New York Journal of Mathematics
Volume 23 (2017) 1185-1203

  

Jason P. Bell, Dragos Ghioca, Zinovy Reichstein, and Matthew Satriano

On the Medvedev-Scanlon conjecture for minimal threefolds of nonnegative Kodaira dimension

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Published: September 9, 2017
Keywords: algebraic dynamics, orbit closures, rational invariants, Medvedev-Scanlon conjecture
Subject: 14E05, 14C05, 37F10

Abstract
Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let F be an algebraically closed field of characteristic 0 and let X be a quasiprojective variety defined over F endowed with a dominant rational self-map ϕ. Then there exists a point x∈ X(F) with Zariski dense orbit under ϕ if and only if ϕ preserves no nontrivial rational fibration, i.e., there exists no nonconstant rational functions f∈ F(X) such that ϕ*(f)=f. The Medvedev-Scanlon conjecture holds when F is uncountable. The case where F is countable (e.g., F=Qbar) is much more difficult; here the Medvedev-Scanlon conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension 0. Our results are most definitive in dimension 3.

Acknowledgements

The authors have been partially supported by Discovery Grants from the National Science and Engineering Board of Canada.


Author information

Jason P. Bell:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
jpbell@uwaterloo.ca

Dragos Ghioca:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
dghioca@math.ubc.ca

Zinovy Reichstein:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
reichst@math.ubc.ca

Matthew Satriano:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
msatriano@uwaterloo.ca