On F-pure thresholds

Using the Frobenius map, we introduce a new invariant for a pair $(R,\text{a})$ of a ring $R$ and an ideal $\text{a}\subset R$, which we call the F-pure threshold $\mathrm{c}(\text{a})$ of $\text{a}$, and study its properties. We see that the F-pure threshold characterizes several ring theoretic properties. By virtue of Hara and Yoshida’s result, the F-pure threshold $\mathrm{c}(\text{a})$ in characteristic zero corresponds to the log canonical threshold $\mathrm{l}\mathrm{c}(\text{a})$ 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 F-pure threshold similar to those of the log canonical threshold with quite simple proofs.