Let $X$ be a del Pezzo surface, say of degree $3$ for concreteness. Then compare:
- the $27$ $(-1)$-curves form a lattice $E_6\subseteq H^2(X;\mathbf{Z})$; the Weyl group is generated by the simple reflections: $s_\delta:x \mapsto x - (x, \delta^\vee) \delta$
- if $X$ is the generic fibre of a Lefschetz pencil with vanishing cycle $\delta$, the monodromy map on $H^2(X;\mathbf{Z})$ is given by the Picard-Lefschetz theorem $T: x \mapsto x+(x,\delta)\delta$.
I'm interested if this is a coincidence or not.
The question: for every $(-1)$-curve $\delta$, is there a naturally arising Lefschetz pencil of del Pezzos with generic fibre $X$ whose monodromy is $s_\delta$? (say $X$ has degree $1-5$)
This question is quite similar to mine, but the answer doesn't make clear to me the relation to the Weyl group.
