 

Jia Jia,
Takahiro Shibata, and
DeQi Zhang
Potential density of projective varieties having an intamplified endomorphism
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Published: 
February 23, 2022. 
Keywords: 
Potential density, Intamplified endomorphism, Arithmetic degree, Dynamical degree. 
Subject: 
37P55, 14G05, 14E30, 08A35. 


Abstract
We consider the potential density of rational points on an algebraic variety defined over a number field K, i.e., the property that the set of rational points of X becomes Zariski dense after a finite field extension of K. For a nonuniruled projective variety with an intamplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an intamplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we prove the potential density for projective varieties with intamplified endomorphisms in dimension less than or equal to 3. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field.


Acknowledgements
The authors would like to thank the referee for very careful reading and the suggestions to improve the paper. The first, second and third authors are supported, from NUS, by the President's scholarship, a Research Fellowship and an ARF, respectively.


Author information
Jia Jia:
Department of Mathematics
National University of Singapore
Singapore 119076, Republic of Singapore
jia_jia@u.nus.edu
Takahiro Shibata:
Department of Mathematics
National University of Singapore
Singapore 119076, Republic of Singapore
mattash@nus.edu.sg
DeQi Zhang:
Department of Mathematics
National University of Singapore
Singapore 119076, Republic of Singapore
matzdq@nus.edu.sg

