On sections of elliptic fibrations

Abstract

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

arXiv comment12 pages, 12 figures