arXiv subjects

Ryo Yamagishi

Publications and source records attributed to Ryo Yamagishi.

10 recordsLinked to original sources

arXiv2026

The Cautis-Logvinenko conjecture

For a finite subgroup $G\subset \operatorname{SL}(3,\mathbb{C})$, the Cautis--Logvinenko conjecture states that for each nontrivial irreducible representation $ρ$ of $G$, the image of the sheaf $\mathcal{O}_0\otimes ρ$ under the derived equivalence of Bridgeland--King--Reid is a pure sheaf on the $G$-Hilbert scheme. We prove a strong form of this conjecture in complete generality, and in doing so, we compute the relevant sheaf explicitly whenever its support is of dimension one. Our main result implies that a matrix defining the Gale dual of the linearisation map is sign-coherent, thereby allowing us to read off the support and cohomological degree of the pure sheaves directly from the matrix.

math.AG↗

arXiv2026

Hilbert schemes of points on canonical surfaces

For $n\geq 1$, we investigate the Hilbert scheme of $n$-points on a surface $S$ with canonical singularities. We generalise the well-known theorem of Fogarty by showing that the underlying reduced subscheme of $\operatorname{Hilb}^n(S)$ is a normal variety of dimension $2n$ with canonical singularities, and for $n\leq 7$, we show that $\operatorname{Hilb}^n(S)$ is reduced. When $S$ has symplectic singularities over $\mathbb{C}$, we show that the underlying reduced subscheme of $\operatorname{Hilb}^n(S)$ also has symplectic singularities, thereby generalising a result of Beauville. Our results build on work of the first author with Gyenge, Gammelgaard and Szendrői that sought to identify the underlying reduced subscheme of the Hilbert scheme of $n$-points on a Kleinian singularity with a Nakajima quiver variety.

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arXiv2025

The Le Bruyn-Procesi theorem following Lusztig

For any quiver $Q$ and dimension vector $v$, Le Bruyn-Procesi proved that the invariant ring for the action of the change of basis group on the space of representations $\text{Rep}(Q,v)$ is generated by the traces of matrix products associated to cycles in the quiver. Lusztig generalised this to allow for vertices where the group acts trivially. Here we provide a simple new proof of Lusztig's theorem and determine the relations between his algebra generators for any quiver with relations.

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arXiv2024

Moduli of $G$-constellations and crepant resolutions II: the Craw-Ishii conjecture

For any given finite subgroup $G\subset SL_3(\mathbb{C})$, we show that every projective crepant resolution $X$ of the quotient variety $\mathbb{C}^3/G$ is isomorphic to the moduli space of $θ$-stable $G$-constellations for a generic stability condition $θ$, as conjectured by Craw and Ishii. We also show that generators of the Cox ring of $X$ can be obtained from semi-invariants for representations of the McKay quiver of $G$.

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arXiv2022

Moduli of $G$-constellations and crepant resolutions I: the abelian case

For a finite abelian subgroup $G\subset SL_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety $\mathbb{C}^n/G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[\mathbb{C}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.

math.AG↗

arXiv2019

Four-dimensional conical symplectic hypersurfaces

We show that every indecomposable conical symplectic hypersurface of dimension four is isomorphic to the known one, namely, the Slodowy slice $X_n$ which is transversal to the nilpotent orbit of Jordan type $[2n-2, 1, 1]$ in the nilpotent cone of $\mathfrak{sp}_{2n}$ for some $n\ge 2$. In the appendix written by Yoshinori Namikawa, conical symplectic varieties of dimension two are classified by using contact Fano orbifolds.

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arXiv2018

Singularities of Fano varieties of lines on singular cubic fourfolds

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.

math.AG↗

arXiv2018

Symplectic resolutions of the Hilbert squares of ADE surface singularities

We study symplectic resolutions of the Hilbert scheme of two points on a surface with one ADE-singularity. We also characterize such singularities by central fibers of their symplectic resolutions. As an application, we show that these singularities are isomorphic to the Slodowy slices which are transversal to the `sub-subregular' orbits in the nilpotent cones of ADE-types.

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arXiv2016

On smoothness of minimal models of quotient singularities by finite subgroups of $SL_n(\mathbb{C})$

We prove that a quotient singularity $\mathbb{C}^n/G $ by a finite subgroup $G\subset SL_n(\mathbb{C})$ has a crepant resolution only if $G $ is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also give a procedure to compute the Cox ring of a minimal model of a given $\mathbb{C}^n/G$ explicitly from information of $G$. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities which admit projective symplectic resolutions.

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arXiv2016

Crepant resolutions of a Slodowy slice in a nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$

One of our results of this article is that every (projective) crepant resolution of a Slodowy slice in a nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$ can be obtained as the restriction of some crepant resolution of the nilpotent orbit closure. We also show that there is a decomposition of the Slodowy slice into other Slodowy slices with good properties. From this decomposition, one can count the number of crepant resolutions.

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