## 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

### Keywords:

elliptic fibrations; singular fiber configuration
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### References:

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