2.1 Group schemes

Let G be a finite group scheme over an algebraically closed field k of characteristic $p\geq 0$ . Since k is perfect, there is a short exact sequence of finite group schemes over k

(2.1) $$ \begin{align} 1\,\to\,G^\circ\,\to\,G\,\to\, G^{\mathrm{\acute{e}t}} \,\to\,1, \end{align} $$

where $G^\circ $ is the connected component of the identity and where $G^{\mathrm {\acute {e}t}}$ is an étale group scheme over k. The reduction $G_{\mathrm {red}}\to G$ provides a canonical splitting of (2.1) and we obtain a canonical semi-direct product decomposition $G\cong G^\circ \rtimes G^{\mathrm {\acute {e}t}}$ . Since k is algebraically closed, $G^{\mathrm {\acute {e}t}}$ is the constant group scheme associated to the finite group $G(k)=G^{\mathrm {\acute {e}t}}(k)$ of k-rational points. Moreover, $G^\circ $ is an infinitesimal group scheme of length equal to some power of p. In particular, if $p=0$ or if the length of G is prime to p, then $G^\circ $ is trivial and then, G is étale.

If M is a finitely generated abelian group, then the group algebra $k[M]$ carries a Hopf algebra structure and the associated commutative group scheme is denoted $D(M):=\mathrm {Spec}\: k[M]$ . By definition, such group schemes are called diagonalisable. For example, we have $D({\mathbf {C}}_n)\cong \boldsymbol {\mu }_n$ , where ${\mathbf {C}}_n$ denotes the cyclic group of order n. We have that $\boldsymbol {\mu }_n$ is étale over k if and only if $p\nmid n$ .

A finite group scheme G over k is said to be linearly reductive if every k-linear and finite-dimensional representation of G is semi-simple. If $p=0$ , then all finite group schemes over k are étale and linearly reductive. If $p>0$ , then, by a theorem that is often attributed to Nagata [Reference NagataNa61, Theorem 2] (but see also [Reference Abramovich, Olsson and VistoliAOV08, Proposition 2.10], [Reference ChinCh92] and [Reference HashimotoHa15, Section 2]), a finite group scheme over k is linearly reductive if and only if it is an extension of a finite and étale group scheme, whose length is prime to p, by a diagonalisable group scheme.

2.2 Abstract groups and canonical lifts

Let G be a finite and linearly reductive group scheme over an algebraically closed field k of characteristic $p>0$ . Following [Reference Liedtke, Martin and MatsumotoLMM21, Section 2], we study the finite group

$$ \begin{align*}G_{\mathrm{abs}} \,:=\, \left(\underline{((G^\circ)^{D}(k))}_{{\mathbb{C}}}\right)^{D}({\mathbb{C}}) \,\rtimes\, G^{\mathrm{\acute{e}t}}(k). \end{align*} $$

Here, $-^D$ denotes $\underline {\mbox {Hom}}(-,{\mathbb {G}}_m)$ , the Cartier dual, of a commutative and finite group scheme. If G is étale, then $G_{\mathrm {abs}}=G(k)$ and if $G=\boldsymbol {\mu }_{p^n}$ , then $G_{\mathrm {abs}}={\mathbf {C}}_{p^n}$ . In any case, the order of $G_{\mathrm {abs}}$ is equal to the length of G.

Lemma 2.2. The functor

$$ \begin{align*}G \,\mapsto\, G_{\mathrm{abs}} \end{align*} $$

establishes an equivalence of categories between the category of finite and linearly reductive group schemes over k and the category of finite groups with a normal and abelian p-Sylow subgroup.

Next, we study lifts to characteristic zero: let $W(k)$ be the ring of Witt vectors of k, let K be its field of fractions and let $\overline {K}$ be an algebraic closure of K. By [Reference Liedtke, Martin and MatsumotoLMM21, Proposition 2.4], there exist lifts of G as finite and flat group scheme over $W(k)$ . More precisely, $G^\circ $ and $G^{\mathrm {\acute {e}t}}$ even lift uniquely to $W(k)$ , but their extension class usually does not, see also [Reference Liedtke, Martin and MatsumotoLMM21, Example 2.6]. However, there is a unique lift ${\mathcal G}_{\mathrm {can}}$ of G to $W(k)$ that is characterised by being a semi-direct product of the unique lift of $G^\circ $ with the unique lift of $G^{\mathrm {\acute {e}t}}$ .

Every other lift of G to some extension of $R\supseteq W(k)$ differs from ${\mathcal G}_{\mathrm {can},R}$ by a twist and thus, there is only one geometric lift of G to $\overline {K}$ up to isomorphism, namely $G_{\mathrm {can},\overline {K}}$ , see [Reference Liedtke, Martin and MatsumotoLMM21, Section 2.2].

Since $\overline {K}$ is algebraically closed and of characteristic zero, $G_{\mathrm {can},\overline {K}}$ is the constant group scheme associated to the finite group $G_{\mathrm {can}}(\overline {K})$ . In fact, we have $G^\circ \cong \prod _i \boldsymbol {\mu }_{p^{n_i}}$ for some $n_i$ ’s and if we set $N:=\max \{n_i\}_i$ , fix a primitive $p^N$ .th root of unity $\zeta _{p^N}$ and set $K_N:=K(\zeta _{p^N})$ , then $G_{\mathrm {can},K_N}$ is the constant group scheme associated to the finite group $G_{\mathrm {can}}(K_N)$ and we have $G_{\mathrm {can}}(K_N)=G_{\mathrm {can}}(\overline {K})$ .

Lemma 2.5. There exist isomorphisms of finite groups

$$ \begin{align*}G_{\mathrm{abs}} \,\cong\, G_{\mathrm{can}}(K_N) \,=\,G_{\mathrm{can}}(\overline{K}). \end{align*} $$

In particular, there exist isomorphisms

$$ \begin{align*}G_{\mathrm{can},K_N} \,\cong\, \left(\underline{G_{\mathrm{abs}}}\right)_{K_N} \quad \mbox{and} \quad G_{\mathrm{can},\overline{K}} \,\cong\, \left(\underline{G_{\mathrm{abs}}}\right)_{\overline{K}} \end{align*} $$

of finite group schemes over $K_N$ and $\overline {K}$ , respectively.

