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On the geometry of singular \(K3\) surfaces with discriminant 3, 4 and 7. (English) Zbl 1491.14057

The present article studies the three singular \(K3\) surfaces \(X_3\), \(X_4\) and \(X_7\), of discriminant \(3\), \(4\) and \(7\) respectively. These \(K3\) surfaces were explicitly constructed by T. Shioda and H. Inose [in: Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 119–136 (1977; Zbl 0374.14006)].
The author shows that the \(K3\) surfaces \(X_3\), \(X_4\) and \(X_7\) admit a realization as a double cover of \(\mathbb{P}^2\) branched over a sextic curve with prescribed singularities. As a consequence, the three surfaces can be viewed as points in a \(3\)-dimensional moduli space of \(K3\) surfaces, namely the moduli space of \(U\oplus A_5^3\)-polarized \(K3\) surfaces.

MSC:

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties

Citations:

Zbl 0374.14006
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References:

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