数理解析研究所講究録1033 巻 1998 年 163-171

163

CYCLIC QUOTIENT SURFACECONSTRUCTING THE ARTIN COMPONENT:$\mathrm{S}\mathrm{l}\mathrm{N}\mathrm{G}\mathrm{U}\mathrm{L}\mathrm{A}\mathrm{R}\mathrm{l}\mathrm{T}\mathrm{I}\mathrm{E}\mathrm{s}$$\mathrm{V}\mathrm{l}\mathrm{A}$THE Mc$\mathrm{K}\mathrm{A}\mathrm{Y}^{-}\mathrm{Q}\mathrm{u}\mathrm{l}\mathrm{V}\mathrm{E}\mathrm{R}$Oswald Riemenschneider (Hamburg)
In memoriam Nobuo Sasakura
As a participant ofin Kyoto and Tokyo in July 1997 where I gave lectures on deformation theoryof rational surface singularities and the result in the present notes I was ceItainlly loohng forward to see Nobuo@asakura again whom I met in Japan five years ago as my host and a little later as my guest in Hamburg. It
was a great shock for me to learn just before my departure to Japan that there will be no discussions with him
anymore.
$\mathrm{t}\mathrm{h}\mathrm{e}\cdot \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{S}\mathrm{h}_{0}\mathrm{p}\mathrm{s}$There exists an extensive literature on the deformation theory of KLElNian singularities, . .
quotient surface singularities of embedding dimension 3 alias rational double points. E. .
P. KRONHEIMER [K] gave an invariant-theoretic construction of the (monodromy trivializing
finite covering of the) versal deformation and the simultaneous resolution of this family. A
quiver-theoretic approach has been proposed by him and was carried out by others (CASSENS[C]; see also SLODOWY [S] and the literature cited there).
$\mathrm{i}$$\mathrm{e}$$\mathrm{g}$As one can easily check in simple examples this construction has no straigthforward
ization to quotient singularitiesa (small) finite subgroup of GL(not in, . . the case of KLElNian singularities) and its associatedSLor AUSLANDER
REITEN-quiver. The purpose of the. present note is to restate the quiver construction for the-singularities in such-a manner that it can be generalized to yield (up to a smooth factor)
the monodromy covering of the ARTIN-component for all cyclic quotients. It is not clear at the
moment (besides for some special cases) how to produce with this method the othernents (not existing in the-cases) or even the whole versal deformation (which also possesses
a monodromy covering;
RIEMENSCHNEIDER [R2] and BROHME- RIEMENSCHNEIDER [BR] .

$\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}-$$X=\mathbb{C}^{2}[\Gamma,$$(2, \mathbb{C})$$(2, \mathbb{C})$$\Gamma$$\mathrm{i}$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{Y}^{-}$$\mathrm{e}$-$\mathrm{A}_{n}$$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{c}\succ$$\mathrm{A}_{n}$$\dot{\mathrm{s}\mathrm{e}}\mathrm{e}$$)$We should point out that our construction uses in a crucial manner the specific combinatorialproperties of cyclic quotients and, hence, gives no hint how to find a conceptual way which
be followed in the general case of quotient surface singuIarities. However, it should bepossible to provide our construction with an invariant-theoretic interpretation since the spcial representations used below have such a description by the work of WUNRAM [W] and theauthor (unpublished) which may lead to a general understanding of the deformation theory of
quotient surface singularities compIetely in terms of the representations of the group F. Noticethat . . the vector space of infinitesimal deformations of quotient surface singularities canbe described in such a way by work of PINKHAM [P]. There should also exist in the cyclic casea direct connection to the tbric structure of these singularities.
$\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{I}\mathrm{d}$$\mathrm{e}$$\mathrm{g}$1. The$\mathrm{A}_{n}$-case.

