Reconstruction algebras of type D (II) Michael Wemyss

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1 Hokkaido Mathematical Journal Vol. 42 (203) p Reconstruction algebras of type D (II) Michael Wemyss (Received July 7, 20; Revised August 9, 202) Abstract. This is the third in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin quivers. This paper is the companion to [W2] and deals with dihedral groups G = D n,q which have rank two special M modules. We show that such reconstruction algebras are described by combining a preprojective algebra of type D e with some reconstruction algebra of type A. Key words: Noncommutative resolutions, M modules, surface singularities.. Introduction Everybody loves the preprojective algebra of an extended Dynkin diagram. Algebraically they provide us with a rich source of non-trivial but manageable examples on which to test the latest theory, whilst geometrically they encode lots of information about the singularities 2 /G where G is a finite subgroup of SL(2, ). However for quotients by finite subgroups of GL(2, ) which are not inside SL(2, ), preprojective algebras are not quite so useful. Fortunately in this case the role of the preprojective algebra is played by a new algebra (the reconstruction algebra) which is built in a similar but slightly modified way. This new algebra is by definition the endomorphism ring of the special M modules in the sense of Wunram [W88], and the recent main result of [W] states that for any finite group G GL(2, ) the quiver of the reconstruction algebra determines and is determined by the dual graph of the minimal resolution of 2 /G, labelled with self-intersection numbers. Note that when the group G is inside SL(2, ) the reconstruction algebra is precisely the preprojective algebra of the associated extended Dynkin diagram. However it should also be noted that in general the above statement is known only on the level of quivers; to give an algebra we need to add in the 200 Mathematics Subject lassification : Primary 34, 4E6, 6S38.

2 294 M. Wemyss extra information of the relations and it is this problem that is addressed in this paper. Although technical in nature, knowledge of the relations of the reconstruction algebra unveils relationships between quotients arising from vastly different group structures and consequently reveals explicit geometric structure that has until now remained unnoticed. In this paper, which should be viewed as a companion to [W2], we deal with dihedral groups D n,q inside GL(2, ) for which n < 2q. The companion deals with the case n > 2q. The main difference between the two is that here rank two special M modules enter the picture, whereas in the n > 2q case all special M modules have rank one. Using knowledge of the special M modules from both [IW08] and [W2] we are able to associate to the dual graph of the minimal commutative resolution of 2 /D n,q a quiver with relations and then prove that it is isomorphic to the endomorphism ring of the special M modules. Like most things involving explicit presentations of algebras the proof is rather technical, however we are able to side-step many issues by using both knowledge of the AR theory and knowledge of the geometry; in particular the non-explicit methods from [W] are used extensively. We note that the reconstruction algebras in this paper naturally split into two subfamilies but it should perhaps be emphasized that the two families are fairly similar, particularly when compared to [W2]. The ν = N family (for notation see Section 2) should be viewed as being geometrically very similar to the dihedral ordinary double points, whereas the examples in loc. cit. should be viewed as being very toric. The remaining 0 < ν < N family sits geometrically somewhere between these two extremes. This paper is organized as follows in Section 2 we define the groups D n,q and recap some combinatorics and results from [W2] and [IW08] that are needed for this paper. Section 3 deals with the case when less than the maximal number of rank two special M modules occurs, whereas Section 4 deals with the case involving the maximal number of rank two special M modules. In Section 5 we illustrate the correspondence between the algebra and the geometry by comparing two examples, making precise some of the above remarks which are of a more philosophical nature. Throughout when working with quivers we shall write ab to mean a followed by b. We work over the ground field but any algebraically closed field of characteristic zero will suffice. The author was supported by a JSPS Postdoctoral Fellowship held at

3 Reconstruction algebras of type D (II) 295 Nagoya University, and the author would like to thank the JSPS for funding this work and Nagoya University for kind hospitality. Thanks also to Osamu Iyama and Alvaro Nolla de elis for useful conversations. 2. Dihedral groups and special M modules In this paper, as in [W2], we follow the notation of Riemenschneider [R77]. Definition 2. For < q < n with (n, q) = define the group D n,q to be D n,q = { ψ2q, τ, ϕ 2(n q) if n q mod 2 ψ 2q, τϕ 4(n q) if n q 0 mod 2 with the matrices ψ k = ( εk 0 ) 0 ε k τ = ( 0 ) ε4 ε 4 0 ( ) εk 0 ϕ k = 0 ε k where ε t is a primitive t th root of unity. The order of the group D n,q is 4(n q)q. By [B68, 2.] the dual graph of the minimal resolution of 2 /D n,q is 2 2 α α N αn where the α s come from the Jung Hirzebruch continued fraction expansion n q = [α,..., α N ]. By definition ν records the number of rank two special M modules. Denoting the dual continued fraction expansion by n n q = [a 2,..., a e ], we now briefly recap some combinatorics. To the above data we define the

