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A matrix realization of Kummer hyperelliptic varieties. (English. Russian original) Zbl 0899.14014
Russ. Math. Surv. 51, No. 2, 319-320 (1996); translation from Usp. Mat. Nauk 51, No. 2, 147-148 (1996).
Let $$\mathcal H$$ be the space of complex symmetric $$(g+2)\times(g+2)$$ matrices $$H=(h_{ij})$$ with $$h_{g+2,g+2}=0$$ and $$h_{g+1,g+2}=2$$. Let $$K\mathcal H=\{H\in\mathcal H:\text{rank}H\leq 3\}$$ be a matrix variety. Let $$C$$ be a smooth hyperelliptic curve of genus $$g$$ defined by an equation $$y^2=4x^{2g+1}+\sum_{k=0}^{2g}\lambda_{2g-k}x^{2g-k}$$. In this paper the authors obtain an explicit realization of the Kummer variety $$\text{Kum}(C)=\text{Jac}(C)/\pm$$ of the hyperelliptic curve $$C$$ in terms of Klein’s $$\sigma$$-functions as a subvariety of $$K\mathcal H$$. As a corollary of this result, there is a new proof of the theorem of Dubrovin and Novikov on the rationality of the universal space of Jacobians of hyperelliptic curves of genus $$g$$ with a distinguished branch point $$\infty$$.
##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14H52 Elliptic curves 14M12 Determinantal varieties 14H40 Jacobians, Prym varieties
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