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Yuya Matsumoto

Publications and source records attributed to Yuya Matsumoto.

15 recordsLinked to original sources
arXiv2025

Linearly Reductive Quotient Singularities

We study isolated quotient singularities by finite and linearly reductive group schemes (lrq singularities for short) and show that they satisfy many, but not all, of the known properties of finite quotient singularities in characteristic zero: (1) From the lrq singularity we can recover the group scheme and the quotient presentation. (2) We establish canonical lifts to characteristic zero, which leads to a bijection between lrq singularities and certain characteristic zero counterparts. (3) We classify subgroup schemes of ${\mathbf{GL}}_d$ and ${\mathbf{SL}}_d$ that correspond to lrq singularities. For $d=2$, this generalises results of Klein, Brieskorn, and Hashimoto. Also, our classification is closely related to the spherical space form problem. (4) F-regular (resp. F-regular and Gorenstein) surface singularities are precisely the lrq singularities by finite and linearly reductive subgroup schemes of ${\mathbf{GL}}_2$ (resp. ${\mathbf{SL}}_2$). This generalises results of Klein and Du Val. (5) Lrq singularities in dimension $\geq 4$ are infinitesimally rigid. We classify lrq singularities in dimension $3$ that are not infinitesimally rigid and compute their deformation spaces. This generalises Schlessinger's rigidity theorem to positive and mixed characteristic. Finally, we study Riemenschneider's conjecture in this context, that is, whether lrq singularities deform to lrq singularities.

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arXiv2025

Torsors over the Rational Double Points in Characteristic $\mathbf{p}$

We study torsors under finite group schemes over the punctured spectrum of a singularity $x\in X$ in positive characteristic. We show that the Dieudonné module of the (loc,loc)-part $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ of the local Picard sheaf can be described in terms of local Witt vector cohomology, making $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ computable. Together with the class group and the abelianised local étale fundamental group, $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ completely describes the finite abelian torsors over $X\setminus\{x\}$. We compute $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ for every rational double point singularity, which complements results of Artin and Lipman, who determined ${π_{\mathrm{loc}}^{\mathrm{et}}}(X)$ and ${\rm Cl}(X)$. All three objects turn out to be finite. We extend the Flenner--Mumford criterion for smoothness of a normal surface germ $x \in X$ to perfect fields of positive characteristic, generalising work of Esnault and Viehweg: If $k$ is algebraically closed, then $X$ is smooth if and only if $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$, ${π_{\mathrm{loc}}^{\mathrm{et}}}(X)$, and ${\rm Cl}(X)$ are trivial. Finally, we study the question whether rational double point singularities are quotient singularities by group schemes and if so, whether the group scheme is uniquely determined by the singularity. We give complete answers to both questions, except for some $D_n^r$-singularities in characteristic $2$. In particular, we will give examples of (F-injective) rational double points that are not quotient singularities.

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arXiv2024

Inseparable Kummer surfaces

We introduce an inseparable version of Kummer surfaces. It is defined as a supersingular K3 surface in characteristic 2 with 16 smooth rational curves forming a certain configuration and satisfying a suitable divisibility condition. The main result is that such a surface admits an inseparable double covering by a non-normal surface $A$ that is similar to abelian surfaces in two aspects: its numerical invariants are the same as abelian surfaces, and its smooth locus admits an abelian group structure.

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arXiv2023

Supersingular reduction of Kummer surfaces in residue characteristic $2$

Given an abelian surface $A$, defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\mathrm{Km}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$.

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arXiv2022

Inseparable maps on $W_n$-valued local cohomology groups of non-taut rational double point singularities and the height of K3 surfaces

We consider rational double point singularities (RDPs) that are non-taut, which means that the isomorphism class is not uniquely determined from the dual graph of the minimal resolution. Such RDPs exist in characteristic $2,3,5$. We compute the actions of Frobenius, and other inseparable morphisms, on $W_n$-valued local cohomology groups of RDPs. Then we consider RDP K3 surfaces admitting non-taut RDPs. We show that the height of the K3 surface, which is also defined in terms of the Frobenius action on $W_n$-valued cohomology groups, is related to the isomorphism class of the RDP.

