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From arXiv

On sections of elliptic fibrations

We find a new relation among right-handed Dehn twists in the mapping class group of a kk-holed torus for 4k94 \leq k \leq 9. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp #(9-k) \cpb with kk base points and twelve singular fibers. By blowing up the base points we get an elliptic Lefschetz fibration on the complex elliptic surface E(1)=E(1)= \cp #9 \cpb S2 \to S^2 with twelve singular fibers and kk disjoint sections. More importantly we can locate these kk sections in a Kirby diagram of the induced elliptic Lefschetz fibration. The nn-th power of our relation gives an explicit description for kk disjoint sections of the induced elliptic fibration on the complex elliptic surface E(n)S2E(n) \to S^2 for n1n \geq 1.
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Published on April 24, 2006
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