4 296 M. Wemyss series c, d, r, i, l, b, and Γ as follows:. The c series, defined as c 2 =, c 3 = 0, c 4 = and c j = a j c j c j 2 for all 5 j e. 2. The d series, defined as d 2 = 0, d 3 =, d 4 = a 3 and d j = a j d j d j 2 for all 5 j e. 3. The r series, defined as r 2 = a 2 (n q) q, r 3 = r 2 (n q), r 4 = (a 3 + )r 3 r 2 and r j = a j r j r j 2 for all 5 j e. 4. the i-series, defined as i 0 = n, i = q and i t = α t i t i t 2 for all 2 t N The l-series, defined as l j = 2 + j p= (α p 2) for j N. 6. The b-series. Define b 0 :=, b ln := N, and further for all t l N 2 (if such t exists), define b t to be the smallest integer b t N such that t b t p= (α p 2). 7. k, defined as k = + k t=ν+ c l t for all ν + k N Γ k defined as Γ k = k t=ν+ d l t for all ν + k N +. Note that in 7 and 8 we use the convention that for k = ν + the sum is empty and so equals zero. Now after setting w := xy and w 2 := (x q + y q )(x q + ( ) a 2 y q ) w 3 := (x q y q )(x q + ( ) a 2 y q ) v 2 := x 2q + ( ) a 2 y 2q v 3 := x 2q + ( ) a 2 y 2q we have the following result. Theorem 2.2 ([R77, Satz 2]) The polynomials w 2(n q) and w r t vc t 2 vd t 3 for 2 t e generate the ring [x, y] D n,q. Alternatively we may take the polynomials w 2(n q) and w r t wc t 2 wd t 3 for 2 t e as a generating set. The next two lemmas are the main combinatorial results that will be used later to determine the relations on the reconstruction algebra. Lemma 2.3 ([W2, 2.3, 2.4]) onsider D n,q. Then for all 2 t e 2, r t+ = r t+2 + i bt. Further r lt = i t i t+ for all ν + t N. Lemma 2.4 ([W2, 2.6]) onsider D n,q. Then for all 2 t e 2, c t+2 = c t+ + bt and d t+2 = d t+ + Γ bt.

5 Reconstruction algebras of type D (II) 297 In particular if ν = N then for all 3 t e c t = t 3 and d t =. Define the rank one M modules W +, W and for each t i ν+ + ν(n q) = q the rank two indecomposable M module V t by the following positions in the AR quiver of [[x, y]] D n,q R V V 2 V 3 V q 3 V q 2 V q W + W i.e. all the V t lie on the diagonal leaving the vertex R, whilst W + and W are the two rank one indecomposable M modules at the bottom of the diagonal. Furthermore for every t n q define the rank one indecomposable M module W t by the following position in the AR quiver: R W 2 W 4 W n q R W W3 Wn q i.e. they all live on the non-zero zigzag leaving R. Note that when n q is even the picture changes slightly since the position of R on the right is twisted (for details we refer the reader to [IW08, Section 6]). It is worth pointing out that V t = ([x, y] ρ t ) D n,q where D n,q acts on both sides of

6 298 M. Wemyss the tensor (on the polynomial ring it acts by inverse transpose) where ρ t is the two-dimensional irreducible representation ψ 2q τ ( ϕ 2(n q) n q odd ( ) ε t 2q 0 0 ε t 2q ) ε t 4 0 ( 0 ε t 4 ε t 2(n q) 0 0 ε t 2(n q) ) ψ 2q ( τϕ 4(n q) n q even ( ) ε t 2q 0 0 ε t 2q 0 ε t 4 εt 4(n q) ε t 4 εt 4(n q) 0 ) For a description of the W s in terms of representations, see [W2, Section 3]. The next two results summarize the classification of the special M modules for the groups D n,q. Below, we often implicitly view M [x, y] D n,q - modules as [x, y] D n,q -summands of the polynomial ring [H78]. Theorem 2.5 ([W2, 3.2]) For any D n,q, the following M modules are special and further they are two-generated as [x, y] D n,q -modules by the following elements: W + x q + y q (xy) n q (x q y q ) x q + y q (xy) n q (x q y q ) W x q y q (xy) n q (x q + y q ) x q y q (xy) n q (x q + y q ) W iν+ (xy) i ν+ w 2 = w ν+ 2 w Γ ν+ 3 (xy) i ν+ v 2 = v ν+ 2 v Γ ν+ 3 W iν+2 (xy) i ν+2 w ν+2 2 w Γ ν+2 3 (xy) i ν+2 v ν W in (xy) i N w N 2 w Γ N 3 (xy) i N v N 2 v Γ N 3 2 v Γ ν+2 where the left column is one such choice of generators, and the right hand column is another choice. Further there are no other non-free rank one special M modules. Theorem 2.6 ([IW08, 6.2]) onsider the group D n,q with n < 2q, then for all 0 s ν V iν+ +s(n q) is special. Furthermore these are all the rank two indecomposable special M modules.

