TITLE: Do varieties without rational curves contain sub-polynomially many rational points?
QUESTION [8 upvotes]: Suppose $V$ is a projective algebraic variety over $\mathbb{Q}$ which does not contain any rational curves. Is it true/provable that the number of rational points on $V$ of Weil height at most $T$ grows slower than $T^{\delta}$ for any $\delta>0$?
I'm not sure what is implied by all the variations of Lang's conjectures out there, so clarifications on this would also be much appreciated.
Thanks!!

REPLY [2 votes]: I found it in an a paper of Mckinnon: https://arxiv.org/pdf/1011.5825.pdf
Apparently this is known as the "rational curve conjecture" and Mckinon almost proves that it follows from Vojta's conjectures. Specifically, assuming Vojta's conjecture he proves that any variety $X$ of non-negative Kodaira-dimension has an open subset $U\subset X$ containing sub-polynomially many points. If one then assumes that all varieties with negatvie Kodaira dimensio are ruled( apparently a conjecture of the minimal model program) one almost gets the result (you get rational curves, but perhaps not defined over $\mathbb{Q}$.