Minkowski decompositions and universal equivariant deformations of toric pairs

Abstract

Let $X_{\sigma }$ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone $\sigma$, and let $m$ be a primitive degree with slice $P_m=\sigma \cap [m=1]$. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all degrees $-jm$, $j\ge 1$, simultaneously. We prove that the irreducible components of the reduced base correspond bijectively to maximal lattice-friendly Minkowski decompositions of $P_m$, and describe the induced family on each component. When $m\in {\sigma }^{\vee }$, we also obtain a formally universal equivariant family for the pair $(X_{\sigma },V({\chi }^m))$. These results extend earlier miniversality and component theorems to arbitrary affine normal toric varieties and arbitrary primitive degrees.