7 Reconstruction algebras of type D (II) 299 For notational convenience denote U s := V iν+ +(ν s)(n q) for all s ν, thus in this new notation the rank two indecomposable special M modules are simply U,..., U ν. Since for dihedral groups D n,q the maximal co-efficient of any curve in the fundamental cycle Z f is two, by combining the previous two results we have full knowledge of all special M modules. We assign to them their corresponding vertices in the dual graph of the minimal resolution in Lemma 3.0 and Lemma 4.7 below. Before proceeding to study the reconstruction algebra, we must first slightly re-interpret the preprojective algebra since this is crucial to later arguments. Throughout the remainder of this section consider only the groups G = D n+,n SL(2, ) and denote their natural representation by V with basis e and e 2. It is very well known that the endomorphism ring of the special M modules in this case (summed without multiplicity) is isomorphic to the preprojective algebra of the extended Dynkin diagram of type D, and furthermore that this is morita equivalent to [x, y]#g. The relations for the preprojective algebra are obtained by choosing a G- equivariant basis for the maps in the McKay quiver; for details see [B98] or [BSW]. Since here for quivers we use the convention that ab is a followed by b, the number of arrows from ρ to σ in [x, y]#g is (see [BSW, p506] for details and notations) dim e ρ (V G)e σ = dim Hom G (Ge ρ, V (Ge σ )) so we interpret arrows from vertex ρ to vertex σ in the preprojective algebra as G-maps from ρ to V σ. Since G acts on [x, y] in the same way as it acts on V, denote the basis of V dual to e and e 2 by x and y. The point is that the G-maps can be written in matrix form in terms of x s and y s, and composition of arrows is given by matrix multiplication. To see this, for example denote the trivial representation of D n+,n by ρ 0 and denote its basis by v 0. Then we have a G-map ρ 0 V V v 0 x e + y e 2 which we write as the matrix ( x y ). Similarly we have a G-map

8 300 M. Wemyss V V ρ 0 e ( y) v 0 e 2 x v 0 which we write as the matrix ( -y x ). The composition of these two maps is (see [BSW, p506]) v 0 (x ( y) + y x) v 0 which is zero in the skew group ring since x y = y x. Thus we view the composition as being matrix multiplication ( x y ) ( -y x ), subject to the relation that x and y commute. Using the basis of the representations given above Theorem 2.5, we can choose the following G-basis for all the maps in the McKay quiver x-y ( y x ) ( -y x ) xy -y 0 0 x... x 0 0 y -y 0 0 x x 0 0 y ` yx ( x -y ) ( x y ) ` -yx and thus we can view the preprojective algebra as being the above algebra subject to the rules of matrix multiplication and the relations induced by the fact that x and y commute. From this picture we may also read off the generators of the M modules corresponding to the vertices. Remark 2.7 Note that the preprojective relations in the above picture at the two branching points are slightly different than the standard ones found elsewhere in the literature. Remark 2.8 The above picture is good for another reason, namely it gives us an explicit description of the Azumaya locus. More precisely for any orbit containing the point 0 (λ x, λ y ) 2, denote by M (λx,λ y ) the representation of the preprojective algebra obtained by placing at the outside vertices and 2 at all other vertices, and further replacing every x

9 Reconstruction algebras of type D (II) 30 in the above picture by λ x and every y by λ y. Then M (λx,λ y ) is the unique simple above the point corresponding to the orbit containing (λ x, λ y ). 3. The reconstruction algebra for 0 < ν < N In this section we define the reconstruction algebra D n,q as a quiver with relations in the case 0 < ν < N and prove that it is isomorphic to the endomorphism ring of the special M modules. onsider, for N N with N 2 and for positive integers α ν+ 3, α t 2 for ν + 2 t N, the labelled Dynkin diagram of type D: αν+... αn Note that the hypothesis 0 < ν < N translates into the condition α = = α ν = 2 with α ν+ 3. We call the left hand vertex the + vertex, the top vertex the vertex and the remaining vertices,..., N reading from left to right. To this picture we add an extended vertex called and double-up as follows: where we have attached to the ν th and N th vertices. Now if N i= (α i 2) 2 add extra arrows in the following way: if α ν+ > 3 then add an extra α ν+ 3 arrows from the (ν + ) st vertex to. If α i > 2 with i ν + 2 then add an extra α i 2 arrows from the i th vertex to.

10 302 M. Wemyss Label the new arrows (if they exist) by k 2,..., k P (α i 2) reading left to right. Name this new quiver Q. Example 3. onsider D 3,8 then 3 8 Q is = [2, 3, 3], ν = and so the quiver g +, f,+ f, g, f 0, a 32 g,2 f c 23 2, g k 2,0 a 03 c30 Example 3.2 onsider D 73,56 then = [2, 2, 2, 5, 2, 3], ν = 3 and so the quiver Q is f, g, g +, f,+ f,2 g 2, f 2,3 g 3,2 f 0,3 g 3,4 a 54 a 65 f c 45 4,3 c 56 g 3,0 k 2 k 4 k 3 c60 a 06 To define the reconstruction algebra we need to first introduce some notation. Throughout, we denote k := g ν,0 and k + P N i= (α i 2) := c N0. Definition 3.3 For all r + N i= (α i 2) define B r to be the number of the vertex associated to the tail of the arrow k r. Notice for all 2 r + N i= (α i 2) it is true that B r = b r where b r is the b-series of n q defined in Section 2, however B b since b = ν + whilst B = ν by the special definition of k. Now define u ν = and further for ν + i N denote

