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Zariski dense orbits for regular self-maps of tori in positive characteristic Saleh, Sina
Abstract
We formulate a variant in characteristic p of the Zariski dense orbit conjecture previously posed by Zhang, Medvedev-Scanlon and Amerik-Campana for rational self-maps of varieties defined over fields of characteristic 0. So, in our setting, let K be an algebraically closed field, which has transcendence degree d β₯ 1 over π½π. Let X be a variety defined over K, endowed with a dominant rational self-map Ξ¦. We expect that either there exists a variety Y defined over a finite subfield π½π² of π½π of dimension at least d + 1 and a dominant rational map Ο: X β€Y such that Ο o π«α΅= FΚ³ o Ο for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field π½π², or either there exists Ξ± β² X(K) whose orbit under π« is well-defined and Zariski dense in X, or there exists a non-constant π : X β€ βΒΉ such that π o π«= π . We explain why the new condition in our conjecture is necessary due to the presence of the Frobenius endomorphism in case X is isotrivial. Then we prove our conjecture for all regular self-maps on πΎα΄Ίm.
Item Metadata
Title |
Zariski dense orbits for regular self-maps of tori in positive characteristic
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Creator | |
Supervisor | |
Publisher |
University of British Columbia
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Date Issued |
2022
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Description |
We formulate a variant in characteristic p of the Zariski dense orbit conjecture previously posed by Zhang, Medvedev-Scanlon and Amerik-Campana for rational self-maps of varieties defined over fields of characteristic 0. So, in our setting, let K be an algebraically closed field, which has transcendence degree d β₯ 1 over π½π. Let X be a variety defined over K, endowed with a dominant rational self-map Ξ¦. We expect that either there exists a variety Y defined over a finite subfield π½π² of π½π of dimension at least d + 1 and a dominant rational map Ο: X β€Y such that Ο o π«α΅= FΚ³ o Ο for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field π½π², or either there exists Ξ± β² X(K) whose orbit under π« is well-defined and Zariski dense in X, or there exists a non-constant π : X β€ βΒΉ such that π o π«= π . We explain why the new condition in our conjecture is necessary due to the presence of the Frobenius endomorphism in case X is isotrivial. Then we prove our conjecture for all regular self-maps on πΎα΄Ίm.
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Genre | |
Type | |
Language |
eng
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Date Available |
2022-04-14
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0412867
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2022-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International