arXiv subjects

Ken-ichi Yoshida

Publications and source records attributed to Ken-ichi Yoshida.

At least 19 recordsLinked to original sources

arXiv2026

Nearly Gorenstein rational surface singularities

In this paper, we show that for any rational surface singularity $A$, the canonical trace ideal $\mathrm{Tr}_A(K_A)$ is an integrally closed ideal, which is represented by the minimal anti-nef cycle $F$ on the minimal resolution of singularities so that $K_X+F$ is anti-nef. Then $F \ge Z_f$ if $A$ is not Gorenstein, where $Z_f$ is the fundamental cycle. As a result, we give a criterion for the rational surface singularity $A$ to be nearly Gorenstein. Moreover, we classify all nearly Gorenstein rational singularities in terms of resolution of singularities in the following cases: (a) the fundamental cycle $Z_f$ is almost reduced; (b) quotient singularities.

math.AG↗

arXiv2026

Nearly Gorenstein blow-up algebras over two-dimensional regular local rings

In this paper, we slightly extend the framework of nearly Gorenstein graded rings introduced by Herzog-Hibi-Stamate, and investigate the nearly Gorenstein property of the blow-up algebras associated with an ideal, namely the Rees algebra, the extended Rees algebra, and the associated graded ring. We establish characterizations of the nearly Gorenstein property for these algebras and clarify the relations among them. In particular, for two-dimensional regular local rings, we obtain a complete classification of the integrally closed ideals for which these blow-up algebras are nearly Gorenstein.

math.AC↗

arXiv2026

Hilbert-Kunz multiplicity of quadrics via Ehrhart theory

We show that the Hilbert-Kunz multiplicity of the d-dimensional non-degenerate quadric hypersurface of characteristic p > 2 is a rational function of p composed from the Ehrhart polynomials of integer polytopes. In consequence, we prove that the Hilbert-Kunz multiplicity of quadrics of fixed characteristic is a decreasing function of dimension and recover results of Trivedi and Gessel-Monsky on the behaviour of said Hilbert-Kunz multiplicity as a function of characteristic.

math.AC↗

arXiv2026

Nearly Gorenstein normal graded rings

We investigate nearly Gorenstein property for a normal graded ring $R = \bigoplus_{n\ge 0}R_n$ finitely generated over a field. For that purpose, we investigate ${K_R}^{-1}$, the inverse of $K_R$ (the canonical module of $R$) and introduce a new invariant $b(R)$ of $R$. We investigate nearly Gorenstein property of $R$ using $a(R)$ and $b(R)$ and $m(R)$, the initial degree of $R$. If $b(R)<0$, (and if $R$ is $\mathbb Q$-Gorenstein), then we believe that $R$ is log-terminal -- this is proved if $\dim R=2$ or $R$ is F-pure (or $F$-pure type). Then we determine the condition for a $2$-dimensional cone singularity over a smooth curve of genus $g\le 3$ to be nearly Gorenstein. We observe that ``almost Gorenstein" property and nearly Gorenstein property are drastically different for such rings.

math.AC↗

arXiv2026

On Gorensteinness of associated graded rings of filtrations

Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the Hilbert coefficients of $\mathcal{F}$ in some cases. As a consequence we recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of $A=S/(f)$ where $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in (x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{b+1}$, and $a, \, b, \, m$ are integers. We show that the normal tangent cone $\overline{G}(\mathfrak{m})$ is Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of the Gorensteinness of $\overline{G}(\mathfrak{m})$.

math.AC↗

arXiv2025

A variant of R{ö}hr's vanishing theorem with an application to the normal reduction number for normal surface singularities

Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field and let $X\to \mathrm{Spec} (A)$ be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the cohomology group of invertible sheaves on $X$ coincides with a natural lower bound. Applying this theorem, we establish upper bounds for the normal reduction number $\bar{\mathrm{r}}(A)$ of $A$. For example, we prove the inequality $\bar{\mathrm{r}}(A) \le p_a(A)+1$, where $p_a(A)$ denotes the arithmetic genus, a fundamental combinatorial (topological) invariant. We introduce the notion of almost cone singularities and give a sharper inequality $\bar{\mathrm{r}}(A) \le p_f(A)+1$ for such singularities, where $p_f(A)$ denotes the fundamental genus. We also show that $\bar{\mathrm{r}}(A)$ is not a combinatorial invariant in general.

math.AG↗

arXiv2025

Gorenstein Normal tangent cones of integrally closed ideals in two-dimensional normal singularities

Let $(A,\mathfrak m)$ be a two-dimensional excellent normal Gorenstein local domain containing an algebraically closed filed. Let $I =H^0(X,\mathcal{O}_X(-Z)) \subset A$ be an $\mathfrak m$-primary integrally closed ideal represented by an anti-nef cycle $Z$ on some resolution $X\to \mathrm{Spec} A$. In this paper, we prove that $\overline{G}(I)$ is Gorenstein if and only if it is Cohen-Macaulay and $(r-1)Z^2+K_XZ=0$, where $r=\bar{\athrm{r}}(I)$ denotes the normal reduction number of $I$ and $K_X$ denotes the canonical divisor on $X$.