2.3 Representation theory

Let k be an algebraically closed field of characteristic $p>0$ , and let $W(k)$ , K and $\overline {K}$ be as in the previous section. The equivalence of Lemma 2.2 can be extended to representations and K-theory. If G is a finite group or a finite group scheme over k, we denote by $\mathrm {Rep}_k(G)$ the category of its k-linear and finite-dimensional representations. Let us first recall [Reference Liedtke, Martin and MatsumotoLMM21, Corollary 2.11].

Proposition 2.6. Let G be a finite and linearly reductive group scheme over k. Then, there exist canonical equivalences of categories

$$ \begin{align*}\mathrm{Rep}_{k}(G) \,\to\, \mathrm{Rep}_{K_N}(G_{\mathrm{can},K_N}) \,\to\, \mathrm{Rep}_{\overline{K}}(G_{\mathrm{can},\overline{K}}) \,\to\, \mathrm{Rep}_{\mathbb{C}}(G_{\mathrm{abs}}), \end{align*} $$

which are compatible with degrees, direct sums, tensor products, duals and simplicity.

Let $K_k(G)$ be the K-group associated to $\mathrm {Rep}_k(G)$ . In fact, $K_k(G)$ has a natural structure of a commutative ring with one, where the sum (resp. product) structure comes from direct sums (resp. tensor products) of representations. A straightforward application of Proposition 2.6 is the following.

Corollary 2.8. There exist isomorphisms of rings

$$ \begin{align*}K_k(G) \,\to\, K_{K_N}(G_{\mathrm{can},K_N}) \,\to\, K_{\overline{K}}(G_{\mathrm{can},\overline{K}}) \,\to\, K_{{\mathbb{C}}}(G_{\mathrm{abs}}). \end{align*} $$

3.1 McKay graph

Let G be a finite and linearly reductive group scheme over an algebraically closed field k of characteristic $p\geq 0$ . Let $\{\rho _i\}_i$ be the finite set of isomorphism classes of k-linear and simple representations of G. Following tradition, we assume that $\rho _0$ is the trivial representation. We fix a representation $\rho :G\to {\mathbf {GL}}(V)$ . If G is a subgroup scheme of ${\mathbf {SL}}_{n,k}$ or ${\mathbf {GL}}_{n,k}$ , then $\rho $ is usually the linear representation corresponding to the embedding of G into this linear algebraic group. By assumption, $\mathrm {Rep}_k(G)$ is semi-simple. Therefore, there exist unique integers $a_{ij}\in {\mathbb {Z}}_{\geq 0}$ for each i, such that we have isomorphisms of k-linear representations

$$ \begin{align*}\rho\otimes\rho_i \,\cong\, \bigoplus_{j}\, \rho_j^{\oplus a_{ij}}. \end{align*} $$

Associated to this data, we define the McKay graph, denoted $\Gamma (G,\{\rho _i\},\rho )$ :

1. - The vertices are the $\{\rho _{i}\}_i$ . (Some sources exclude the the trivial representation $\rho _0$ .)

2. - There are $a_{ij}$ edges from the vertex corresponding to $\rho _i$ to the vertex corresponding to $\rho _j$ .

We now establish a couple of elementary properties of this graph, which are well-known in the classical case and which immediately carry over to the linearly reductive situation. We leave the proof of the first lemma to the reader.

Lemma 3.1. We have

$$ \begin{align*}a_{ij} \,=\, \dim_k\, \mathrm{Hom}(\rho_i,\rho_j\otimes\rho). \end{align*} $$

In particular, if $\rho $ is self-dual, that is, $\rho \cong \rho ^\vee $ , then $a_{ij}=a_{ji}$ for all $i,j$ . In this case, we can consider $\Gamma (G,\{\rho _i\},\rho )$ as an undirected graph.

Let k be an algebraically closed field of characteristic $p>0$ , and let $W(k)$ , K and $\overline {K}$ be as in Section 2.2. There, we also discussed the canonical lift $G_{\mathrm {can}}$ of G over K and we saw that there exists an isomorphism of finite groups $G_{\mathrm {abs}}\cong G_{\mathrm {can}}(\overline {K})$ . Proposition 2.6 implies the following result.

Proposition 3.5. Let G be a finite and linearly reductive group scheme over k. Let $\{\rho _i\}_i$ be the set of isomorphism classes of simple representations of G. Let $\rho $ be a finite-dimensional representation of G.

1. 1. Proposition 2.6 yields sets of representations $\{\rho _{\mathrm {can},i}\}_i$ and $\{\rho _{\mathrm {abs},i}\}_i$ of $G_{\mathrm {can}}$ and $G_{\mathrm {abs}}$ , respectively, which are the sets of isomorphism classes of simple representations of $G_{\mathrm {can}}$ and $G_{\mathrm {abs}}$ , respectively.

2. 2. Proposition 2.6 yields representations $\rho _{\mathrm {can}}$ and $\rho _{\mathrm {abs}}$ of $G_{\mathrm {can}}$ and $G_{\mathrm {abs}}$ , respectively.

This data leads to a bijection of the three McKay graphs

$$ \begin{align*}\Gamma(G,\{\rho_i\}_i,\rho),\quad \Gamma(G_{\mathrm{can}},\{\rho_{\mathrm{can},i}\}_i,\rho_{\mathrm{can}}),\quad \Gamma(G_{\mathrm{abs}},\{\rho_{\mathrm{abs},i}\}_i,\rho_{\mathrm{abs}}). \end{align*} $$

3.2 McKay correspondence

We now run through the classical McKay correspondence [Reference McKayMc80] in our setting, where we follow [Reference KirillovKi16, Section 8.3]. Let k be an algebraically closed field of characteristic $p\geq 0$ and let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ .

1. 1. The K-group $K_k(G)$ carries a symmetric bilinear form

   $$ \begin{align*}([V],[W])_0 \,:=\, \dim_k\,\mathrm{Hom}_G(V,W). \end{align*} $$

2. 2. We consider the closed embedding $G\to {\mathbf {SL}}_{2,k}$ as a 2-dimensional representation $\rho :G\to {\mathbf {SL}}_{2,k}\to {\mathbf {GL}}_{2,k}$ and define an operator

   $$ \begin{align*}A\,:\,K_k(G)\,\to\, K_k(G),\qquad [V]\mapsto [V]\otimes\rho\,. \end{align*} $$

   Since $\rho $ is self-dual by Lemma 3.2, it follows that A is symmetric with respect to $(-,-)_0$ , see [Reference KirillovKi16, Lemma 8.12].

3. 3. Using $\rho $ , we define a symmetric bilinear form on $K_k(G)\otimes _{\mathbb {Z}}{\mathbb {R}}$ by

   $$ \begin{align*}([V],[W]) \,:=\, \left([V],\, (2-A)[W]\right)_0, \end{align*} $$
   which is positive semi-definite. The class $\delta \in K_k(G)$ of the regular representation of G generates the radical of $(-,-)$ .

4. 4. Let $\{\rho _i\}$ be the set of isomorphism classes of simple representations of G.

   1. (a) The McKay graph $\widehat {\Gamma }=\Gamma (G,\{\rho _i\}_i,\rho )$ is an affine Dynkin diagram of type $\widehat {A}_n$ , $\widehat {D}_n$ with $n\geq 4$ , $\widehat {E}_6$ , $\widehat {E}_7$ or $\widehat {E}_8$ .

   2. (b) After removing the vertex corresponding to the trivial representation $\rho _0$ from $\widehat {\Gamma }$ , we obtain a finite Dynkin diagram of type $A_n$ , $D_n$ with $n\geq 4$ , $E_6$ , $E_7$ or $E_8$ , respectively.

If G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then these statements are part of the classical McKay correspondence, see [Reference McKayMc80] or [Reference KirillovKi16, Section 8.3]. In our setting of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ , the above claims immediately follow from the classical McKay correspondence together with the lifting results Proposition 2.6 and Proposition 3.5. We then obtain the following analog of McKay’s theorem [Reference McKayMc80].

Theorem 3.6. Let k be an algebraically closed field of characteristic $p\geq 0$ . There exists a bijection between finite, non-trivial and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ up to conjugation and affine Dynkin diagrams of type

$$ \begin{align*}\begin{array}{lcl} \widehat{A}_n, \widehat{D}_n, \widehat{E}_6, \widehat{E}_7, \widehat{E}_8 &\quad \mbox{if} \quad & p=0 \quad \mbox{or} \quad p\geq7, \\ \widehat{A}_n, \widehat{D}_n, \widehat{E}_6, \widehat{E}_7 &\quad \mbox{if} \quad & p=5,\\ \widehat{A}_n, \widehat{D}_n &\quad \mbox{if} \quad & p=3 \mbox{ and} \\ \widehat{A}_n &\quad \mbox{if} \quad &p=2. \end{array} \end{align*} $$

4.1 Quotient singularities

Let k be an algebraically closed field of characteristic $p\geq 0$ . If G is a finite and linearly reductive group scheme over k, if V is a finite-dimensional k-vector space and if $\rho :G\to {\mathbf {GL}}(V)$ is a linear representation, then we define the $\lambda $ -invariant of $\rho $ as in [Reference Liedtke, Martin and MatsumotoLMM21, Definition 2.7] to be

$$ \begin{align*}\lambda(\rho) \,:=\, \max_{\{{\mathrm{id}}\}\neq\boldsymbol{\mu}_n\subseteq G} \dim V^{\boldsymbol{\mu}_n}, \end{align*} $$

where $V^{\boldsymbol {\mu }_n}$ denotes the subspace of $\boldsymbol {\mu }_n$ -invariants. As explained in [Reference Liedtke, Martin and MatsumotoLMM21, Remark 2.8], the representation $\rho $ is faithful if and only if $\lambda (\rho )\neq \dim V$ . Moreover, $\rho $ contains no pseudo-reflections if and only if $\lambda (\rho )\leq \dim V-2$ and in this case, the representation $\rho $ is said to be small.

Note that in dimension two the notions of small and very small coincide. We refer to [Reference Liedtke, Martin and MatsumotoLMM21, Section 3] for the classification of finite and linearly reductive group schemes that admit a very small representation. By [Reference Liedtke, Martin and MatsumotoLMM21, Proposition 6.5], $\lambda (\rho )$ is equal to the dimension of the non-free locus of the induced G-action on the spectrum of the formal power series ring $k[[V^\vee ]]=(k[V^\vee ])^\wedge $ . Following [Reference Liedtke, Martin and MatsumotoLMM21, Definition 6.4] and using the linearisation result [Reference Liedtke, Martin and MatsumotoLMM21, Proposition 6.3], we define:

Properties of these singularities have been studied in [Reference Liedtke, Martin and MatsumotoLMM21]: for their local étale fundamental groups, class groups and F-signatures, we refer to [Reference Liedtke, Martin and MatsumotoLMM21, Section 7].

By [Reference Liedtke, Martin and MatsumotoLMM21, Theorem 8.1], a linearly reductive quotient singularity determines the finite and linearly reductive group scheme G together with the very small representation $\rho :G\to {\mathbf {GL}}(V)$ uniquely up to isomorphism and conjugacy, respectively. It thus makes sense to refer to $\rho (G)\subset {\mathbf {GL}}(V)$ as the finite and linearly reductive subgroup scheme of ${\mathbf {GL}}(V)$ associated to the linearly reductive quotient singularity $x\in X=U/G$ . In particular, the classification of linearly reductive quotient singularities in dimension d is “the same” as the classification of very small, finite and linearly reductive subgroup schemes of ${\mathbf {GL}}_{d,k}$ up to conjugacy. We refer to [Reference Liedtke, Martin and MatsumotoLMM21, Section 3] for details and this classification.