11 Reconstruction algebras of type D (II) 303 { u i := max j : 2 j + { v i := min j : 2 j + N } (α i 2) with b j = i i= N } (α i 2) with b j = i if such things exist (i.e. vertex i has an extra arrow leaving it). Also define W ν+ := ν and for every ν + 2 i N define where W i = i= { ν if Si is empty the maximal number in S i else S i = {vertex j : j < i and j has an extra arrow leaving it}. Thus W i is defined for ν + i N. The idea behind it is that W i records the closest vertex to the left of vertex i which has a k leaving it; since we have defined k := g ν,0 this is always possible to find. Now define, for all ν + i N, V i := u Wi. Thus V i records the number of the largest k to the left of the vertex i, where since k := g ν,0 and u ν = it always exists. Denote also F,ν := { e if ν = f,2... f ν,ν if ν > G ν, := { e if ν = g ν,ν... g 2, if ν > where e is the trivial path at vertex, and define A 0t := a 0N... a t+t for every ν + t N. Definition 3.4 For n q = [α,..., α N ] with 0 < ν < N define D n,q, the reconstruction algebra of type D, to be the path algebra of the quiver Q defined above subject to the relations f 0,ν g ν,ν+ = 2A 0ν+ g +, f,+ = 0 f 0, g,0 = 0 If ν = g, f, = 0 f 2, g,2 = 0 g,0 f 0, g,2 f 2, = f,+ g +, f, g,

12 304 M. Wemyss g +, f,+ = 0 f 0,ν g ν,0 = 0 g, f, = 0 f ν+,ν g ν,ν+ = 0 If ν > f,+ g +, f, g, = 2f,2 g 2, g t,t f t,t = f t,t+ g t+,t for all 2 t ν g ν,0 f 0,ν g ν,ν+ f ν+,ν = 2g ν,ν f ν,ν together with the relations defined algorithmically as Step ν + If α ν+ = 3 c ν+ν+2 a ν+2ν+ = p ν+ν+ If α ν+ > 3 k 2 A 0ν+ = p ν+ν+, A 0ν+ k 2 = 0ν k k t 0ν+ = k t+ A 0ν+, 0ν+ k t = A 0ν+ k t+ 2 t < u ν+ k uν+ 0ν+ = c ν+ν+2 a ν+2ν+ Step i If α i = 2 c ii+ a i+i = a ii c i i. If α i > 2 k vi A 0i = a ii c i i, A 0i k vi = 0BVi k Vi k t 0i = k t+ A 0i, 0i k t = A 0i k t+ v i t < u i k ui 0i = c ii+ a i+i. Step N If α N = 2 c N0 a 0N = a NN c N N, a 0N c N0 = 0BVN k VN If α N > 2 k vn a 0N = a NN c N N, a 0N k vn = 0BVN k VN k t 0N = k t+ a 0N, 0N k t = a 0N k t+ v N t < u N The only notation that remains to be defined are the p s and the s, and it is these which change according to the presentation, namely p ν+ν+ = 0ν = { fν+,ν G ν, f,+ g +, F,ν g ν,ν+ 2 f ν+,νg ν, (f,+ g +, + f, g, )F,ν g ν,ν+ { f0,ν G ν, f, g, F,ν 2 f 0,νG ν, (f,+ g +, + f, g, )F,ν

13 Reconstruction algebras of type D (II) 305 0ν+ = { f0,ν G ν, f,+ g +, F,ν g ν,ν+ 2 f 0,νG ν, (f,+ g +, + f, g, )F,ν g ν,ν+ where for the moduli presentation we take the top choices and for the symmetric presentation we take the bottom choices. We also define, for all ν + 2 t N, 0t = 0ν+ c ν+ν+2... c t t. Remark 3.5 As in [W2] we should explain why we give two presentations. The symmetric case is pleasing since it treats the two ( 2) horns equally, so that the algebra produced is independent of how we view the dual graph. On the other hand the relations in the moduli presentation are more binomial and so this makes the explicit geometry easier to write down in Section 5. We show in Theorem 3. that the two presentations yield isomorphic algebras, but for the moment denote D n,q for the moduli presentation and D n,q for the symmetric presentation. Remark 3.6 The algorithmic relations are just the relations from the reconstruction algebra of Type A. Thus the reconstruction algebra here should be viewed as a preprojective algebra of type D with a reconstruction algebra of type A stuck onto the side. Remark 3.7 The ν = and ν > split in the above definition is not natural in the sense that in both cases they are just the preprojective relations. We only split them for notational ease. Furthermore in the remaining relations, in both presentations note the existence of a relation that is not a cycle, and also note the fact that at any vertex in the reconstruction algebra corresponding to a ( 2) curve there is only one relation, a preprojective relation. Example 3.8 For the group D 3,8 the symmetric presentation of the reconstruction algebra is the quiver in Example 3. subject to the relations f 0, g,2 = 2a 03 a 32 g +, f,+ = 0 g, f, = 0 f 0, g,0 = 0 f 2, g,2 = 0

