On F-pure Thresholds

Using the Frobenius map, we introduce a new invariant for 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 several 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 the F-pure threshold, we prove some ring theoretic properties of three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of the F-pure threshold similar to those of the log canonical threshold with quite simple proofs.