The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5

Abstract: We determine the vanishing cycles of Xiao's genus-two fibrations $X_d\to P<sup>1$, $d=3,4,5$, whose fibers admit degree-$d$ maps to a fixed elliptic curve $E$. We tile the base $P<sup>1$ by triangles and realize the surface $X_d$ as a degree-$d$ branched cover of $P^1\times E$. The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For $d=4,5$, these give new positive factorizations of the identity in the genus-two mapping class group, of types $(6,7)$ and $(12,19)$.