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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.