  We undertake a detailed study of the geometry of Bottacin's Poisson
structures on Hilbert schemes of points in Poisson surfaces, i.e. smooth
complex surfaces equipped with an effective anticanonical divisor. We focus on
three themes that, while logically independent, are linked by the interplay
between (characteristic) symplectic leaves and deformation theory. Firstly, we
construct the symplectic groupoids of the Hilbert schemes and develop the
classification of their symplectic leaves, using the methods of derived
symplectic geometry. Secondly, we establish local normal forms for the Poisson
brackets, and combine them with a toric degeneration argument to verify that
Hilbert schemes satisfy our recent conjecture characterizing holonomic Poisson
manifolds in terms of the geometry of the modular vector field. Finally, using
constructible sheaf methods, we compute the space of first-order Poisson
deformations when the anti-canonical divisor is reduced and has only
quasi-homogeneous singularities. (The latter is automatic if the surface is
projective.) Along the way, we find a tight connection between the Poisson
geometry of the Hilbert schemes and the finite-dimensional Lie algebras of
affine transformations, which is mediated by syzygies. In particular, we find
that the Hilbert scheme has a natural subvariety that serves as a global
counterpart of the nilpotent cone, and we prove that the Lie algebras of affine
transformations have holonomic dual spaces -- the first such series of Lie
algebras to be discovered.
