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On Jacobian Kummer surfaces. (English) Zbl 1318.14042

It is well-known that the Kummer surface of the Jacobian of a curve of genus 2 is a complete intersection of 3 quadric in \(\mathbb{P}^5\). F. Klein [Clebsch Ann. II, 198–226 (1869); Gött. Nachr. 1869, 258–276 (1869; JFM 02.0605.01)] explicit equation for this in terms of the Weierstrass points of the curve.
The present paper gives explicit equations in terms of the 10 even theta constants of the corresponding principally polarized abelian surface. The method of proof consists in applying Riemann’s theta relations. Moreover, explicit equations are given for the total space of the fibration of Kummer surfaces over the Satake compactification of the Siegel modular threefold of level \((2,4)\). This can be regarded as a higher-dimensional analogue of Shioda’s elliptic modular surfaces.

MSC:

14K25 Theta functions and abelian varieties
14J28 \(K3\) surfaces and Enriques surfaces

Citations:

JFM 02.0605.01
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References:

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