 

Jason P. Bell, Dragos Ghioca, Zinovy Reichstein, and Matthew Satriano
On the MedvedevScanlon conjecture for minimal threefolds of nonnegative Kodaira dimension view print


Published: 
September 9, 2017

Keywords: 
algebraic dynamics, orbit closures, rational invariants, MedvedevScanlon 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 selfmap 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 MedvedevScanlon conjecture holds when F is uncountable. The case where
F is countable (e.g., F=Qbar) is much more difficult; here
the MedvedevScanlon conjecture has only been proved in a small number
of special cases. In this paper we show that
the MedvedevScanlon 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