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arXiv2022

On $μ_{n}$-actions on K3 surfaces in positive characteristic

In characteristic $0$, symplectic automorphisms of K3 surfaces (i.e.\ automorphisms preserving the global $2$-form) and non-symplectic ones behave differently. In this paper we consider the actions of the group schemes $μ_{n}$ on K3 surfaces (possibly with rational double point singularities) in characteristic $p$, where $n$ may be divisible by $p$. We introduce the notion of symplecticness of such actions, and we show that symplectic $μ_{n}$-actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order $n$ in characteristic $0$. We also study local $μ_n$-actions on rational double points.

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arXiv2022

Degeneration of K3 surfaces with non-symplectic automorphisms

We prove that a K3 surface with an automorphism acting on the global $2$-forms by a primitive $m$-th root of unity, $m \neq 1,2,3,4,6$, does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the $l$-adic cohomology groups of the surface.

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arXiv2021

$μ_p$- and $α_p$-actions on K3 surfaces in characteristic $p$

We consider $μ_p$- and $α_p$-actions on RDP K3 surfaces (K3 surfaces with rational double point singularities allowed) in characteristic $p > 0$. We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of $μ_p$- and $α_p$-actions are analogous to those of $\mathbb{Z}/l\mathbb{Z}$-actions (for primes $l \neq p$) and $\mathbb{Z}/p\mathbb{Z}$-quotients respectively. We also show that conversely an RDP K3 surface with a certain configuration of singularities admits a $μ_p$- or $α_p$- or $\mathbb{Z}/p\mathbb{Z}$-covering by a "K3-like" surface, which is often an RDP K3 surface but not always, as in the case of the canonical coverings of Enriques surfaces in characteristic $2$.

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arXiv2021

Canonical coverings of Enriques surfaces in characteristic $2$

Let $\bar{Y}$ be a normal surface that is the canonical $μ_2$- or $α_2$-covering of a classical or supersingular Enriques surface in characteristic $2$. We determine all possible configurations of singularities on $\bar{Y}$, and for each configuration we describe which type of Enriques surfaces (classical or supersingular) appear as quotients of $\bar{Y}$.

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arXiv2021

Extendability of automorphisms of K3 surfaces

A K3 surface $X$ over a $p$-adic field $K$ is said to have good reduction if it admits a proper smooth model over the ring of integers of $K$. Assuming this, we say that a subgroup $G$ of $\mathrm{Aut}(X)$ is extendable if $X$ admits a proper smooth model equipped with $G$-action (compatible with the action on $X$). We show that $G$ is extendable if it is of finite order prime to $p$ and acts symplectically (that is, preserves the global $2$-form on $X$). The proof relies on birational geometry of models of K3 surfaces, and equivariant simultaneous resolutions of certain singularities. We also give some examples of non-extendable actions.

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arXiv2017

On automorphisms of Enriques surfaces and their entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.

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arXiv2017

Good Reduction of K3 Surfaces

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study group actions and their quotients on models of varieties.

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arXiv2017

Good reduction criterion for K3 surfaces

We prove a Neron--Ogg--Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its $l$-adic cohomology group is unramified. We also prove a $p$-adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic.

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arXiv2012

On good reduction of some K3 surfaces related to abelian surfaces

The Neron--Ogg--Safarevic criterion for abelian varieties tells that whether an abelian variety has good reduction or not can be determined from the Galois action on its l-adic etale cohomology. We prove an analogue of this criterion for some special kind of K3 surfaces (those which admit Shioda--Inose structures of product type), which are deeply related to abelian surfaces. We also prove a p-adic analogue. This paper includes Ito's unpublished result for Kummer surfaces.

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