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

Operator Dimensions from Moduli

We consider the operator spectrum of a three-dimensional ${\cal N} = 2$ superconformal field theory with moduli spaces of one complex dimension, such as the fixed point theory with three chiral superfields $X,Y,Z$ and a superpotential $W = XYZ$. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large $R$-charge $J$. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly $J+1$, a fact that cannot be seen directly from the $XYZ$ Lagrangian. The third-lowest scalar primary lies in a long multiplet with dimension $J+2 - c_{-3} \, J^{-3} + O(J^{-4})$, where $c_{-3}$ is an unknown positive coefficient. The coefficient $c_{-3}$ is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of $c_{-3}$ does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large $J$, this proves the existence of scalar semi-short states at all values of $J$. Thus the combination of ${\cal N}=2$ superconformal symmetry with the large-$J$ expansion is more powerful than the sum of its parts.Comment: 48 pages, 8 figures, LaTeX, typos correcte

神経芽腫培養細胞株のPolyphyllin D(Paris polyphylla)におけるアポトーシスとネクロトーシスについて

藤田保健衛生大

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