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##### References:
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