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Group Theoretical Characterizations of Rationality
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Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X, with equality occurring if and only if X is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving conjectures by Popov and Furter-Poloni.
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Borel subgroups of the automorphism groups of affine toric surfaces
The automorphism group of each cyclic quotient X_{d,e} of the affine plane has either one or two conjugacy classes of Borel subgroups, depending on whether e^2 is congruent to 1 modulo d.
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