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Kohsuke Shibata

Publications and source records attributed to Kohsuke Shibata.

13 recordsLinked to original sources
arXiv2026

A characterization of virtually free actions via arc spaces and its application to the lower semi-continuity conjecture

In this paper, we study the precise inversion of adjunction (PIA) conjecture and the lower semi-continuity (LSC) conjecture for hyperquotient singularities. Previously known results for these conjectures in this setting required the singularity to be klt, and without this assumption, a counterexample to the PIA conjecture is known to exist. To resolve this obstacle, we introduce a localized notion of virtually free actions and characterize it via the arc spaces of quotient varieties. Utilizing this characterization, we establish a necessary and sufficient condition for the PIA conjecture to hold for arbitrary hyperquotient singularities, thereby clarifying the mechanism of the counterexample. Furthermore, as an application of this insight, we unconditionally establish the LSC conjecture for arbitrary hyperquotient singularities.

math.AG↗
arXiv2026

Inversion of adjunction for quotient singularities III: semi-invariant case

We prove the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, we prove the semi-continuity of minimal log discrepancies for them. These results extend the results in our first paper, where we prove the same results for complete intersection varieties defined by ``invariant equations".

math.AG↗
arXiv2025

$F$-rationality of two-dimensional graded rings with a rational singularity

It is known that a two-dimensional $F$-rational ring has a rational singularity. However a two-dimensional ring with a rational singularity is not $F$-rational in general. In this paper, we investigate $F$-rationality of a two-dimensional graded ring with a rational singularity in terms of the multiplicity. Moreover, we determine when a two-dimensional graded ring with a rational singularity and a small multiplicity is $F$-rational.

math.AC↗
arXiv2025

A characterization of binomial Macaulay dual generators for complete intersections

We characterize a binomial such that the Artinian algebra whose Macaulay dual generator is the binomial is a complete intersection. As an application, we prove that the Artinian algebra with a binomial Macaulay dual generator has the strong Lefschetz property in characteristic 0 if the Artinian algebra is a complete intersection.

math.AC↗
arXiv2024

Inversion of adjunction for quotient singularities II: Non-linear actions

We prove the precise inversion of adjunction formula for quotient singularities. As an application, we prove the semi-continuity of minimal log discrepancies for hyperquotient singularities. This paper is a continuation of arXiv:2011.07300, and we generalize the previous results to non-linear group actions.

math.AG↗
arXiv2021

Inversion of adjunction for quotient singularities

We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.

math.AG↗
arXiv2020

Characterization of two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic

In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustaţǎ-Nakamura.

math.AG↗
arXiv2019

Minimal log discrepancies in positive characteristic

We show the existence of prime divisors computing minimal log discrepancies in positive characteristic except for a special case. Moreover we prove the lower semicontinuity of minimal log discrepancies for smooth varieties in positive characteristic.

math.AG↗
arXiv2019

The core of a module and the adjoint of an ideal over a two dimensional regular local ring

The core of an module is the intersection of all its reductions. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two dimensional regular local ring is the product of the module and the adjoint of an ideal. This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules $M$ and $N$ over a two-dimensional regular local ring with $M\subset N$ and $M^{**}=N^{**}$, the core of $M$ is contained in the core of $N$.

math.AC↗