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? 



Del Pezzo surfaces and Picard-Lefschetz theory
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](https://arxiv.org/abs/math/0309012 "Seidel 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](https://link.springer.com/chapter/10.1007/978-3-0348-9217-9_5)). 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.


