On Tropical Compactifications

We give an affirmative answer to a conjecture proposed by Tevelev in characteristic 0 case: any variety contains a sch"on very affine open subvariety. Also we show that any fan supported on the tropicalization of a sch"on very affine variety produces a sch"on compactification. Using toric schemes over a discrete valuation ring, we extend tropical compatifications to the non-constant coefficient case.