A generalization of the dual Kummer surface. (English) Zbl 1186.11036

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Let \(X\) be a proper, geometrically integral variety defined over field \(k.\) Here \(k\) is a perfect field of characteristic different from 2. Let further \(G= \text{Gal}({\bar k}/k)\) and \({\bar X}= X{\times}_{k}{\bar k}\). Denote by \(\text{Pic }\,X\) the classes in \(\text{Pic }\,{\bar X}\) that contain a divisor defined over \(k.\) If \(V\) is an abelian variety then the Kummer variety \(\mathcal K\) associated to \(V\) is defined to be a quotient variety of \(V\) by the involution \(x\rightarrow -x\). When \(V\) is the Jacobian of a hyperelliptic curve \(\mathcal C\), then \(\mathcal K\) is called the Kummer variety belonging to \(\mathcal C.\) In order to find rational divisor classes the authors study the exact sequence \[ \begin{tikzcd} 0\rar & \operatorname{Pic }X \rar & (\operatorname{Pic }{\bar X})^G \rar["\delta"] & \mathrm{Br}(k) \end{tikzcd} \] in the case where \(X=\mathcal C\) is a smooth hyperelliptic curve of genus \(g.\) This leads to a construction of a variety \({\mathcal K}^{*}\) with a hyperplane section \(H_m\) whose complement \({\mathcal K}_m^{*}\) parametrizes classes of divisors on \(\mathcal C\) in general position modulo the involution \(\pm Y\). This variety is showed to be birational to the Kummer variety belonging to \(\mathcal C\). The results extend these of [Manuscr. Math. 120, No. 4, 403–413 (2006; Zbl 1185.11043)].


11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14L30 Group actions on varieties or schemes (quotients)
11G10 Abelian varieties of dimension \(> 1\)
14L17 Affine algebraic groups, hyperalgebra constructions


Zbl 1185.11043
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