Finite $p$-groups of birational automorphisms and characterizations of rational varieties

Xu, Jinsong

doi:10.48550/arxiv.1809.09506

Cited by 1 publication

(3 citation statements)

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“…Our proof uses a finiteness criterion for the automorphism group of a projective variety proven in [39,Theorem 1.6] and properties of quasi-minimal models of projective non-uniruled varieties. Our approach is inspired by the work of Jinsong Xu [77].…”

Section: On the Finiteness Of The Group Of Birational Self-mapsmentioning

confidence: 99%

“…That is, Aut 1 (X) is the group of birational self-maps X X which are an automorphism outside a subset of codimension at least two. We refer the reader to [45], [62, §4] and [77] for a discussion of such birational self-maps.…”

Section: On the Finiteness Of The Group Of Birational Self-mapsmentioning

confidence: 99%

“…If L is a line bundle on X, we let [L] denote the class of L in NS(X) Q = NS(X) ⊗ Z Q. The following lemma is well-known (see for instance [77,Lemma 2.4]). We include a short proof for the reader's convenience.…”

Section: On the Finiteness Of The Group Of Birational Self-mapsmentioning

confidence: 99%

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Finiteness properties of pseudo-hyperbolic varieties

Javanpeykar

Xie

2019

Preprint

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Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi-Ochiai's finiteness result for varieties of general type, and thereby generalize Noguchi's theorem (formerly Lang's conjecture). Our proof here relies on a deformation-theoretic result for surjective maps of normal varieties due to Hwang-Kebekus-Peternell. Contents 19 8. Dominant self-maps of pseudo-Mordellic varieties 21 9. Dominant self-maps of pseudo-1-bounded varieties 22 References 24

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Section: On the Finiteness Of The Group Of Birational Self-mapsmentioning

confidence: 99%

Section: On the Finiteness Of The Group Of Birational Self-mapsmentioning

confidence: 99%