F-thresholds of hypersurfaces
arXiv:0705.1210
Abstract
We continue our study of F-thresholds begun in math/0607660 by an in depth analysis of the hypersurface case. We use the D--module theoretic description of generalized test ideals which allows us to show that in any F--finite regular ring the F-thresholds of hypersurfaces are discrete and rational (in math/0607660 the finite type over a field case was shown for arbitrary ideals). Furthermore we show that any limit of F-pure thresholds of principal ideals in bouneded dimension is again an F-pure-threshold, hence in particular the limit is rational. The study of the set of F-pure-thresholds leads to natural analogs of conjectures of Shokurov and Kollár (for log canonical thresholds) in the case of F-pure-thresholds.
19 pages; v.2: a slight modification of the argument allowed us to extend our results to the case of an arbitrary regular F-finite ring; v.3: final version, to appear in Transactions of the AMS