We let the group($*\rangle$$\mathbb{Z}_{n+1}=\mathbb{Z}/(n+1)\mathbb{Z},$$nrightarrow\zeta_{n+1}u,$$vrightarrow\zeta_{n+1}^{-1}v$,$n\geq 1$, act on$\zeta_{n+1}$$\mathbb{C}[u, v]$a primitiveby$(n+1)\mathrm{s}\mathrm{t}$root of unity,

164

with (generating) invariants$x_{0}=u^{n+1},$ $x_{1}=uv,$ $x_{2}=v^{n+1}$

and (generating) equation$x_{0}x_{2}=(uv)^{n+1}=x_{1}^{n+1}$The$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{v}$-quiver is of the form
$M_{\cap}$

where the $n+1$ vertices are the$\mathrm{o}\mathrm{n}\propto \mathrm{c}\mathrm{h}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$$\lambda/I_{j}:urightarrow\zeta_{n+1}^{\check{J}}u$

and tensoring$M_{j}$,

representations$u\in \mathbb{C}$,$j=0_{\}}\ldots,$ $n$with the natural representation on$M_{\mathrm{j}}\otimes N=M_{\mathrm{j}-1}\oplus M_{j}+1$$N=\mathbb{C}^{2}$,,

as in $(*)$ yields$j\in \mathbb{Z}_{n+1}$

to $M_{j+1}$ and backwards. Interpreting
which explains the two arrows from, with a number
as a linear homomorphism we can identify it, after fixing a basis for each, and$\Lambda f_{j}arrow M_{j+1}$$M_{j}$$M_{j}$$u_{j}$$U=$
describes the corresponding endomorphism$U\in$

End $M$ ,Similarly, we have an endomorphism$M=M_{0}\oplus\cdots\oplus M_{n}$$V\in \mathrm{E}\mathrm{n}\mathrm{d}M$

with

$V=$

.

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The endomorphisms $U^{n+1}$ , $UB$ and$U^{n+1}=\mathfrak{B}\mathfrak{g}\ldots u_{n}E_{n}+1$,$UV=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}$Take the entries of these matrices,$u_{\mathrm{O}}\ldots u_{n}$as generators of a subalgebra of$\mathrm{v}\mathrm{i}\mathrm{z}$,have diagonal form:$B^{n+\mathrm{I}}$$(v_{0}u_{n}, v_{10}u, .

.., v_{n}u_{n-}1)$,$V^{n+1}=v\mathrm{o}\cdots v_{n}E_{n+}1$.

.

$v_{\mathrm{O}}u_{n},$$v_{1}u_{0}\ldots,$$vnu_{n-}1,$$\mathbb{C}[u_{0,\ldots,0\cdot\cdot n}u_{n}, v,..v]$$v_{\mathrm{O}}\ldots$un. This, then, is presented by$\mathbb{C}[x_{\mathrm{O}}, x^{(0)\ldots,\mathrm{t}^{n}1}x1’ 1’ x_{2}]/(x_{\mathrm{O}}x_{2}=x_{1}^{(0)}\cdot\ldots\cdot x^{(n)})1$. It is well-known that this$x_{2}=v_{0}\ldots v_{n},$withversal deformation ([R1],ofthecovering(monodromy).
space
of
the
the
total
describes
algebra[S] of the given singularity. Notice that the corresponding deformation can be written in theform$(j\rangle$$x_{\mathrm{O}}=u_{\mathrm{O}}\ldots u_{n},$$=v_{\mathrm{j}}u_{j-1},$$x_{1}$$j\in \mathbb{Z}_{n+1}$$\cdot$$)$$[U, V]=UV-VU=\dot{\mathrm{d}}$2. Statement of the result in theTake now $0<q<n$ and$\mathrm{A}_{n,q}$$\mathbb{C}^{2}/\mathbb{Z}_{n}$$\mathrm{A}_{n+1,n}-\mathrm{t}\mathrm{y}\mathrm{P}\mathrm{e}$.

$\langle\lambda_{\mathrm{O}}$, ...,$\lambda_{n}\rangle$,$\sum_{j=0}^{n}\lambda_{\mathrm{j}}=0$.

