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Tatsuki Yamaguchi

Publications and source records attributed to Tatsuki Yamaguchi.

7 recordsLinked to original sources
arXiv2026

BCM-regularity of diagonal hypersurfaces and plus-pure thresholds in mixed characteristic

We introduce a new method for computing plus-pure thresholds, a mixed-characteristic analogue of both log canonical thresholds and $F$-pure thresholds. We obtain some necessary conditions and some sufficient conditions for BCM-regularity of Fermat-type hypersurfaces. We also establish lower bounds for plus-pure thresholds of diagonal hypersurfaces in mixed characteristic. Furthermore, we give bounds for plus-pure thresholds of hypersurfaces in mixed characteristic $(0,2)$ using splitting-order sequences, introduced by Yoshikawa. As an application, we classify BCM-regular diagonal hypersurfaces in mixed characteristic $(0,2)$.

math.AC↗
arXiv2025

On the uniform positivity of $F$-signature under reduction modulo $p$

Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. In this paper, we give an affirmative answer to this conjecture in the case of pure subrings of regular local rings--for example, reductive quotient singularities. We also show that the conjecture can be reduced to the Gorenstein case. Finally, we discuss the connection with $F$-alpha invariants--a characteristic $p$ analog of Tian's alpha invariants introduced by Pande--for log Fano pairs.

math.AG↗
arXiv2025

On the behavior of adjoint ideals under pure morphisms

We characterize adjoint ideal sheaves via ultraproducts and, utilizing this characterization, study their behavior under pure morphisms. In particular, given a pure morphism $f:Y \to X$ between normal quasi-projective complex varieties, a reduced divisor $D$ and an effective $\mathbb{Q}$-Weil divisor $Γ$ on $X$ without common components, we have the following result: if the cycle-theoretic pullback $E:=f^{\natural}D$ is reduced and $(Y, E+f^*Γ)$ is of plt type along $E$, then $(X, D+Γ)$ is of plt type along $D$. This provides an affirmative answer to a question posed by Z. Zhuang.

math.AG↗
arXiv2025

Ultra-test ideals in rings with finitely generated anti-canonical algebras

When anti-canonical rings are finitely generated, we give a characterization of adjoint ideals using ultra-Frobenii, a characteristic zero analogue of Frobenius morphisms. This characterization enables us to give an alternative proof of a result of Zhuang, which states that if a ring is of klt type, then so is any of its pure subrings.

math.AG↗
arXiv2023

A characterization of multiplier ideals via ultraproducts

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens' non-standard tight closure, ultra-tight closure, on ideals of a local domain $R$ essentially of finite type over $\mathbb{C}$. We prove that the ultra-test ideal $τ_{\rm u}(R,\mathfrak{a}^t)$, the annihilator ideal of all ultra-tight closure relations of $R$, coincides with the multiplier ideal $\mathcal{J}(\operatorname{Spec} R,\mathfrak{a}^t)$ if $R$ is normal $\mathbb{Q}$-Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions.

math.AC↗
arXiv2023

Big Cohen-Macaulay test ideals in equal characteristic zero via ultraproducts

Utilizing ultraproducts, Schoutens constructed a big Cohen-Macaulay algebra $\mathcal{B}(R)$ over a local domain $R$ essentially of finite type over $\mathbb{C}$. We show that if $R$ is normal and $Δ$ is an effective $\mathbb{Q}$-Weil divisor on $\operatorname{Spec} R$ such that $K_R+Δ$ is $\mathbb{Q}$-Cartier, then the BCM test ideal $τ_{\hat{\mathcal{B}(R)}}(\hat{R},\hatΔ)$ of $(\hat{R},\hatΔ)$ with respect to $\hat{\mathcal{B}(R)}$ coincides with the multiplier ideal $\mathcal{J}(\hat{R},\hatΔ)$ of $(\hat{R},\hatΔ)$, where $\hat{R}$ and $\hat{\mathcal{B}(R)}$ are the $\mathfrak{m}$-adic completions of $R$ and $\mathcal{B}(R)$, respectively, and $\hatΔ$ is the flat pullback of $Δ$ by the canonical morphism $\operatorname{Spec} \hat{R}\to \operatorname{Spec} R$. As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.

math.AC↗