Monodromy factorizations of lines on del Pezzo surfaces

Abstract

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 $d\le 4$. 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.

26 pages, 5 Figures, 7 Tables. Last replacement: the title is modified, new statements, 2.4.2, 2.4.3 and 2.6.4 are added and more details have been supplied in the proofs of 1.2.2 and 1.2.3