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Takehiko Yasuda

Publications and source records attributed to Takehiko Yasuda.

53 records · Page 3Linked to original sources
arXiv2013

The $p$-cyclic McKay correspondence via motivic integration

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ like in the tame case. Also, we link a crepant resolution with a count of Artin-Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

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arXiv2011

Universal flattening of Frobenius

For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme.

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arXiv2011

F-blowups of normal surface singularities

We study F-blowups of non-F-regular normal surface singularities. Especially the cases of rational double points and simple elliptic singularities are treated in detail.

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arXiv2011

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal Cohen-Macaulay; second, to see that in that situation, a different, but Morita equivalent, noncommutative crepant resolution can be constructed by using Frobenius morphisms; finally, to study the relation between Frobenius morphisms of noncommutative rings and the finiteness of global dimension. As a byproduct, we will obtain a result on wild quotient singularities: If the smooth cover of a wild quotient singularity is unramified in codimension one, then the singularity is not strongly F-regular.

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arXiv2009

Frobenius morphisms of noncommutative blowups

We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong's theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety.

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arXiv2009

Noncommutative resolution, F-blowups and D-modules

We explain the isomorphism between the $G$-Hilbert scheme and the F-blowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of $D$-modules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic.

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arXiv2009

On monotonicity of F-blowup sequences

For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the $e+1$-th blowup dominates the $e$-th, locally or globally. It is shown that the answer is affirmative (globally for any $e$) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowups. We also give a sufficient condition for local domination.

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arXiv2007

Non-adic formal schemes

Our purpose is to make a contribution to the foundation of the theory of formal scheme. We are interested particularly in non-Noetherian or non-adic formal schemes, which have been little studied. We redefine the formal scheme as a proringed space and study its basic properties. We also find several examples of non-adic formal schemes.

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arXiv2006

Flag higher Nash blowups

In his previous paper, the author has defined a higher version of the Nash blowup and considered it a possible candidate for the one-step resolution. In this paper, we will introduce another higher version of the Nash blowup and prove that it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions.

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arXiv2006

Higher Nash blowups

For each non-negative integer $n$, we define the $n$-th Nash blowup of an algebraic variety, and call them all higher Nash blowups. When $n=1$, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its $n$-th Nash blowup with $n$ large enough.

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arXiv2006

On subschemes of formal schemes

We think about what the subscheme of the formal scheme is. Differently form the ordinary scheme, the formal scheme has different notions of ``subscheme''. We lay a foundation for these notions and compare them. We also relate them to singularities of foliations.

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arXiv2003

Twisted jets, motivic measure and orbifold cohomology

We introduce the notion of twisted jet on a Deligne-Mumford stack. and study the motivic measure on the space of the twisted $\infty$-jets on a smooth Deligne-Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology with Hodge structure.

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arXiv2003

Jet schemes, log discrepancies and Inversion of Adjunction

We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal and Q-Gorenstein. As a first application, we prove a precise version of Inversion of Adjunction, in the case when the ambient variety is smooth. Another application concerns the semicontinuity of minimal log discrepancies on smooth varieties.

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arXiv2002

Dimensions of jet schemes of log singularities

The aim of the paper is to characterize Kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes. It is a generalization of Mustata's result.

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