On sections of elliptic fibrations

We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. 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 $n \geq 1$.