Toric hyperk\"{a}hler varieties and Q-factorial terminalizations

A toric hyperk\"{a}hler variety is determined by combinatorial data A and $\alpha$. Here A is an integer valued matrix and $\alpha$ is a character of an algebraic torus $T^d$. $Y(A, \alpha)$ is a crepant partial resolution of an affine toric hyperk\"{a}hler variety $Y(A,0)$. However, $Y(A, \alpha)$ is not generally a Q-factorial terminalization of $Y(A,0)$ even if $\alpha$ is generic. In this article, we realize $Y(A,0)$ as another toric hyperk\"{a}hler variety $Y(A^{\sharp}, 0)$ so that $Y(A^{\sharp}, \alpha^{\sharp})$ is a Q-factorialization of $Y(A^{\sharp}, 0)$ for a generic $\alpha^{\sharp}$. As an application, we give a necessary and sufficient condition for $Y(A,0)$ to have a crepant resolution. Moreover, we construct explicitly the universal Poisson deformation of $Y(A,0)$ in terms of $A^{\sharp}$.