14 306 M. Wemyss g,0 f 0, g,2 f 2, = f,+ g +, f, g, 2 f 2,(f,+ g +, + f, g, )g,2 = c 23 a 32 a 32 c 23 = k 2 (a 03 ) 2 f 0,(f,+ g +, + f, g, )g,0 = (a 03 )k 2 k 2 ( 2 f 0,(f,+ g +, + f, g, )g,2 c 23 ) = c 30 (a 03 ) ( 2 f 0,(f,+ g +, + f, g, )g,2 c 23 )k 2 = (a 03 )c 30 Example 3.9 For the group D 73,56 the moduli presentation of the reconstruction algebra is the quiver in Example 3.2 subject to the relations f 0,3 g 3,4 = 2a 06 a 65 a 54 g +, f,+ = 0 g, f, = 0 f 0,3 g 3,0 = 0 f 4,3 g 3,4 = 0 f,+ g +, f, g, = 2f,2 g 2, g 2, f,2 = f 2,3 g 3,2 g 3,0 f 0,3 g 3,4 f 4,3 = 2g 3,2 f 2,3 f 4,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 = k 2 (a 06 a 65 a 54 ) f 0,3 g 3,2 g 2, f, g, f,2 f 2,3 g 3,0 = (a 06 a 65 a 54 )k 2 k 2 (f 0,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 ) = k 3 (a 06 a 65 a 54 ) (f 0,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 )k 2 = (a 06 a 65 a 54 )k 3 k 3 (f 0,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 ) = c 45 a 54 a 54 c 45 = c 56 a 65 a 65 c 56 = k 4 (a 06 ) (f 0,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 )k 3 = (a 06 )k 4 k 4 (f 0,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 c 45 c 56 ) = c 60 (a 06 ) (f 0,3 g 3,2 g 2, f,+ g +, f,2 f 2,3 g 3,4 c 45 c 56 )k 4 = (a 06 )c 60

15 Reconstruction algebras of type D (II) 307 Before we show that the endomorphism ring of the special M modules is isomorphic to this quiver with relations we must first justify the positions of the special M modules relative to the dual graph. To do this, recall that given the exceptional curves {E i } i I in the minimal resolution, the fundamental cycle Z f = i I r ie i (with each r i ) is defined to be the unique smallest element such that Z f E i 0 for all i I. Furthermore, we denote the canonical cycle by Z K. It the rational cycle defined by the condition Z K E i = K ex E i for all i I. By adjunction this means that Z K E i = Ei for all i I. Lemma 3.0 onsider D n,q with 0 < ν < N. Then the special M modules correspond to the dual graph of the minimal resolution in the following way αν+ αn W W + U U ν W iν+ W in Proof. Denote by E t the curve corresponding to vertex t for t {+,,,..., N}. The assumption 0 < ν < N translates into the condition α ν+ 3 which then forces Z f = E + + E + ν p= 2E p + N t=ν+ E t. Hence Z f E t (Z K Z f ) E t 0 0 t {+,,,..., ν } t = ν α ν+ 3 t = ν + α t 2 0 ν + < t < N α N t = N and so by [W, or 3., Lemma 3.2] the quiver of the endomorphism ring of the special M modules coincides with Q as defined above. But it is clear that the following composition of maps in the AR quiver do not factor through other special M modules

16 308 M. Wemyss R W Uν U W + W where in the above picture the positions of R and W are fixed, however since U ν = V iν+ in general the length of the distance between R and U ν depends on i ν+. Also the distance between the other special M modules changes depending on n q, however none of these changes in distance effect the irreducibility of the highlighted arrows. The above irreducible maps thus fix the positions of W, W +, W and the U s, and the rest follow by noting the factorization of (xy) i ν+ as The following is the main result of this section. Theorem 3. For a group D n,q with parameter 0 < ν < N, denote R = [x, y] D n,q and let T n,q = R W + W ν s= U s N t=ν+ W it be the sum of the special M modules. Then D n,q = EndR (T n,q ) = D n,q. Proof. We prove both statements at the same time by making different choices for the generators of the special M modules. As stated in the proof

17 Reconstruction algebras of type D (II) 309 of Lemma 3.0, the quiver of the endomorphism ring of the special M modules is precisely Q defined above. We first find representatives for the known number of arrows. Set m := n q. By choosing a G-equivariant basis as explained in Section 2, the following are always homomorphisms between the special M modules The proof that these maps are actually homomorphisms (equivalently G- equivariant) splits into two cases depending on the parity of n q since by definition both G and the representations depend on n q; we suppress the details. By inspection they do not factor through any of the other special M modules and so can be chosen as representatives. Note that with these choices f ν+,ν G ν, f,+ g +, F,ν g ν,ν+ = (xy) r 3 w c 3 2 wd 3 3 = (xy)r 3 w 3 p ν+ν+ = 2 f ν+,νg ν, (f,+ g +, + f, g, )F,ν g ν,ν+ = (xy) r 3 v c 3 2 vd 3 3 = (xy)r 3 v 3 f 0,ν G ν, f, g, F,ν g ν,0 = (xy) r 3 w c 3 2 wd 3 3 = (xy)r 3 w 3 0ν k = 2 f 0,νG ν, (f,+ g +, + f, g, )F,ν g ν,0 = (xy) r 3 v c 3 2 vd 3 3 = (xy)r 3 v 3 { f0,ν G ν, f,+ g +, F,ν g ν,ν+ = w 2 0ν+ = 2 f 0,νG ν, (f,+ g +, + f, g, )F,ν g ν,ν+ = v 2 We must now choose representatives of the remaining arrows, and it is different choices of these that give the two different presentations. As in the proof of the above lemma it is clear that the anti-clockwise arrows are a N0 = xy and a tt = (xy) i t i t. Also, since we must reach the generators of the