-case.

$\mathrm{g}\mathrm{c}\mathrm{d}(n, q)=1$. We have a natural action of$u\mapsto\zeta_{n}u$

We calliag,$vrightarrow\zeta_{n}^{q}v$

the quotient singularity of type$\mathrm{A}_{n,q}$$\mathbb{Z}_{n}$on$\mathbb{C}[u,$

$v1$

by

.

. Remark that$\mathrm{A}_{n}$-type is the same as$M_{n-1}$ , has now arrows: $M_{j}arrow M_{j+1}$ and-quiver, with verticesThe$M_{j}arrow
M_{j+q},$,endomorphisms,$n-1$
giving$j=0,$
all indices taken modulo:$M_{0},$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{Y}$$v_{j}$$u_{j}$$\ldots,$$n$$\ldots,$$U,$$V\in \mathrm{E}\mathrm{n}\mathrm{d}M$,$M=M_{0}\oplus\cdots\oplus\dot{M}_{n-1}$

It is well-known [R1} that the equations of an$(n, q)$ and its continued fraction expansion$\mathrm{A}_{n,q}$.

-singularity are determined by the pair$\frac{n}{n-q}=a_{1}-\propto 1a_{2}$ -. $..-1\propto a_{\Gamma}$,$a_{\rho}\geq 2$:

if we set

, , $j_{1}=1,$$i_{p+1}=a_{\rho}\iota_{\rho}’-i_{\rho}-1$;$j_{0}=0$$i_{\mathrm{O}}>i_{1}>\cdots>i_{r}=1>\dot{i}_{r+1}=0$,$j_{0}=0<j_{1}<\cdots<j_{r+1}=n$ ,$i_{0}=n,\dot{i}_{1}=n-q,$$j_{\rho+1}=a_{\beta}j_{\rho}-j\rho-1$,

we get $r-+2$ numbers

and the monomials
the equations$x_{\rho}=u^{i_{\rho}}v^{\overline{J}\rho},$$x_{0}x_{2}=x_{1}^{a_{1}},$$r+1a_{2}$ ’ generate the invariant algebra. They satisfy$\rho=0,$ $\ldots,$$x_{1}x_{3}=x_{2},$ $\ldots,$ $xr-1X\Gamma+1=x_{\mathrm{r}}^{a_{r}}$plus some other equations which166

can compactly be written in quasideterminantal form, . . as the 2-quasimatrix
2$\mathrm{i}$$\mathrm{e}$of the$\mathrm{x}2-quasimino\Gamma S$$\mathrm{x}(r+1)$

Recall moreover that the exceptional divisor in the minimal resolution of such a singularity is$\sigma=1,$, wherea string of smooth rational curves with selfintersection numbers$-b_{\sigma},$$s$$\ldots,$$\frac{n}{q}=b_{1}-\propto 1b_{2}$ -.

..

$-1\propto b_{S}$,$b_{\sigma}\geq 2$$s$.

As we mentioned at the beginning, our construction uses special representations. Recall thata-correspondence yields for any binary polyhedral groupSL
the famous
bijective mapping between the set of (nontrivial) irreducible complex representations ofand the set of irreducible components of the exceptional set in the minimal resolution of thethis is no longer true;
singularity. For the other finite (small) subgroups
in fact there are in these cases always more nontrivial irreducible representations than curves.
But by the work of WUNRAM [W] one can associate to each curve a well-defined nontrivial
representation. We call these (together with the trivial one) special representations. They forma (proper) subset of the set of vertices in the-quiver. For cyclic quotient singularities
this set can be easily described [W]: The special representations are precisely thewith$\sigma=0,$, where$\Gamma\in$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{Y}$$(2, \mathbb{C})$$\Gamma$$\mathbb{C}^{2}/\Gamma$$\Gamma\in \mathrm{G}\mathrm{L}(2, \mathbb{C})$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{Y}$$M_{\alpha}$$\alpha=t_{\sigma},$$\ldots,$$s$$t_{\mathrm{O}}:=n,$$t_{1}:=q,$ $\ldots,$ $t_{\sigma+1}=b_{\sigma}t_{\sigma}-t_{\sigma-1},$ $\ldots,$ $t_{s}=1$$(t_{s+1}=0)$, $1\leq\sigma\leq s$ , to the curve with label
Here, of course,
belongs forto have these numbers in their natural ordering, . . (modulo )$M_{\alpha}$$\alpha=t_{\sigma}$$\mathrm{e}$,$\sigma=0,$

