Hilbert-Kunz multiplicity of three-dimensional local rings

In this paper, we investigate a lower bound (say $s_{HK}(p,d)$) on Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ with characteristic $p>0$. Especially, we focus three-dimensional local rings. In fact, as a main result, we will prove that $s_{HK}(p,3) = 4/3$ and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degnerate quadric hyperplanes $k[[X,Y,Z,W]]/(X^2+Y^2+Z^2+W^2)$ under mild conditions. Furthermore, we pose a generalization of the main theorem to the case of $\dim A \ge 4$ as a conjecture, and show that it is also true in case of $\dim A = 4$ using the similar method as in the proof of the main theorem.Comment: about 21 pages, LaTe

Minimal Hilbert-Kunz multiplicity

In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we calculate this invariant for quotient singularities and for the coordinate ring of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$, respectively.Comment: about 22 pages, AMS-Te

Good ideals and pgp_g-ideals in two-dimensional normal singularities

In this paper, we introduce the notion of $p_g$-ideals and $p_g$-cycles, which inherits nice properties of integrally closed ideals on rational singularities. As an application, we prove an existence of good ideals for two-dimensional Gorenstein normal local rings. Moreover, we classify all Ulrich ideals for two-dimensional simple elliptic singularities with small degree.Comment: 21 pages. The proof of Proposition 5.6 of the first version has been revise

Rees algebras and pgp_g-ideals in a two-dimensional normal local domain

The authors introduced the notion of $p_g$-ideals for two-dimensional excellent normal local domain over an algebraicaly closed field in terms of resolution of singularities. In this note, we give several ring-theoretic characterization of $p_g$-ideals. For instance, an $m$-primary ideal $I \subset A$ is a $p_g$-ideal if and only if the Rees alegbra $\mathcal{R}(I)$ is a Cohen-Macaulay normal domain.Comment: 10 page

Normal reduction numbers for normal surface singularities with application to elliptic singularities of Brieskorn type

In this paper, we give a formula for normal reduction number of an integrally closed $\mathfrak m$-primary ideal of a $2$-dimensional normal local ring $(A,\mathfrak m)$ in terms of the geometric genus $p_g(A)$ of $A$. Also we compute the normal reduction number of the maximal ideal of Brieskorn hypersurfaces. As an application, we give a short proof of a classification of Brieskorn hypersurfaces having elliptic singularities.Comment: 14 page

A characterization of two-dimensional rational singularities via Core of ideals

The notion of $p_g$-ideals for normal surface singularities has been proved to be very useful. On the other hand, the core of ideals has been proved to be very important concept and also very mysterious one. However, the computation of the core of an ideal seems to be given only for very special cases. In this paper, we will give an explicit description of the core of $p_g$-ideals of normal surface singularities. As a consequence, we give a characterization of rational singularities using the inclusion of the core of integrally closed ideals.Comment: 16 pages; various corrections and improvements. To appear in Journal of Algebr

The strong Rees property of powers of the maximal ideal and Takahashi-Dao's question

In this paper, we introduce the notion of the strong Rees property (SRP) for $\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal $\mathfrak{m}$ has its property if the associated graded ring $G$ of $\mathfrak{m}$ satisfies $\text{depth} \ G \ge 2$. As its application, we characterize two-dimensional excellent normal local domains so that $\mathfrak{m}$ is a $p_g$-ideal. Finally we ask what $\mathfrak{m}$-primary ideals have SRP and state a conjecture which characterizes the case when $\mathfrak{m}^n$ are the only ideals which have SRP

Ulrich ideals and modules over two-dimensional rational singularities

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen-Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.Comment: 30 page

Irregular Oscillatory-Patterns in the Early-Time Region of Coherent Phonon Generation in Silicon

Coherent phonon (CP) generation in an undoped Si crystal is theoretically investigated to shed light on unexplored quantum-mechanical effects in the early-time region immediately after the irradiation of ultrashort laser pulse. One examines time signals attributed to an induced charge density of an ionic core, placing the focus on the effects of the Rabi frequency $\Omega_{0cv}$ on the signals; this frequency corresponds to the peak electric-field of the pulse. It is found that at specific $\Omega_{0cv}$'s where the energy of plasmon caused by photoexcited carriers coincides with the longitudinal-optical phonon energy, the energetically {\it resonant } interaction between these two modes leads to striking anticrossings, revealing irregular oscillations with anomalously enhanced amplitudes in the observed time signals. Also, the oscillatory pattern is subject to the Rabi flopping of the excited carrier density that is controlled by $\Omega_{0cv}$. These findings show that the early-time region is enriched with quantum-mechanical effects inherent in the CP generation, though experimental signals are more or less masked by the so-called coherent artifact due to nonlinear optical effects.Comment: 5 pages, 4 figure

Polaronic-Quasiparticle Picture for Generation Dynamics of Coherent Phonons in Semiconductors: Transient and Non-Linear Fano Resonance

We examine generation dynamics of coherent phonons (CPs) in both of polar and non-polar semiconductors -- such as GaAs and Si -- based on a polaronic-quasiparticle (PQ) model. In the model concerned, the PQ operator is composed of two kinds of operators. One is a quasiboson operator -- defined as a linear combination of a set of pairs of electron operators -- and the other is a longitudinal optical (LO) phonon operator. The problem of transient and non-linear Fano resonance (FR) is tackled in particular; the vestige of this quantum interference effect was observed exclusively in lightly $n$-doped Si immediately after carriers were excited by an ultrashort pulse-laser [M. Hase et. al., Nature 426, 51 (2003)], though not observed yet in GaAs. It is shown that the phonon energy state is embedded in a continuum state formed by a set of adiabatic eigenstates of the quasiboson. This result implies the possibility of manifestation of the transient FR in the present optically-non-linear system.Comment: 28 pages, 13 figure