A Note on Families of Hyperelliptic Curves Sergey Gorchinskiy and Filippo Viviani
نویسنده
چکیده
We give a stack-theoretic proof for some results on families of hyperelliptic curves.
منابع مشابه
A Note on Families of Hyperelliptic Curves Sergey Gorchinskiy and Filippo Viviani
We give a stack-theoretic proof for some results on families of hyperelliptic curves.
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The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the non-existence of a tautological family over the coarse moduli space.
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In a recent paper [1], Arsie and Vistoli have shown, as a byproduct of their study of moduli of cyclic covers of projective spaces, that the Picard group of the moduli stack Hg of smooth hyperelliptic curves of genus g ≥ 2 is finite cyclic, and that its order is 8g + 4 for odd g, and 4g + 2 for even g. In another recent paper [3], Gorchinskiy and Viviani have given a geometric construction of g...
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In 1963 Manin proved the Mordell conjecture for function fields (see [Man63]): Let K be a function field and let X be a nonisotrivial curve of genus at least 2 defined over K. Then X has finitely many K-rational points. Some years later Parshin (in the case of a complete base, [Par68]) and Arakelov (in the general case, [Ara71]) proved the Shafarevich conjecture for function fields: Let B be a ...
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Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic...
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