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Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (in the case of a cubic surface the corresponding root system is $E_6$). In particular this relation manifests itself in the action of the Weyl group on $Pic(X)$ permuting the classes of exceptional curves.

I suspect that one could see this action via Picard-Lefschetz theory. For that one could consider a moduli space of del Pezzo surfaces and the universal family of del Pezzo surfaces, fibered over it. Then the monodromy action on the middle cohomology of the fiber should be generated by orthogonal reflections, according to Picard-Lefschetz theory.

I wonder how to make this picture precise. In particular, what moduli space one should consider here?

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  • $\begingroup$ A useful reference might be "Galois groups of enumerative problems" by Joe Harris. $\endgroup$ – Pop Aug 30 '18 at 14:41
  • $\begingroup$ You should know that moduli spaces of surfaces are extremely hard to understand, at least if you want them to be compact (surfaces parametrized by the boundary can have semi-log canonical singularities). $\endgroup$ – Tabes Bridges Aug 30 '18 at 18:59
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Indeed you can see it this way. This is my symplectic geometer's perspective on it (I blame Paul Seidel's Lecture notes on four-dimensional Dehn twists).

Consider the $n$-point blow-up of $\mathbf{CP}^2$ at $n$ general points; you get a moduli space by varying the points and you get isomorphic varieties if the point configurations are related by the action of $PGL(3,\mathbf{C})$, so take as your moduli space the quotient of the space of general point configurations by this $PGL(3)$-action. This is not a fine moduli space because a surface can have automorphisms: if $n\geq 4$ then this automorphism group is finite (the $PGL(3)$-stabiliser of four general points is $S_4$), and if $n\geq 5$ then the automorphism group is generically trivial, so for simplicity let's focus on $n\geq 5$ and just throw away all the surfaces with automorphisms. You get a universal family over this moduli space; using a relatively ample bundle (suitable power of the anticanonical bundle) you get a map from the total space of the family to some fixed projective space. Pull back a Fubini-Study form to get a closed 2-form $\Omega$ on the universal family whose restriction to fibres is a symplectic form.

Now you can do symplectic parallel transport along paths in the moduli space: you get a symplectic connection on the total space by taking the $\Omega$-orthogonal complement to the fibres; parallel transport maps preserve the symplectic form on fibres; contractible loops give Hamiltonian monodromies. You therefore get a map from $\pi_1$ of your moduli space to the symplectic mapping class group ($Symp(X)/Ham(X)$). You can further compose this with a map $Symp(X)/Ham(X)\to Aut(H^2(X))$ to the group of automorphisms of cohomology; the image of this latter map will be your Weyl group (because that is the automorphism group of the cohomology lattice preserving its intersection form).

How to see the link with Picard-Lefschetz theory? As your $n$ points vary, there is a complex codimension one thing they can fail to be in general position: namely, one of them can pass through the complex line connecting two others. Other things can also happen: a sixth point can pass through the complex conic connecting five others, for example. When one of these things happen, you get a $-2$-sphere by taking the proper transform of the line/conic/whatever. Your relatively ample bundle fails to be ample for this blowup because the anticanonical class annihilates the class of a $-2$-sphere (by adjunction: $-K.C=C^2+2=0$) so the family of Del Pezzos develops a nodal singularity as this degeneration occurs (the minimal resolution of a nodal singularity has exceptional locus a single $-2$-sphere). In terms of symplectic geometry, there is a Lagrangian 2-sphere in the smooth Del Pezzo which is collapsed to the node if you follow the symplectic parallel transport; this is called the vanishing cycle.

As this phenomenon is complex codimension one, there is a loop in the moduli space where your $n$ points skirt around this degenerate configuration. The monodromy around this loop is a symplectic Dehn twist in the Lagrangian sphere (as was observed by Arnold). The Dehn twist acts as a reflection in cohomology in the class of the $-2$-sphere: if you have a homology class disjoint from the Lagrangian sphere then it is unaffected by the twist, which is supported near the sphere; the homology class of the sphere gets reversed because the Dehn twist is the antipodal map on the sphere. This is precisely the Picard-Lefschetz formula.

Note that the homology class of the Lagrangian sphere and the homology class of the holomorphic $-2$-sphere in the minimal resolution of the singular guy can be identified; I have an old blog-post explaining how this works using small resolutions:

http://www.homepages.ucl.ac.uk/~ucahjde/blog/kronheimer-argument.html

For a symplectic geometer, the more interesting fact is that you can go beyond the homological monodromy action: if you take the moduli space of $n$ ordered points then there is a universal family (no automorphisms when $n\geq 4$) and the homology action is trivial because the monodromy is generated by squared Dehn twists (when you go around one of the loops I discussed before, you switch two points, so to get back to the identity in homology you need to go around the loop twice) and Picard-Lefschetz tells you that a squared Dehn twist acts as the identity. Nonetheless, the monodromy gives a map from $\pi_1$ of the moduli space to the symplectic mapping class group, and Paul Seidel's early work showed that this is often injective (not only for Del Pezzos). You don't see anything at the level of ordinary smooth mapping class groups (the squared Dehn twist is smoothly isotopic to the identity) so symplectic geometry is remembering more about the algebraic geometry here.

Like I said, I learned all of this from Seidel's Lectures on Dehn twists: well worth the read.

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