## 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)].

### MSC:

 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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