one may find them by reversing the ordering of the. If one wants$n$$\mathrm{i}$$\overline{t_{\sigma}}:=t_{\delta+1-}\sigma$$\sigma$.

$b_{\sigma}$$\ldots,$$s$,: writing$\beta_{\sigma}=b_{s+1-}\sigma$, $\sigma=1,$$\ldots,$$s$,yields the continued fraction expansion$\beta_{1}-1\propto\beta_{2}$and the nonnegative integers$\overline{t_{\sigma}}$$\overline{t_{\mathrm{O}}}=0$–$-1 \propto\beta_{s}=\frac{n}{q\sim},$$qq\equiv\sim 1$

mod

$n$

,

are determined by,$\sim t_{1}=1$,$\overline{t}_{\sigma+1}=\beta_{\sigma\sigma\sigma-1}t-\sim\overline{t}$.

We are now able to state our Theorem. It says in particular that in order to understand thedeformation theory of cyclic quotient surface singularities one has to consider the underlying
quiver seriously as the-quiver, not just as an abstract one ! In particular the .yclic
symmetry of the abstract quiver has to be broken in the general case.
$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{Y}$$\mathrm{c}$167

Theorem For given $(n, q)$ , take in $\mathbb{C}[u_{0,\ldots,n-1}u, v_{0}, \ldots, v_{n-1}]$ the subalgebra generated
by those elements of the diagonal matrices,$U^{i_{\rho}}V^{J^{\vee}\rho}$$p=0,$$\ldots,$$r+1$ ,

which belong to special representations. Then this algebra is canonically isomorphic to thealgebra of the total space of the (monodromy covering) of the ARTIN component of thesingularity up to . smooth factor.

$\mathrm{A}_{n,q^{-}}$$a$Remark. Since in theone.

$\mathrm{A}_{n}$-case all representations are special this statement includes the former

3. More explicit statement of the result.
In order to prepare the grounds for proving the theorem we derive now an explicit formulation.
$r+1$ . These are$p=0,$
We first have to compute the entries of the matrices
diagonal matrices of type$U^{i_{\rho}j_{\rho}}V,$$\ldots,$

diag$(v\mathrm{O}^{\cdot}v\cdot.\cdot vq\cdot\cdot \mathrm{t}j\rho^{-}1$)$qu_{\langle-}j_{\rho}1$$\ldots)$

1$ )$\rho^{-}q+i_{\rho},$) $q+1u_{1j\rho^{-})+}1q2\ldots\cdot u\mathrm{t}g\prime,

where the next entries are obtained from the first one by permutating the indices cyclicallymodulo .
$n$, $\sigma=0,$
The theorem advices us to choose only the entries belonging to the indicessuppresstheweby-factorstheSince these monomiaLs are complete.ly determined
write for them$\sim t_{\sigma}$

$v$

.

)

$v_{\overline{\mathrm{e}}_{\sigma}}v_{\overline{f}_{\sigma}+qt}$$\ldots\cdot v\vee\sigma+\mathrm{t}j_{\rho}-1$

It now seems,$\sigma=0,$ $\ldots,$ $s$

.

and

.

sonable. to make the coordinate change(all indices taken modulo$n$,$v_{1}rightarrow v_{q}$,$v_{2}rightarrow v_{2q},$$\ldots$, i.e.

