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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.

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