A map to a toric variety and a toric degeneration

Lara Bossinger, Takuya Murata

Published 2022-10-24Version 1

We study maps to a toric variety. A torus embedding gives a simple example and, conversely, such maps generalize a torus embedding in that they are extensions of monomial maps. The motivation comes from a toric degeneration, a flat family whose special fiber is a not-necessarily-normal toric variety. In that case (and more generally), we show, away from a base locus, there is a dominant morphism from a general fiber of a toric degeneration to the special fiber. In the process, we give a negative answer to the question of I. Dolgachev and K. Kaveh as to whether a projective toric degeneration can be constructed by degeneration by projection. In the classical topology over $\mathbb{C}$, following Goresky and MacPherson, we construct a continuous surjective map from a general fiber of a toric degeneration to the special fiber that is a covering over each orbit. We then explain how to recover and extend the construction of integral systems in M. Harada and K. Kaveh. "Integrable systems, toric degenerations and okounkov bodie" to the singular locus (using the structure of a symplectic stratified space).