Given a birational map ϕ among normal projective varieties, a geometric realization of ϕ is a normal projective C*-variety such that the birational map among geometric quotients parametrizing general orbits coincides with ϕ. Geometric realizations can be thought of as a projective algebraic version of the notion of cobordism coming from Morse theory. After recalling basic facts about Mori dream spaces, we show that any birational map among Mori dream spaces admits a geometric realization. In the context of toric varieties, we present a SageMath function to explicitly compute the polytope of the geometric realization. We conclude providing explicit examples of this construction.