4.2 Minimal resolution of singularities

If $x\in X$ is a normal surface singularity, then it admits a unique minimal resolution of singularities

$$ \begin{align*}\pi\,:\, Y\,\to\,X. \end{align*} $$

If $x\in X$ is moreover a rational singularity, then the exceptional locus of $\pi $ is a union of ${\mathbb {P}}^1$ ’s, whose dual intersection graph $\Gamma $ contains no cycles. The graph $\Gamma $ is called the type of $x\in X$ . If the type determines the singularity up to analytic isomorphism, then the singularity is said to be taut.

Over ${\mathbb {C}}$ , taut singularities have been classified by Laufer [Reference LauferLa73]. For example, two-dimensional finite quotient singularities over ${\mathbb {C}}$ are taut. Since two-dimensional linearly reductive quotient singularities over algebraically closed fields of positive characteristic are F-regular (see [Reference Liedtke, Martin and MatsumotoLMM21, Proposition 7.1]), they are taut by Tanaka’s theorem [Reference TanakaTa15].

4.3 The Ishii-Ito-Nakamura resolution

In [Reference Ito and NakamuraIN99], Ito and Nakamura showed that if G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then a minimal resolution of singularities of ${\mathbb {C}}^2/G$ is provided by the G-Hilbert scheme $G\text{-}\mathrm {Hilb}({\mathbb {C}}^2)$ . Ishii [Reference IshiiIs02] extended this to quotient singularities ${\mathbb {C}}^2/G$ for G a finite and very small subgroup of ${\mathbf {GL}}_2({\mathbb {C}})$ and Ishii and Nakamura [Reference Ishii and NakamuraIN19] extended this to quotient singularities $U/G$ for G a finite and very small subgroup of ${\mathbf {GL}}_2(k)$ of order prime to p.

Let k be an algebraically closed field of characteristic $p\geq 0$ . Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ . Set $U:={\mathbb {A}}^2_k$ or $\widehat {{\mathbb {A}}}_k^2$ and let $x\in X:=U/G$ be the associated linearly reductive quotient singularity. By [Reference BlumeBl11], there exists a G-Hilbert scheme $G\text{-}\mathrm {Hilb}(U)$ over k that parametrises zero-dimensional G-invariant subschemes $Z\subset U$ (so-called clusters), such that the G-representation on $H^0(Z,{\mathcal O}_Z)$ is the regular representation. Taking a cluster Z to its G-orbit (see, for example, [Reference BlumeBl11, Remark 3.3]) induces a morphism

$$ \begin{align*}\pi\,:\,Y\,:=\,G\text{-}\mathrm{Hilb}(U) \,\to\, U/G\,=\,X \end{align*} $$

over k.

4.4 Canonical lifts

Let k be an algebraically closed field of characteristic $p>0$ , let $W(k)$ be the ring of Witt vectors, let K be its field of fractions and let $\overline {K}$ be an algebraic closure. Let $x\in X=U/G$ be a d-dimensional linearly reductive singularity, where $U={\mathbb {A}}^d_k$ or $U=\widehat {{\mathbb {A}}}_k^d$ . Next, we set ${\mathcal U}:={\mathbb {A}}^d_{W(k)}$ or $\widehat {{\mathbb {A}}}^d_{W(k)}$ , respectively. In Section 2.2, we recalled the canonical lift ${\mathcal G}_{\mathrm {can}}$ of G over $W(k)$ . By [Reference Liedtke, Martin and MatsumotoLMM21, Proposition 2.9], there exists a unique lift of the G-action from U to ${\mathcal U}$ . From this, we obtain a flat family

(4.1) $$ \begin{align} \mathcal{X}_{\mathrm{can}}\,:=\,{\mathcal U}/{\mathcal G}_{\mathrm{can}} \,\to\, \mathrm{Spec}\: W(k)\,. \end{align} $$

of linearly reductive quotient singularities over $W(k)$ , whose special fibre over k is $x\in X=U/G$ .

By the Lefschetz principle, the geometric generic fibre of $\mathcal {X}_{\mathrm {can}}$ can be identified with a finite quotient singularity of the form ${\mathbb {C}}^d/G_{\mathrm {abs}}$ , where $G_{\mathrm {abs}}$ is the abstract group associated to G and the embedding of $G_{\mathrm {abs}}\to {\mathbf {GL}}_d({\mathbb {C}})$ corresponds to the embedding $G\to {\mathbf {GL}}_{d,k}$ provided by Proposition 2.6. The canonical lift is unique in the following sense:

4.5 Simultaneous resolution of singularities

If $\mathcal {X}\to S$ is a deformation of a rational double point singularity $x\in X$ , then it admits a simultaneous resolution of singularities, but usually only after some finite base-change $S'\to S$ , see [Reference ArtinAr74]. In the case where $\mathcal {X}\to S$ is a family of rational surface singularities, then such a finite $S'\to S$ exists if S maps to the so-called Artin component inside the versal deformation space of $x\in X$ . Moreover, due to the existence of flops, these simultaneous resolutions (if they exist) are not unique in general.

In the special case where $x\in X$ is a two-dimensional linearly reductive quotient singularity over some algebraically closed field k of characteristic $p>0$ and where $\mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ is the canonical lift of $x\in X$ , we will now show that there exists a simultaneous and minimal resolution of singularities over $W(k)$ and that it is unique. This simultaneous resolution can be most elegantly constructed using the Ishii-Ito-Nakamura resolution from Section 4.3 in families.

