Limits of log canonical thresholds

Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n lies in T_{n-1}, proving in this setting a conjecture of Koll\'{a}r. We also show that T_n is a closed subset in the set of real numbers; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurov's ACC Conjecture for all T_n, it is enough to show that 1 is not a point of accumulation from below of any T_n. In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.Comment: 26 pages; revised version, to appear in Ann. Sci. Ecole Norm. Su

Log canonical thresholds of del Pezzo surfaces

We study global log canonical thresholds of del Pezzo surfaces.Comment: 16 page

A Running Time Improvement for Two Thresholds Two Divisors Algorithm

Chunking algorithms play an important role in data de-duplication systems. The Basic Sliding Window (BSW) algorithm is the first prototype of the content-based chunking algorithm which can handle most types of data. The Two Thresholds Two Divisors (TTTD) algorithm was proposed to improve the BSW algorithm in terms of controlling the variations of the chunk-size. In this project, we investigate and compare the BSW algorithm and TTTD algorithm from different factors by a series of systematic experiments. Up to now, no paper conducts these experimental evaluations for these two algorithms. This is the first value of this paper. According to our analyses and the results of experiments, we provide a running time improvement for the TTTD algorithm. Our new solution reduces about 7 % of the total running time and also reduces about 50 % of the large-sized chunks while comparing with the original TTTD algorithm and make average chunk-size closer to the expected chunk-size. These significant results are the second important value of this project

Log canonical thresholds of Del Pezzo Surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Koll\'ar-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degree. As an application we conclude that non-singular del Pezzo surfaces in finite characteristic of degree lower or equal than $4$ are K-semistable.Comment: 21 pages. Thorough rewrite following referee's suggestions. To be published in Manuscripta Mathematic