math.AC↗

arXiv2025

Ulrich ideals on rational triple points of dimension two

In this paper, we prove the canonical trace ideal trace(omega_A) is an Ulrich ideal for any two-dimensional rational triple point A. Using this, we classify all Ulrich ideals on rational triple points. Moreover, we show that if (A, m) is a two-dimensional quotient singularity with the multiplicity e \ge 4 then m is the unique Ulrich ideal of A. As a result, we can classify all Ulrich ideals of A if A is either a rational triple point or a quotient singularity.

math.AC↗

arXiv2025

On vanishing of higher direct images of the structure sheaf

We show the vanishing of the first direct image of the structure sheaf of a normal scheme $X$ which is mapped properly and birationally over a regular scheme of any dimension. On the other hand, for any dimension greater than two, we show examples of a proper birational morphism from a normal and Cohen-Macaulay scheme to a regular scheme such that the second direct image does not vanish and has an isolated support.

math.AG↗

arXiv2024

A Geometric description of almost Gorensteinness for two-dimensional normal singularities

Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field. Then $A$ is called an elliptic singularity if $p_f(A)=1$, where $p_f$ denotes the fundamental genus. On the other hand, the concept of almost Gorenstein rings was introduced by Barucci and Fröberg for one-dimensional local rings and generalized by Goto, Takahashi and Taniguchi to higher dimension. In this paper, we describe almost Gorenstein rings in geometric language using resolution of singularities and give criterions to be almost Gorenstein. In particular, we show that elliptic singularities are almost Gorenstein. Also, for every integer $g\ge 2$, we provide examples of singularities that is almost Gorenstein (resp. not almost Gorenstein) with $p_f(A)=g$. We also provide several examples of determinantal singularities associated with $2\times 3$ matrices, which include both almost Gorenstein singularities and non-almost Gorenstein singularities.

math.AC↗

arXiv2024

Gorensteinness for normal tangent cones of elliptic ideals

Let $A$ be a two-dimensional excellent normal Gorenstein local domain. In this paper, we characterize elliptic ideals $I \subset A$ for its normal tangent cone $\overline{G}(I)$ to be Gorenstein. Moreover, we classify all those ideals in a Gorenstein elliptic singularity in the characteristic zero case.

math.AC↗

arXiv2022

Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper we study the elliptic and the strongly elliptic ideals of $A$ with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and by Yau. In analogy with the rational singularities, in the main result we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$-primary ideals of $A$. Unlike $p_g$-ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary and sufficient conditions for being normal are given. In the last section we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

math.AC↗

arXiv2022

Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert--Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.

math.AC↗

arXiv2021

The normal reduction number of two-dimensional cone-like singularities

Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} = Q\overline{I^n}\}, \quad \bar{r}(I) = \min\{n \;|\; \overline{I^{N+1}} = Q\overline{I^N}, \forall N\ge n\}$, $\mathrm{nr}(A)$, and $\bar{r}(A)$, where $\mathrm{nr}(A)$ (resp. $\bar{r}(A)$) is the maximum of $\mathrm{nr}(I)$ (resp. $\bar{r}(I)$) for all $\mathfrak m$-primary integrally closed ideals $I\subset A$. Then we have that $\bar{r}(A) \le p_g(A) + 1$, where $p_g(A)$ is the geometric genus of $A$. In this paper, we give an upper bound of $\bar{r}(A)$ when $A$ is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if $A$ is a hypersurface singularity defined by a homogeneous polynomial of degree $d$, then $\bar{r}(A)= \mathrm{nr}(\mathfrak m) = d-1$. Also we give an example of $A$ and $I$ so that $\mathrm{nr}(I) = 1$ but $\bar{r}(I)= \bar{r}(A) = p_g(A) +1=g+1$ for every integer $g \ge 2$.

math.AC↗

arXiv2018

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.

math.AC↗

arXiv2017

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.

math.AG↗

arXiv2017

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.

math.AC↗

arXiv2017

Almost Gorenstein Rees algebras of $p_g$-ideals, good ideals, and powers of the maximal ideals

Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K \cong A/{\mathfrak m}$ and $I$ is a $p_g$-ideal; (2) $(A,{\mathfrak m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity and $I={\mathfrak m}^{\ell}$ for all $\ell \ge 1$; (3) $(A,{\mathfrak m})$ is a regular local ring of dimension $d \ge 2$ and $I={\mathfrak m}^{d-1}$. Conversely, if ${\mathcal R}({\mathfrak m}^{\ell})$ is an almost Gorenstein graded ring for some $\ell \ge 2$ and $d \ge 3$, then $\ell=d-1$.

math.AC↗