$v_{j}rightarrow v_{jq}$). -We have the following

If $k_{0}=0,$ $.k_{1}=1,$$\ldots,$$k_{\sigma+1}=b_{\sigma}k_{\sigma}-k_{\sigma}-1,$$\overline{t_{\mathrm{t}^{s}+1)-}}\sigma=qk_{\sigma}$. For$\rho=0,$ $\ldots,$ $r+1$$s$$\mathrm{r}\mathrm{e}.\mathrm{a}$$v_{0}rightarrow v_{0}$

Lemma

,

$q$

$\ldots,$$u’ \mathrm{s}$

mod

$n$

,

$\sigma=1,$ $\ldots,$ $s,$$\sigma=0,$ $\ldots,$ $s+1$$(k_{s+1}=n)$, then.

and $k_{0}=0$ , for $\sigma=s-1$ it follows that $t_{1}\sim=t_{1}=$, we have$q,$ $k_{1}=1$ . The rest follows by induction:$Pj_{OO}f$$\sigma=0$$\overline{t_{s+1}}=n$$qk_{\sigma+1}-\sim t_{\mathrm{t}^{s+}}1)-_{\mathrm{t}^{\sigma+}}1)=q(b_{\sigma}k_{\sigma}-k_{\sigma-1})-(\beta_{(s+1)\sigma}-t\sim(_{S}+1)-\sigma-\sim t_{(s+1)-_{\mathrm{t}^{\sigma}1)}}-)$$=b_{\sigma}(qk_{\sigma}-\sim t_{(s+1)-}\sigma’)-(qk_{\sigma-1}-\sim t_{\langle+1)-\mathrm{t}1)}-)S\sigma$$\equiv 0$

mod

$n$

.

Combining these results we get the following description of the algebra under discussion as the$v_{n-1}1$ generated by the elementsof the polynomial ring
subalgebra$\overline{A}_{\mathrm{e}\iota,q}$$\mathbb{C}[v_{0},$$\ldots,$$x_{\rho}^{(\sigma)}=v_{k_{\sigma}}v_{k_{\sigma}+1}\cdot\ldots\cdot vk\sigma+(j_{\rho}-1)$,$\rho=0,$ $\ldots,$ $r+1$,$\sigma=0,$ $\ldots,s$

.

168

and
Notice thattheorem is equivalent to the following claim:$x_{\gamma+1}^{\{\sigma)}=v_{0}v_{1}\cdot\ldots\cdot v_{n-1}$$x_{0}^{\langle\sigma)}=1$

for all$\sigma$. –Hence, the statement of theis, up to a smooth factor, the coordinate ring of the total space of the monThe algebra.
odromy covering of the ARTIN eomponent of the$\tilde{A}_{ti,q}$$S\dot{i}ngula\gamma ityX_{n,q}$

4. Some examples.
Before sketching a general proof we discuss some examples. We start with the simplest non-quiver is of the form-case, : the case $(n, q)=(3,1)$ . The$\mathrm{A}_{n}$$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{A}\mathrm{Y}$$\mathrm{v}\mathrm{i}\mathrm{z}$$h\ell_{\cap}$$\mathit{1}V\mathit{1}2$$\mathit{1}V\mathit{1}1$

We have to take into consideration the endomorphisms$U^{3},$$U^{22}V,$$UV,$$V^{3}$(there is just one component in the exceptionalandThe special representations are-singularity). Thus, we have to regard the invariants
set of the minimal resolution of the$M_{0}$$\mathrm{A}f_{1}$$\mathrm{A}_{3,1}$$u_{0}u1u2,$$v_{01}uu_{2},$$u_{01}vu_{2},$$v_{0}v_{1}u_{2},$$u0v1v2,$ $v_{0}v_{1}v_{2}$which satisfy the determinantal relations

and no others.

