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 \subset R, which we call the F-pure threshold \mathrm{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, the F-pure threshold \mathrm{c}(\a) in characteristic zero corresponds to the log canonical threshold \mathrm{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 F-pure threshold similar to those of the log canonical threshold with quite simple proofs.Comment: 19 pages; v.2: minor changes, to appear in J. Algebr

Multiplicity bounds in graded rings

The $F$-threshold c^J(\a) of an ideal \a with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of \a with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \cite{HMTW} by the same authors together with M. Musta\c{t}\u{a}, which bounds c^J(\a) in terms of the multiplicities e(\a) and $e(J)$, when \a and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when \a and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We also prove a similar inequality involving, instead of the $F$-threshold, the jumping number for the generalized parameter test submodules introduced in \cite{ST}.Comment: 19 pages; v.2: a new section added, treating a comparison of F-thresholds and F-jumping number