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.

QUANTITATIVE OSCILLATION ESTIMATES FOR ALMOST-UMBILICAL CLOSED HYPERSURFACES IN EUCLIDEAN SPACE

We prove${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of${\it\delta}$on${\it\epsilon}$within the class of uniformly convex hypersurfaces with bounded volume.