Remark. Considering all invariants together with the obvious 4-parameter famiIy given by$[U^{2}, V]=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda_{1}, \lambda_{2}, \lambda_{3})$with$\sum^{3}j\simeq 1x- J=\theta,$

formation of it$\sum_{j=\mathrm{g}^{\lambda}}^{6}J’=0$

yieitk,[ $U,$ $V^{2}1=$ diag$(\lambda_{4},$$\lambda_{5},$$\mathrm{t}\mathrm{b}\rho \mathrm{c}\mathrm{Q}\mathrm{r}\mathrm{r}\wp i\mathrm{S}\dot{\iota}t\iota \mathrm{g}\nu 1\beta\Gamma i\mathrm{t}\mathrm{y}_{\mathrm{O}}\mathrm{V}\mathrm{p}\mathrm{r}\mathrm{t}\iota\wp$$\lambda_{6}\rangle$

origin, but no flat de-$\backslash |$with endomorphisms
This example can immediately be generalized to the cone$U^{n-1}V,$ $Un-2V^{2},$yielding the correct invariants andand special modules
equations$\mathrm{A}_{n,1}$$U^{n},$$V^{n}$$\ldots,$$M_{0},$$M_{1}$

169

We discuss another somewhat more complicated example,
we have$\mathrm{v}\mathrm{i}\mathrm{z}$. the case $(n, q)=(46,27)$ . Here$\frac{n}{q}=2-\lrcorner 1\overline{4}-\underline{1}\overline{2}-\underline{1}\ulcorner 3^{-}-\lrcorner 1\overline{2}$,hence $s=5$ , and$\frac{n}{n-q}=\frac{46}{19}=3-\lrcorner 1\overline{2}-\underline{1}\ulcorner 4-\lrcorner 1\overline{3}$are given by 46, 19, 11, 3, 1, andandsuch that $e=r+2=6$ . The sequencesby 46, 27, 8, 5, 2, 1, . Therefore, we have to$0,1,3,5,17,46$ , resp.; the sequence):
consider the following monomials (where we delete, as mentioned above, the variables$0$$(j_{\rho})$$(\dot{i}_{\rho})$$0$$(t_{\sigma})$$u_{j}$$x_{0}^{(0)}=1$$(0)$$\langle 0)$$\langle 0)$$\langle \mathrm{O}\}$$=v_{0}$$x_{1}$(1)$x_{1}$$\langle$(3)$x_{1}$(4)$x_{1}$(5)$x_{1}$(1)$=v_{1}$$x_{2}$(2)$2)$$x_{1}$$x_{2}$$=v_{2}$$x_{2}$$=v_{01}vv_{2}$(1)$=v_{1}v_{2}V_{3}$(2)$=v_{2}v_{3}v_{4}$$x_{2}$(4)$=v_{12}$$x_{2}$$\langle$$=v_{29}$5)$x_{2}$$\langle$$x_{3}${2)$x_{4}$$=v_{2}\cdot\ldots\cdot v_{6}$(3)$=v_{7}v_{8^{V_{9}}}$$=v\iota 2^{U}13V14$$=v_{29}v_{30^{v}31}$$x_{4}$(4)(4}$=v_{12}\cdot\ldots\cdot v_{16}$$x_{3}$(5)$.x_{3}$$=v_{1}\cdot\ldots\cdot v_{17}$$=v_{2}\cdot\ldots\cdot v_{18}$$(3\rangle$$=v_{7}\cdot\ldots\cdot v_{11}$$x_{3}$$=v_{0}\cdot\ldots\cdot v_{16}$1)$x_{4}$$=v_{1}\cdot\ldots\cdot v_{5}$$x_{3}$(3)$=v_{7}$$x_{4}$$=v_{0}\cdot\ldots\cdot v_{4}$$x_{3}$$x_{4}$$=v_{7}$ ’ $\ldots\cdot v_{23}$$=v_{12}\cdot.,$(5}$=v_{29}\cdot\ldots\cdot v_{33}$$x_{4}$.

