# Del Pezzo surfaces and Picard--Lefschetz theory

Let $$X$$ be a del Pezzo surface, say of degree $$3$$ for concreteness. Then compare:

1. the $$27$$ $$(-1)$$-curves form a lattice $$E_6\subseteq H^2(X;\mathbf{Z})$$; the Weyl group is generated by the simple reflections: $$s_\delta:x \mapsto x - (x, \delta^\vee) \delta$$
2. 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.