Let k be an algebraically closed field of characteristic $p>0$ . Let G be a finite and linearly reductive group scheme over k and let $\rho :G\to {\mathbf {GL}}_{2,k}$ be a very small representation. Let ${\mathcal G}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ be the canonical lift of G and let $\widetilde {\rho }:{\mathcal G}_{\mathrm {can}}\to {\mathbf {GL}}_{2,W(k)}$ be the lift of $\rho $ to $W(k)$ . Let ${\mathcal U}:={\mathbb {A}}^2_{W(k)}$ and let

$$ \begin{align*}\mathcal{X}_{\mathrm{can}} \,:=\, {\mathcal U}/{\mathcal G}_{\mathrm{can}} \,\to\, \mathrm{Spec}\: W(k) \end{align*} $$

be the canonical lift of $x\in X$ . By [Reference BlumeBl11], there exists a ${\mathcal G}_{\mathrm {can}}$ -Hilbert scheme

$$ \begin{align*}{\mathcal G}_{\mathrm{can}}\text{-}\mathrm{Hilb}({\mathcal U}) \,\to\,\mathrm{Spec}\: W(k) \end{align*} $$

that parametrises ${\mathcal G}_{\mathrm {can}}$ -invariant subschemes $Z\subset {\mathcal U}$ that are finite and flat over $W(k)$ (so-called ${\mathcal G}_{\mathrm {can}}-$ clusters) and such that the ${\mathcal G}_{\mathrm {can}}$ -representation on $H^0(Z,{\mathcal O}_Z)$ is the regular representation. Taking such a cluster Z to its ${\mathcal G}_{\mathrm {can}}$ -orbit (see, for example, [Reference BlumeBl11, Remark 3.3]) induces a morphism

(4.2) $$ \begin{align} \widetilde{\pi}\,:\, \mathcal{Y}\,:=\,{\mathcal G}_{\mathrm{can}}\text{-}\mathrm{Hilb}({\mathcal U}) \,\to\, {\mathcal U}/{\mathcal G}_{\mathrm{can}}\,=\,\mathcal{X}_{\mathrm{can}} \end{align} $$

over $W(k)$ .

Theorem 4.10. Keeping the assumptions and notations

$$ \begin{align*}\widetilde{\pi} \,:\, \mathcal{Y} \,\to\, \mathcal{X}\,\to\,\mathrm{Spec}\: W(k) \end{align*} $$

is a simultaneous minimal resolution of singularities of the canonical lift $\mathcal {X}\to \mathrm {Spec}\: W(k)$ of the linearly reductive quotient singularity $x\in X=U/G$ .

1. 1. The simultaneous resolution $\widetilde {\pi }$ is unique up to isomorphism.

2. 2. The exceptional locus of $\widetilde {\pi }$ is a union of ${\mathbb {P}}^1_{W(k)}$ ’s meeting transversally and we denote by $\Gamma $ its dual intersection graph.

3. 3. The special fibre and the generic fibre are linearly reductive quotient singularities of type $\Gamma $ and $\widetilde {\pi }$ identifies the components of the exceptional loci of $\widetilde {\pi }_k$ and $\widetilde {\pi }_K$ .

Proof. We recall that we defined $G_{\mathrm {can}}:={\mathcal G}_{\mathrm {can},K}$ , which will simplify the notation in the following. The generic and special fibre of (4.2) over K and k are isomorphic to

$$ \begin{align*}\begin{array}{cccccc} &G_{\mathrm{can}}\text{-}\mathrm{Hilb}({\mathcal U}_K) &\to& {\mathcal U}_K/G_{\mathrm{can}} &\to&\mathrm{Spec}\: K\\ \mbox{and} \quad& G\text{-}\mathrm{Hilb}(U) &\to& U/G &\to&\mathrm{Spec}\: k, \end{array} \end{align*} $$

respectively, by [Reference BlumeBl11, Remark 3.1]. Now, $G_{\mathrm {can},\overline {K}}\text{-}\mathrm {Hilb}({\mathcal U}_{\overline {K}}) \to {\mathcal U}_{\overline {K}}/G_{\mathrm {can},\overline {K}}$ and $G\text{-}\mathrm {Hilb}(U) \to U/G$ , are minimal resolutions of singularities by Theorem 4.5. Thus, $G_{\mathrm {can}}\text{-}\mathrm {Hilb}({\mathcal U}_K) \to {\mathcal U}_K/G_{\mathrm {can}}$ is a resolution of singularities, which is minimal since it is minimal over $\overline {K}$ . Thus, $\mathcal {Y}\to \mathcal {X}\to \mathrm {Spec}\: W(k)$ is a simultaneous minimal resolution of singularities.

The exceptional fibres $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ of $\widetilde {\pi }$ over K and k are unions of ${\mathbb {P}}^1$ ’s that intersect transversally. The types, that is dual resolution graphs, associated to $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ are the same (see also Remark 4.4) and we denote this graph by $\Gamma $ . In particular, the exceptional fibres of $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ have the same numbers of irreducible components. In particular, the specialisation maps of Néron-Severi lattices

$$ \begin{align*}\mathrm{NS}(\mathcal{Y}_K) \,\leftarrow\, \mathrm{NS}(\mathcal{Y}) \,\to\, \mathrm{NS}(\mathcal{Y}_k) \end{align*} $$

are isometries of lattices. These identify the components of the exceptional loci of $\widetilde {\pi }_K$ and $\widetilde {\pi }_k$ . Moreover, given such a component of the exceptional locus of $\widetilde {\pi }_k$ , it is isomorphic to ${\mathbb {P}}^1_k$ and it extends uniquely to a ${\mathbb {P}}^1_{W(k)}$ in the exceptional locus of $\widetilde {\pi }$ . This establishes claims (2) and (3).

