The Lefschetz Principle in Birational Geometry: Birational Twin Varieties

On the Weak Lefschetz Principle in Birational Geometry

“…The following definition focuses on two possible forms of the weak Lefschetz principle in this context. They were introduced and explored in [13], [10]. We say that and are birational twins if they are Lefschetz divisorially equivalent Mori dream spaces and in addition * MCD( ) = MCD( ).…”

“…Recently in [12][13][14][15] the birational geometry of classical spaces, called spaces of complete forms, have been studied from the perspective of Definition 4. We finish up the article by mentioning some of the ingredients of such a study.…”

“…We finish up by citing our contribution to the vast realm of Lefschetz-type theorems: the spaces of complete quadrics and complete collineations are Lefschetz divisorially equivalent, for all > 1, via the embedding ∶ ( ) ↪ ( ) [13, Theorem A]. Furthermore, in the case of complete quadric surfaces, we have that (3) and (3) are birational twins [13,Theorem B]. Example 2.…”

“…The following definition focuses on two possible forms of the weak Lefschetz principle in this context. They were introduced and explored in [LHM20], [LMR20].…”

On the birational geometry of spaces of complete forms I: collineations and quadrics

“…An obvious but not very satisfactory answer would be the following: when X → Y yields an isomorphism of groups Pic(Y ) → Pic(X), and an isomorphism Cox(Y ) → Cox(X) of graded K-algebras with respect to a fixed grading on Pic(Y ) ∼ = Pic(X). For an introduction to this topic we refer to [32].…”