18 30 M. Wemyss special M modules as paths from (i.e. the vertex corresponding to R), by Theorem 2.5 we can choose c t t = w c l t 2 w d l t 3 for all ν + 2 t N. Alternatively, also by Theorem 2.5, we can choose c t t = v c l t 2 v d l t 3 for all ν + 2 t N (these choices will give the symmetric presentation). As explained in [W2, 4.] we may choose c N0 = w c l N 2 w d l N 3 = w c e 2 w d e 3 (respectively v c e 2 v d e 3 ). Thus we have proved that we may take as representatives of the arrows with c t t = w c l t 2 w d l t 3 for all ν + 2 t N and c N0 = w c e 2 w d e 3. Now if N i= (α i 2) 2 then for the moduli presentation we take k t = (xy) r t+2 w c t+ 2 w d t+ 3 for all 2 t (α i 2) as representatives. To see why we can do this, for example if α ν+ 4 consider (xy) r 4 w c 3 2 wd 3 3 : W iν+ R. First, it does not factor through maps we have already chosen, since if it does then since f ν+,ν has a (xy) r 3 factor and c ν+ν+2 = w c l ν+ 2 w d l ν+ 3 we may write (xy) r 4 w c 3 2 wd 3 3 = (xy) r 3 F + w c l ν+ 2 w d l ν+ 3 h for some polynomials F and h. But by looking at xy powers we know (xy) r 4 divides h and so after cancelling factors we may write w c 3 2 wd 3 3 = (xy)r 3 r 4 F + w c l ν+ 2 w d l ν+ 3 h. After cancelling w c 3 2 wd 3 3 (which F must be divisible by) = (xy) r 3 r 4 F + w c l ν+ c 3 2 w d l ν+ d 3 3 h which is impossible since the assumption α ν+ 4 means that the right hand side cannot have degree zero terms. Second, (xy) r 4 w c 3 2 wd 3 3 does not factor as a map W iν+ R followed by a non-scalar invariant since this would contradict the embedding dimension. The proof for the remainder of the arrows is similar; in fact the proof is identical to [W2, 4.]. The symmetric presentation is the same, but we replace everywhere w 2 by v 2 and w 3 by v 3.

19 Reconstruction algebras of type D (II) 3 The relations for the reconstruction algebra are now induced by the relations of matrix multiplication such that x and y commute. Using Lemma 2.3, Lemma 2.4 and the polynomial expressions for p ν+ν+, 0ν k, and 0ν+ above, it is straightforward to verify that the relations in Definition 3.4 (which we denote by S ) are satisfied by these choices of polynomials, since the algorithmic relations are just the pattern in the cycles in D (respectively D 2 ) from [W2, 3.6]. The remainder of the proof is now identical to [W2, 4.]: we first work in the completed case (so we can use [W, or 3., Lemma 3.2] and [BIRS, 3.4]) and we prove that the completion of the endomorphism ring of the special M modules is given as the completion of Q (denoted ˆQ) modulo the closure of the ideal S (denoted S ). The non-completed version of the theorem then follows by simply taking the associated graded ring of both sides of the isomorphism. Denote the kernel of the surjection ˆQ End [[x,y]] G(T n,q ) := Λ by I, denote the radical of ˆQ by J and further for t {, +,,,..., N} denote by S t the simple corresponding to the vertex t of Q. In Lemma 3.2 below we show that the elements of S are linearly independent in I/(IJ + JI). Thus we may extend S to a basis S of I/(IJ + JI) and so by [BIRS, 3.4(a)] I = S. But by combining [BIRS, 3.4(b)], [W, or 3., Lemma 3.2] and inspecting our set S we see that #(e a ˆQe b ) S = #(e a ˆQe b ) S for all a, b {, +,,,..., N}, proving that the number of elements in S and S are the same. Hence S = S and so I = S, as required. The proof of the following lemma is very similar to that of [W2, 4.] and so we only prove the parts in which the subtle differences arise. Lemma 3.2 With notation from the above proof, the members of S are linearly independent in I/(IJ + JI). Proof. Following [W2, 4.] we say that a word w in the path algebra ˆQ satisfies condition (A) if ( i ) It does not contain some proper subword which is a cycle. ( ii ) It does not contain some proper subword which is a path from to ν +.

