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Classification of all Jacobian elliptic fibrations on certain \(K3\) surfaces. (English) Zbl 1105.14055

The author classifies all the configurations of singular fibers of elliptic fibrations on the double cover of \(\mathbb{P}_2\) ramified along six lines in general position. The minimal resolution of the double cover is a \(K3\) surface. Since choosing six distinct lines in \(\mathbb{P}_2\) gives four moduli, also the family of \(K3\) surfaces is four dimensional. The author shows that the generic \(K3\) surface \(X\) in the family has Picard number 16.
Let \(L_1,\dots,L_6\) be the six lines in \(\mathbb{P}_2\) and \(l_i\) be the divisor on \(X\) such that \(2l_i\) is the strict transform of the pullback of the \(L_i\)’s. To find the singular fibers the author distinguishes two cases: when the \(l_i\)’s are components of the singular fibers and when there is a \(l_i\) which do not satisfy this condition. By using general results on elliptic fibrations, he find the following four possible configurations of singular fibers: \(I_{10}\), \(I_2\), \(aII\), \(bI_1\); \(I_8\), \(I_4\), \(aII\), \(bI\); \(2I_6\), \(aII\), \(bI_1\); \(IV^*\), \(I_4\), \(aII\), \(bI_1\) in any case with \(2a+b=12\). He computes also the rank of the Mordell-Weil group which is \(4,4,4\) resp. \(5\).

MSC:

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28 \(K3\) surfaces and Enriques surfaces
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References:

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