# Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).

K3 surfaces is obtained by removing a fiber from two copies of $E$ and gluing along the boundaries.

How do we realize 22 second homology classes of K3, in terms of 10 second homology classes of $E$. I know this is classic but I could not find a reference.

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The construction is mentioned in arxiv.org/pdf/1201.0930.pdf. They reference the string theory paper arxiv.org/pdf/math/0308106v1.pdf. –  René Oct 3 '12 at 22:22
They have a discussion before gluing (on the union), I could not find anything there discussing what happens after gluing. I remember once Szabo told us about that and he was using some standard book on surfaces. –  Mohammad F. Tehrani Oct 3 '12 at 23:00
Try to look at the book of Gompf and Stipsicz –  Paolo Ghiggini Oct 3 '12 at 23:42
This might be a good solution. I should try it tomorrow. –  Mohammad F. Tehrani Oct 3 '12 at 23:45