Abstract
Inspired by the Weak Lefschetz Principle, we study when a smooth projective
variety fully determines the birational geometry of some of its subvarieties.
In particular, we consider the natural embedding of the space of complete
quadrics into the space of complete collineations and we observe that their
birational geometry, from the point of view of Mori theory, fully determines
each other. When two varieties are related in this way, we call them birational
twins. We explore this notion and its various flavors for other embeddings
between Mori dream spaces.
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