20 32 M. Wemyss We know from the intersection theory and [W, or 3., Lemma 3.2] that the ideal I is generated by one relation from to ν +, whereas all other generators are cycles. onsequently if a word w satisfies (A) then w / IJ + JI. It is also clear that f 0,ν g ν,ν+ 2A 0ν+ / IJ + JI. All members of S are either cycles at some vertex or paths from to ν +, so to prove that the members of S are linearly independent in I/(IJ + JI) we just need to show that. the elements of S that are paths from to ν + are linearly independent in e (I/(IJ + JI))e ν+. 2. for all t {, +,,,..., N}, the elements of S that are cycles at t are linearly independent in e t (I/(IJ + JI))e t. The first condition is easy, since the only relation in S from to ν + is f 0,ν g ν,ν+ 2A 0ν+ and we have already noted that it does not belong to IJ + JI, thus it is non-zero and so linearly independent in e (I/(IJ + JI))e ν+. Note also that there are only two paths of minimal grade from to ν + (namely f 0,ν g ν,ν+ and A 0ν+ ) and by inspection of the polynomials they represent we do not have any other relation from to ν+ of this grade. For the second condition, we must check t case by case: ase t = + and t =. There is only one relation in S which is a cycle at that vertex and so to prove linearly independence requires only that the relation is non zero in I/(IJ + JI). But clearly g +, f,+ / IJ + JI and g, f, / IJ + JI since they both satisfy condition (A). ase t =,... ν. Again there is only one relation, and since each word satisfies condition (A) it does not belong to IJ + JI and so the relation is non-zero. For example if f,+ g +, f, g, 2f,2 g 2, IJ + JI then f,+ g +, = f, g, + 2f,2 g 2, + u for some u IJ + JI in the free algebra ˆQ which is impossible since f,+ g +, cannot appear in the right hand side. ase t = ν +. If α ν+ = 3 then there are only two relations in S from ν + to ν +. To prove linear independence suppose that λ (c ν+ν+2 a ν+2ν+ p ν+ν+ ) + λ 2 (f ν+,ν g ν,ν+ ) = 0 in e ν+ (I/(IJ + JI))e ν+. Then λ c ν+ν+2 a ν+2ν+ + λ 2 f ν+,ν g ν,ν+ = λ p ν+ν+ + u

21 Reconstruction algebras of type D (II) 33 in the free algebra ˆQ for some u IJ + JI. But c ν+ν+2 a ν+2ν+ satisfies condition (A) and so cannot appear on the right hand side, forcing λ = 0. Similarly since f ν+,ν g ν,ν+ satisfies condition (A) λ 2 = 0. This proves the case α ν+ = 3 and so we can assume that α ν+ > 3. Now there are α ν+ relations in S from ν + to ν +, and suppose that λ (f ν+,ν g ν,ν+ ) + λ 2 (k 2 A 0ν+ p ν+ν+ ) u ν+ + p=2 λ p+ (k p+ A 0ν+ k p 0ν+ ) + λ uν+ +(c ν+ν+2 a ν+2ν+ k uν+ 0ν+ ) = 0 in e ν+ (I/(IJ + JI))e ν+ (where the sum may be empty). Then u ν+ λ f ν+,ν g ν,ν+ + λ p+ k p+ A 0ν+ + λ uν+ +c ν+ν+2 a ν+2ν+ p= u ν+ = λ 2 p ν+ν+ + λ p+ k p 0ν+ + λ uν+ +k uν+ 0ν+ + u p=2 in the free algebra ˆQ for some u IJ + JI. Now f ν+,ν g ν,ν+ and c ν+ν+2 a ν+2ν+ satisfy condition (A) forcing λ = 0 and λ uν+ + = 0. Thus λ uν+ k uν+ A 0ν+ λ 2 p ν+ν+ + terms starting with k of strictly smaller index mod IJ + JI. But since k uν+ A 0ν+ does not have any subwords which are cycles, the only way we can change it mod IJ + JI is to bracket as k uν+ (A 0ν+ ) and use the relation in I from to ν +. Doing this we get k uν+ A 0ν+ 2 k u ν+ f 0,ν g ν,ν+. This still does not start with p ν+ν+ or k of strictly lower index, so we must again use relations in IJ + JI to change the terms. But k uν+ f 0,ν g ν,ν+ does not contain any subwords which are cycles, which means the only way to change it mod IJ + JI is to use a relation for f 0,ν g ν,ν+. But there is only one relation for f 0,ν g ν,ν+ in I and so using it we arrive back at

