Hypertoric variety $Y(A, \alpha)$ is a (holomorphic) symplectic variety, which is defined as Hamiltonian reduction of complex vector space by torus action. By definition, there exists projective morphism $\pi:Y(A, \alpha) \to Y(A, 0)$, and for generic $\alpha$, this gives a symplectic resolution of affine hypertoric variety $Y(A, 0)$.
In general, for conical symplectic variety and it's symplectic resolution, Namikawa showed the existence of universal Poisson deformation space of them.
We construct universal Poisson deformation space of hypertoric varieties $Y(A, \alpha)$, $Y(A, 0)$. We will explain this construction and concrete description of Namikawa-Weyl group action in this case. If time permits, We will also talk about some classification results of affine hypertoric variety. This talk is based on my master thesis.