Monodromy factorizations of lines on del Pezzo surfaces

We give a list of monodromy factorizations in the pure mapping class group $PMod(T_{d+1})$ of a torus with d+1 marked points that represent lines on a del Pezzo surface Y of degree $d\le4$.

These factorizations are lifts of a certain fixed monodromy factorization in $PMod(T_d)$ that represents Y.

In the case d=1, discussed in more detail, we give an explicit correspondence between such factorizations and the 240 roots of $E_8=K^\perp$, the orthogonal complement in $H_2(Y)$ of the canonical class.