It remains to prove claim (1): Let $\mathcal {Y}'\to \mathcal {X}\to \mathrm {Spec}\: W(k)$ be a simultaneous resolution of singularities that coincides with the minimal resolution on special and generic fibres, respectively. Let $\alpha :\mathcal {Y}^{\prime }_K\,\to \,\mathcal {Y}_K$ be an isomorphism over $\mathcal {X}_K$ and choose a relatively (to $\mathcal {X}$ ) ample invertible sheaf $\mathcal {L}$ on $\mathcal {Y}_K$ . Then, $\alpha ^*\mathcal {L}$ is relatively ample on $\mathcal {Y}_K$ . Since the types of the singularities of $\mathcal {X}_K$ and $\mathcal {X}_k$ are the same, the fibres of $\mathcal {Y}^{\prime }_K\to \mathcal {X}_K$ and $\mathcal {Y}^{\prime }_k\to \mathcal {Y}_k$ contain the same number of exceptional divisors. Thus, the specialisation map of Néron-Severi lattices $\mathrm {NS}(\mathcal {Y}^{\prime }_K)\to \mathrm {NS}(\mathcal {Y}^{\prime }_k)$ is an injective map between two lattices of the same rank and the same discriminant, whence an isometry. Similarly, the specialisation map of Néron-Severi lattices $\mathrm {NS}(\mathcal {Y}_K)\to \mathrm {NS}(\mathcal {Y}_k)$ is an isometry. In particular, we can identify the components of the exceptional loci of $\mathcal {Y}_K$ and $\mathcal {Y}_k$ . (If the types of $\mathcal {X}_k$ and $\mathcal {X}_K$ differ, then this specialisation map is usually only injective, there are usually more $(-2)$ -curves in the special fibre and this is also the place where the difficulties with flops begin.) In particular, $\mathcal {L}$ and $\alpha ^*\mathcal {L}$ extend to relative ample invertible sheaves on $\mathcal {Y}$ and $\mathcal {Y}'$ , respectively. By [Reference KovacsKo09, Theorem 5.14], the isomorphism $\alpha $ extends to an isomorphism $\mathcal {Y}'\to \mathcal {Y}$ over $\mathcal {X}$ and the claimed uniqueness follows.

4.6 Rational double point singularities and Klein singularities

We specialise Definition 4.2 to the following case.

In particular, a Klein singularity is a two-dimensional and linearly reductive quotient singularity, it is a rational surface singularity, and since $\det (\rho )=1$ , it is Gorenstein. Quite generally, rational and Gorenstein surface singularities are precisely the rational double point singularities [Reference ArtinAr66]. We refer to [Reference DurfeeDu79] or [Reference DemazureSSS] for more background on surface singularities and rational double point singularities.

If $x\in X$ is a rational double point, then its type $\Gamma $ is a simply-laced finite Dynkin graph. In characteristic zero, every rational double point singularity is taut. In positive characteristic, a Klein singularity is F-regular and thus, taut. In the case of rational double points, this also follows from Artin’s explicit classification [Reference ArtinAr77]. On the other hand, rational double point singularities that are not F-regular need not be taut. We have the following relation between rational double point singularities and Klein singularities in positive characteristic, see [Reference Liedtke, Martin and MatsumotoLMM21, Theorem 11.2].

A Klein singularity $x\in X=U/G$ is a linearly reductive quotient singularity and thus, determines G and $\rho :G\to {\mathbf {SL}}_{2,k}$ up to isomorphism and conjugacy, respectively. Thus, the classification of Klein singularities boils down to the classification of finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_{2,k}$ and we refer to Remark 4.4 for this classification.

5.1 The Ito-Nakamura resolution revisited

We first slightly extend the setup of Section 4.3. Let k be an algebraically closed field of characteristic $p>0$ and let $x\in X=U/G$ be a Klein singularity as in Definition 4.11, that is, we have $U={\mathbb {A}}^2_k$ and G a very small, finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ .

Let $\{\rho _i\}_{i\in I}$ be the set of isomorphism classes of simple representations of G. Given a finite-dimensional representation $\rho $ of G, there exist non-negative integers $\nu _i\in {\mathbb {Z}}_+$ , such that $\rho $ is isomorphic to $\bigoplus _i\rho _i^{\oplus \nu _i}$ and we combine these into a multi-index $\nu =(\{\nu _i\})\in {\mathbb {Z}}_{\geq 0}^I$ . We set $\dim (\nu ):=\sum _i\nu _i\dim \rho _i$ , which is the dimension of the representation associated to $\nu $ .

For any integer $n\geq 1$ , the G-action on U induces a G-action on the Hilbert scheme $\mathrm {Hilb}^n(U)$ , which parametrises zero-dimensional subschemes of length n of U. We consider the fixed point scheme

$$ \begin{align*}\mathrm{Hilb}^{n,G}(U) \,:=\, \left( \mathrm{Hilb}^n(U) \right)^G, \end{align*} $$

that is, the largest subscheme of $\mathrm {Hilb}^n(U)$ on which G acts trivially. It parametrises G-invariant and zero-dimensional subschemes of length n of U. Given $\nu =(\{\nu _i\})_i\in {\mathbb {Z}}_{\geq 0}^I$ with $\dim (\nu )=n$ , we define

$$ \begin{align*}H^\nu\,=\,\left\{ Z\,\in\,\mathrm{Hilb}^{n,G}(U)\,|\, H^0(Z,{\mathcal O}_Z)\,\cong\, \bigoplus \rho_i^{\oplus \nu_i}\mbox{ as }G\mbox{-representation} \right\}, \end{align*} $$

which defines a subscheme of $\mathrm {Hilb}^{n,G}(U)$ . Adapting [Reference KirillovKi16, Lemma 12.4], which follows [Reference Ito and NakamuraIN99, Section 9], to our situation, we have the following.

