Manin’s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.
"On the geometry of thin exceptional sets in Manin’s conjecture." Duke Math. J. 166 (15) 2815 - 2869, 15 October 2017. https://doi.org/10.1215/00127094-2017-0011