Fibred toric varieties in toric hyperk\"{a}hler varieties

Abstract

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these fibred toric varieties also arise naturally in the variation of hyperk\"{a}hler varieties, namely, the fibred toric varieties are contained in the exceptional sets of the hyperk\"{a}hler natural morphisms and the Mukai flops.