$\cdot v_{2\S}$$=v_{29}\cdot\ldots\cdot v_{45}$(O)$x_{5}$$=v_{0}v_{1\cdot\cdot 45}..\cdot v$

A computation by hand or via [GPS] yields the following equations in quasideterminantal form,
leaving the remaining variables untouched:$(x_{0}^{(\mathrm{o})}x^{(\mathrm{O})}1^{\cdot}$$x_{1}^{\langle \mathrm{I})}$$x_{2}^{(0)}x_{1}^{()}2$$x_{2}^{(2)}x_{3}^{(0_{\rangle}}$$x_{3}^{\mathrm{t}2)}x_{3}^{\mathrm{t}3)}$$x_{3}^{(4)}x_{4}^{(0)}$$x_{4}^{(4)}$$x_{4}^{(5}x_{5}(0)1)$

5. Sketch of proof.
is- up to a smooth factor- the $gene\dot{N}C$The main point is to show that the algebra
deformation of the given singularity in its canonical quasideterminantal format ([R1], [Ro]).
The last example gives the right idea. Recall [R1] that$\overline{A}_{n,q}$$e-3=r-1= \sum_{\sigma=1}^{s}(b_{\sigma}-2)$

and, dually,$s-1= \sum_{\ell=1}\mathrm{f}(a_{\ell}-2)$.

Define now$s_{p}= \sum_{\ell=1}\beta(a_{t}-2)+1$,$\rho=1,$ $\ldots,$ $r$

,

170

such that $s_{1}=a_{1}-1\leq s_{2}\leq\cdots\leq s_{r}=s$ , and select the following variables out of all$x_{\rho}^{(0)}$$(\sigma)$$x_{1}$$\{\sigma)$$x_{p}$,$p=0,$.

,$\sigma=1\ldots s_{1}’$

,.

$.’$

$\ldots,$$x_{\rho}^{(\sigma 1}$:$r+1$ ,.

$\sigma=S\mathit{8}\rho-1,$$\ldots,\rho$ ’$\rho=2,$ $\ldots,$ $r$.

variables. Obviously, it is sufficient to show
Notice that they form a system ofof variables satisfies the following (generating) quasideterminantalthat the complete set
relations:$e+ \sum_{\beta}^{r}=1(a_{\rho}-1)$$x_{\rho}^{(\sigma)}$which is just the generic deformation of the quasideterminantal format in question.
In order to do this, one can use induction with respect to the number , the case $r=1$ being
trivial. Therefore, we start with a fixed number $r\geq 1$ and. a continued fraction expansion
$r$

where
$\frac{n}{n-q}:=a_{1}-\infty 1a_{2}$ $–...-\infty 1a_{r}$.

Similarly, we provide the indices referring to the pair $(n’, q’)$ by an accent; in particular, we$\rho=1,$$r+1,$$=n$ and$p=1,$$r,$have $r^{f}=r+1,$$=a_{r+1}$ and, it follows that$j_{r+1}’,=n’$ . Moreover, invoking the,,in$a_{p}’=a_{\rho},$$j_{p}’=j_{\rho},$$a_{r}’,$$\ldots,$$\mathrm{d}\mathrm{o}\mathrm{t}"-\mathrm{d}i\mathrm{a}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{m}$$\ldots,$$j_{r}’,$$\mathrm{t}^{\mathrm{R}1]}\ulcorner$$s’=s+(a_{r+1}-2)$ ,$b_{\sigma}’=b_{\sigma}$, $\sigma=1,$$s-1$ ,$\ldots,$$b_{S}’=b_{s}+1$,$\sigma=s+1,$ $\ldots,$ $s’$$b_{\sigma}’=2,$.

