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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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