Bounded Volume Denominators and Bounded Negativity

In this paper, we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular, we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our 1st result shows that boundedness of volume denominators is equivalent to primitive bounded negativity, which in turn is implied by bounded negativity. We connect this result to the study of semi-effective orders of divisors: our 2nd result shows that negative classes exist, which become effective only after taking an arbitrarily large multiple.