These identities imply$k_{\sigma}’=k_{\sigma},$$\sigma=.1,$$\ldots,$$s$,$k_{s+\sigma}’=k_{s}+\sigma n,$$\sigma=1,$$\ldots,$$s’-s$ .

and the following
Lemma$k_{s}=n-j_{f}$Proof. We have by induction$a_{\gamma+}1j_{\Gamma}+1-j_{r}-j_{+},1=n-j_{r}’’,$$k_{s}’,$.

.

$=(a_{r+1}-2)n+k_{s}=(a_{r+1}-1)n-j_{r}=a_{r+1}n-jr-n=$Moreover, we get by the same reasoning$x_{r}^{(s_{r}\}}=v_{n-j_{r}}\cdot\ldots\cdot v_{n-1}$,$x_{r+1}^{(s_{r}})$.

.

$\ldots\cdot X_{\Gamma+}(s_{r+\iota,1}\}=v_{n-j\mathrm{r}n’-1}$$\ldots\cdot v$

and hence the relation$X_{r}\langle s_{\mathrm{r}})x_{r+}(0)2=X_{f+}\cdot x..x_{r}(0)1r+1+(Sr)..\cdot(s_{r}1+\iota 1$

,

171

etc. This shows that the variables chosen satisfy the quasidetermisince (O)nantal equations as above.
$x_{r+1}=v\mathrm{o}\cdot\ldots\cdot v_{n-1}$

The only problem left is to show that there are no other relations and moreover only trivial
ones with respect to the remaining variables. We leave this as an exercise to the reader.

References[BR]

S. Brohme, O. Riemenschneider. In preparation.

[C]

H. Cassens. Lineare Modifikationen algebraischer Quotienten, Darstellungen des
McKay-K\"ochers und Kleinsche Singularit\"aten. Dissertation. Fachbereich Mathematik der Universit\"at Hamburg: Hamburg 1994.

[GPS]

G.-M. Greuel, G. Pfister, H. Schoenemann. Singular. A Computer Algebra System
for Commutative Algebra, Algebraic Geometry and Singularity Theory. Version 1.0.0.
University of Kaiserslautern 1986-97.

[K]

P. Kronheimer. The construction of ALE spaces as hyper-K\"ahler quotients. J. Diff.
Geom. 29, 665-697 (1989).

[P]

H.C. Pinkham. Deformations of quotient surface singularities. Symposia in
Mathematics, Vol. 10. Providence: AMS 1976.

[R1]

O. Riemenschneider. Deformationen von Quotientensingularit\"aten (nach zyklischen
Gruppen). Math. Ann. 209, 211-248 (1974).

[R2]

O. Riemenschneider. Special surface singularities. A survey on the geometry and combinatorics of their deformations. In: Analytic varieties and singularities. Sunrikaiseki
Kenkyuosho Kokyuroku (RIMS Symposium Report) Nr. 807, pp. 93-118 (1992).

[Ro]

A. R\"ohr. Formate rationaler. Dissertation, Fachbereich Mathematik der Universit\"at Hamburg, Hamburg 1992.

[S]

P. Slodowy. Algebraic Groups and Resolutions of Kleinian singularities. Hamburger
Beitr\"age zur Mathematik (aus dem Mathematischen Seminar), Heft 45, Hamburg
University 1996. (Also available as RIMS preprint).

[W]

J. Wunram. Reflexive Modules on Quotient
583-598 (1988).

$\mathrm{P}\mathrm{u}\dot{\mathrm{r}}\mathrm{e}$$\mathrm{F}\mathrm{l}\ddot{\mathrm{a}}\mathbb{C}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\ddot{\mathrm{a}}\mathrm{t}\mathrm{e}\mathrm{n}$$\mathrm{S}_{\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}}\mathrm{C}\mathrm{e}$Singularities. Math. Ann. 279,

Address of author: Prof. Dr. Oswald Riemenschneider, Math. Seminar der Universit\"at Ham: riemenschneider@math.uni-hamburg.deburg, BundeSstra&\acute 5, D-20146 HAMBURG.
$\mathrm{e}-\mathrm{m}\mathrm{a}\mathrm{i},1$