2 00 3 on F - Pure Thresholds

Using the Frobenius map, we introduce a new invariant of a pair (R, a) of a ring R and an ideal a ⊂ R, which we call the F-pure threshold c(a) of a, and study its properties. We see that the F-pure threshold characterizes some ring theoretic properties. By virtue of Hara and Yoshida's result [HY], the F-pure threshold c(a) in characteristic zero corresponds to the log canonical threshold lc(a) which is an important invariant in birational geometry. Using F-pure thresholds, we prove some ring theoretic properties for three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of F-pure thresholds similar to those of log canonical thresholds with quite simple proofs.