Proof. Let $x\in X^G$ . Passing to the completion of the local ring ${\mathcal O}_{X,x}$ , using that X is smooth over k and passing to coordinates such that the G-action is linear (this is always possible by [Reference SatrianoSa12, proof of Corollary 1.8]), we may assume that

$$ \begin{align*}\widehat{{\mathcal O}}_{X,x} \,\cong\, k[[u_1,...,u_d]] \end{align*} $$

and that the G-action is linear. In this description it is easy to see that the G-invariant subscheme of $\mathrm {Spec}\:\widehat {{\mathcal O}}_{X,x}$ is smooth (see also the proof of [Reference Ito and NakamuraIN99, Lemma 9.1]), which implies that $X^G$ is smooth near x.

An important special case is the regular representation of G, where we have $\nu _i=\dim \rho _i$ for all i and in this case, we will write $\delta $ for the corresponding multi-index. The dimension of $\delta $ is equal to the length of G. In this case, we have

$$ \begin{align*}\pi\,:\, H^\delta \,=\, G\text{-}\mathrm{Hilb}(U) \,\to\, U/G \,=\,X, \end{align*} $$

which we already studied in Section 4.3. There, we saw that $\pi $ is a minimal resolution of singularities of the Klein singularity $x\in X=U/G$ .

5.2 Hecke correspondences

We keep the assumptions and notations of the previous section. We let n be the length of G and let $\delta $ be the multi-index corresponding to the regular representation. For $i\in I$ , we set $\alpha _i:=(0,...,0,1,0...)$ with the non-zero entry in the i.th position. We define

$$ \begin{align*}B_i \,:=\, \left\{ J_1 \,\in\, H^{\delta-\alpha_i},\quad J_2\,\in\, H^\delta\quad |\quad J_2\subseteq J_1 \right\} \,\subseteq\, H^{\delta-\alpha_i}\times H^\delta, \end{align*} $$

which is a Hecke correspondence, see [Reference NakajimaNa96], [Reference NakajimaNa01, Section 6.1] or [Reference KirillovKi16, Section 12.4]. We let

$$ \begin{align*}E_i \,:=\, \mathrm{pr}_2(B_i)\,\subseteq\, H^\delta \,=\, G\text{-}\mathrm{Hilb}(U) \end{align*} $$

be the image under the projection $\mathrm {pr}_2$ onto the second factor.

In characteristic zero, the following result is due to Nakajima [Reference NakajimaNa96] and independently to Ito and Nakamura [Reference Ito and NakamuraIN99], see also [Reference KirillovKi16, Section 12.4].

Here is a second proof of this theorem that uses our lifting results:

5.3 Local McKay correspondence

We now extend work of Ishii and Nakamura [Reference IshiiIs02, Reference Ishii and NakamuraIN19] to our linearly reductive setting. Let G be a very small, finite and linearly reductive subgroup scheme of ${\mathbf {GL}}_{2,k}$ and let $x\in X=U/G$ be the associated two-dimensional linearly reductive quotient singularity. Until the end of this section, we do not require it to be a Klein singularity. Note that in dimension two, linearly reductive quotient singularities are the same as F-regular singularities, see Remark 4.6.

Let $\{\rho _i\}_i$ be the set of simple representations of G. Let $\rho _0$ be the trivial representation and we choose our numbering of the $\rho _i$ ’s to be the one of [Reference Ishii and NakamuraIN19, Theorem 3.6]. Let $\pi :Y\to X$ be its minimal resolution of singularities and let $\{E_i\}$ be the irreducible components of the exceptional divisor $\textrm {Exc}(\pi )$ of $\pi $ . Then, there is exists a connection between the $\{\rho _i\}$ and the exceptional divisors of $\pi $ as follows - this is yet another version of the McKay correspondence.

Theorem 5.4. Keeping assumptions and notations, let ${\mathfrak m}\subset {\mathcal O}_U$ be the maximal ideal corresponding to the origin, let $y\in Y$ be a closed point, let $Z_y$ be the G-invariant cluster of U corresponding to y and let ${\mathcal I}_{Z_y}\subset {\mathcal O}_U$ be its ideal sheaf. Then, the G-representation on ${\mathcal I}_{Z_y}/{\mathfrak m}{\mathcal I}_{Z_y}$ is given by

$$ \begin{align*}\left\{ \begin{array}{ll} \rho_i\,\oplus\,\rho_0 &\quad \mbox{if }y\in E_i\,\backslash\,\bigcup_{j\neq i}E_j\\ \rho_i\,\oplus\,\rho_j\,\oplus\,\rho_0 & \quad \mbox{if }y\in E_i\cap E_j \quad \mbox{with} \quad i\neq j. \end{array} \right. \end{align*} $$

5.4 Reflexive sheaves on the minimal resolution

We end this section by shortly digressing on work of Wunram [Reference WunramWu88], Ishii and Nakamura [Reference Ishii and NakamuraIN19], which generalises work of Artin and Verdier [Reference Artin and VerdierAV85]. We keep the assumptions and notations from Section 5.3. Let $F\subset Y$ be the fundamental divisor of $\pi $ , see [Reference ArtinAr66].

Theorem 5.6. Keeping assumptions and notations, there exists a bijection between

1. 1. the set of irreducible components $\{E_i\}$ of $\mathrm {Exc}(\pi )$ and

2. 2. the set of non-trivial indecomposable full ${\mathcal O}_Y$ -modules $\{{\mathcal M}_i\}$ , special in the sense that $H^1(Y,{\mathcal M}_i^{\vee })=0$

This correspondence ${\mathcal M}_i\mapsto E_i$ is defined by

$$ \begin{align*}c_1({\mathcal M}_i)\cdot E_j \,=\, \delta_{i,j}\,. \end{align*} $$

The rank of ${\mathcal M}_i$ is equal to $c_1({\mathcal M}_i)\cdot F$ , the multiplicity of $E_i$ in F.

