Flag varieties, toric varieties, and suspensions: Three instances of infinite transitivity

We say that a group $G$ acts infinitely transitively on a set $X$ if for every $m\in\mathbb N$ the induced diagonal action of $G$ is transitive on the cartesian $m$th power $X^m\setminus\Delta$ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups.
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