Here, a full ${\mathcal O}_Y$ -module is as defined in [Reference Ishii and NakamuraIN19, Definition 2.4]. By [Reference Ishii and NakamuraIN19, Corollary 2.5], the assignment ${\mathcal M}\mapsto \pi _*{\mathcal M}$ sets up a bijection between the set of (indecomposable) full ${\mathcal O}_Y$ -modules with the set of (indecomposable) reflexive ${\mathcal O}_X$ -modules. In particular, the above theorem yields a bijection between the set of irreducible components of $\mathrm {Exc}(\pi )$ and the set of non-trivial, indecomposable and reflexive ${\mathcal O}_X$ -modules.

6.1 Conjugacy classes

Given a finite group $G_{\mathrm {abs}}$ , a representation $\rho :G_{\mathrm {abs}}\to {\mathbf {GL}}_n({\mathbb {C}})$ and an element $g\in G_{\mathrm {abs}}$ , we have the trace $\mathrm {tr}(\rho (g))\in {\mathbb {C}}$ . This pairing $(\rho ,g)\mapsto \mathrm {tr}(\rho (g))$ induces a non-degenerate pairing between isomorphism classes of simple representations of $G_{\mathrm {abs}}$ and conjugacy classes. Thus, conjugacy classes can be thought of as being “dual” to semi-simple representations, which can be used to give an unusual definition of conjugacy classes. This definition generalises to finite and linearly reductive group schemes. More precisely, we make the following definition, which is inspired by a result of Serre [Reference SerreSe77, Section 11.4] and which we discuss in detail in Appendix B.2.

We discuss several approaches to conjugacy classes for finite group schemes in Appendix B - there, we hope to convince the reader that Definition 6.1 is the best for the purposes of this article. For example, by Proposition B.2 it is compatible with canonical lifts and there is a natural bijection with the set of conjugacy classes of the abstract group $G_{\mathrm {abs}}$ associated to G. Concerning the choice of field ${\mathbb {C}}$ in Definition 6.1, we refer to Remarks B.3.

6.2 An explicit bijection

For a finite and linearly reductive subgroup scheme G of ${\mathbf {SL}}_{2,k}$ , we have an associated embedding of the finite group $G_{\mathrm {abs}}$ into ${\mathbf {SL}}_2({\mathbb {C}})$ . Let $\{\rho _i\}_i$ be the set of isomorphism classes of semi-simple representations of G and let $\{\rho _{\mathrm {abs},i}\}_i$ be the corresponding set for $G_{\mathrm {abs}}$ obtained by Proposition 2.6. The associated McKay graphs $\Gamma (G,\rho ,\{\rho _i\}_i)$ and $\Gamma (G_{\mathrm {abs}},\rho _{\mathrm {abs}}, \{\rho _{\mathrm {abs},i}\}_i)$ coincide by Proposition 3.5 and we denote both by $\widehat {\Gamma }$ . Next, the group $G_{\mathrm {abs}}$ admits a presentation of the form

$$ \begin{align*}G_{\mathrm{abs}} \,=\, \left\langle A,B,C \,|\, A^r=B^s=C^t=ABC \right\rangle \end{align*} $$

for suitable non-negative integers $r,s,t$ . The non-trivial conjugacy classes of $G_{\mathrm {abs}}$ can be uniquely represented by the following elements

$$ \begin{align*}ABC,\quad A^i \mbox{ with }1\leq i\leq r-1, \quad B^i \mbox{ with }1\leq i\leq s-1, \quad C^i \mbox{ with }1\leq i\leq t-1. \end{align*} $$

This allows to give an explicit bijection between the conjugacy classes of $G_{\mathrm {abs}}$ and the vertices of $\widehat {\Gamma }$ . We refer to [Reference KirillovKi16, Section 2] for details. Using Proposition B.2, we obtain an explicit bijection between the conjugacy classes of G and the vertices of $\widehat {\Gamma }$ .

6.3 The Ito-Reid correspondence

If G is a finite subgroup of ${\mathbf {SL}}_2({\mathbb {C}})$ , then Ito and Reid [Reference Ito and ReidIR96] (see also [Reference ReidRe02]) constructed a canonical bijection. Let us run through [Reference ReidRe02, Section 2] and show that this can be carried over to our linearly reductive setting: let k be an algebraically closed field of characteristic $p>0$ and let G be a finite and linearly reductive subgroup scheme of ${\mathbf {SL}}_{2,k}$ .

1. 1. Associated to G, we have the abstract group $G_{\mathrm {abs}}$ . By Proposition 2.6, an embedding $G\to {\mathbf {SL}}_{n,k}$ yields an embedding $G_{\mathrm {abs}}\to {\mathbf {SL}}_n({\mathbb {C}})$ . By Proposition B.2, we can identify conjugacy classes of G (in the sense of Definition 6.1) and conjugacy classes of $G_{\mathrm {abs}}$ , which allows us to define the age of a conjugacy class of G via the corresponding notion for $G_{\mathrm {abs}}$ as, for example, in [Reference ReidRe02, Section 2]. A conjugacy class of age 1 is called junior and if $n=2$ , then all conjugacy classes are junior.

1. 2. Let $x\in X:=U/G$ be the associated Klein singularity. We have the canonical lift $\mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ and the simultaneous resolution of singularities $\widetilde {\pi }:\mathcal {Y}\to \mathcal {X}_{\mathrm {can}}\to \mathrm {Spec}\: W(k)$ by Theorem 4.10. Passing to geometric generic fibres and using the Lefschetz principle, we obtain the minimal resolution of singularities of ${\mathbb {C}}^2/G_{\mathrm {abs}}$ . The special fibre of $\widetilde {\pi }$ , the geometric generic fibre of $\widetilde {\pi }$ and the minimal resolution of ${\mathbb {C}}^2/G_{\mathrm {abs}}$ are crepant in the sense of [Reference ReidRe87] and we can identify the exceptional divisors of these three resolutions with each other, see Theorem 4.10. This way, we obtain an Ito-Reid correspondence between junior conjugacy classes of G and crepant divisors of the resolution [Reference ReidRe02, Theorem 2.1].