22 34 M. Wemyss k uν+ A 0ν+ 2 k u ν+ f 0,ν g ν,ν+ k uν+ A 0ν+ mod IJ + JI. Thus mod IJ + JI it is impossible to transform k uν+ A 0ν+ into an expression involving p ν+ν+ or k terms with strictly lower index, thus we must have λ uν+ = 0. Now re-arranging we get λ uν+ k uν+ A 0ν+ λ 2 p ν+ν+ + terms starting with k of strictly smaller index mod IJ + JI and so by induction all the λ s are zero, as required. ase t with ν + 2 t N. The proof here is identical to the 2 t N case in the proof of [W2, 4.]. ase t =. This is proved using a very similar argument as in ase t = ν + above. 4. The reconstruction algebra for ν = N In this section we define the reconstruction algebra D n,q when ν = N and prove that it is isomorphic to the endomorphism ring of the special M modules. This is in fact very similar to the previous section, but here the vertex is connected in a slightly different way and also the number of extra arrows out of the vertex ν + is α ν+ 2 compared to α ν+ 3 previously. onsider, for N N with N 2 and for a positive integer α N 2, the labelled Dynkin diagram of type D: αn We call the left hand vertex the + vertex, the top vertex the vertex and the remaining vertices,..., N reading from left to right. To this picture we add an extended vertex and double-up as follows:

23 Reconstruction algebras of type D (II) 35 f, g, g +, f,+ f,2 f N 2,N... g 2, g N,N 2 g N,N f N,N f 0,N g N,0 This is precisely the underlying quiver of the preprojective algebra of type D. Now if α N > 2 then add an extra α N 2 arrows from the N th vertex to and label them k,..., k αn 2 reading from left to right. all this quiver Q. Example 4. onsider D 7,4 then 7 4 = [2, 4] and so Q is f, g, g +, f,+ f 0, g,0 g,2 f 2, k k 2 Example 4.2 onsider D 7,5 then 7 5 = [2, 2, 3] and so Q is f, g, g +, f,+ f,2 g 2, f 0,2 g 2,0 g 2,3 f 3,2 k

24 36 M. Wemyss Definition 4.3 For n q = [α,..., α N ] with ν = N define D n,q, the reconstruction algebra of type D, to be the path algebra of the quiver Q defined above subject to the relations g +, f,+ = 0 f 0, g,0 = 0 If N = 2 g, f, = 0 f 2, g,2 = 0 g,0 f 0, g,2 f 2, = f,+ g +, f, g, g +, f,+ = 0 f 0,N g N,0 = 0 g, f, = 0 f N,N g N,N = 0 If N > 2 f,+ g +, f, g, = 2f,2 g 2, g t,t f t,t = f t,t+ g t+,t for all 2 t N 2 g N,0 f 0,N g N,N f N,N = 2g N,N 2 f N 2,N Further if α N > 2 we add, in each case, the relations k A 0N = p NN, A 0N k = p 00 k t 0N = k t+ A 0N, A 0N k t+ = 0N k t for all t α N 3 where A 0N := 2 f 0,N g N,N. The only notation that remains to be defined are the p s and s, and it is these which change according to the presentation, namely p 00 = p NN = 0N = { f0,n G N, f, g, F,N g N,0 2 f 0,N G N, (f,+ g +, + f, g, )F,N g N,0 { fn,n G N, f,+ g +, F,N g N,N 2 f N,N G N, (f,+ g +, + f, g, )F,N g N,N { f0,n G N, f,+ g +, F,N g N,N 2 f 0,N G N, (f,+ g +, + f, g, )F,N g N,N where for the moduli presentation we take the top choices and for the symmetric presentation we take the bottom choices. Remark 4.4 The relations are very similar to before, but now all relations are cycles. As in the previous section for the moment denote D n,q for the

25 Reconstruction algebras of type D (II) 37 moduli presentation and D n,q for the symmetric presentation; we shall see in Theorem 4.9 that they are isomorphic. Example 4.5 For the group D 7,4 the moduli presentation of the reconstruction algebra is the quiver in Example 4. subject to the relations g +, f,+ = 0 g, f, = 0 f 0, g,0 = 0 f 2, g,2 = 0 g,0 f 0, g,2 f 2, = f,+ g +, f, g, f 2, f,+ g +, g,2 = k ( 2 f 0,g,2 ) ( 2 f 0,g,2 ) k = f 0, f, g, g,0 k (f 0, f,+ g +, g,2 ) = k 2 ( 2 f 0,g,2 ) ( 2 f 0,g,2 ) k2 = (f 0, f,+ g +, g,2 )k Example 4.6 For the group D 7,5 the symmetric presentation of the reconstruction algebra is the quiver in Example 4.2 subject to the relations g +, f,+ = 0 g, f, = 0 f 0,2 g 2,0 = 0 f 3,2 g 2,3 = 0 f,+ g +, f, g, = 2f,2 g 2, g 2,0 f 0,2 g 2,3 f 3,2 = 2g 2, f,2 2 f 3,2g 2, (f,+ g +, + f, g, )f,2 g 2,3 = k ( 2 f 0,2g 2,3 ) 2 f 0,2g 2, (f,+ g +, + f, g, )f,2 g 2,0 = ( 2 f 0,2g 2,3 )k We now justify the positions of the special M modules relative to the dual graph. Lemma 4.7 onsider D n,q with ν = N. Then the special M modules correspond to the dual graph of the minimal resolution in the following way

26 38 M. Wemyss αn W W + U... U N W in Proof. Now Z f = E + +E + N p= 2E p+e N as in the SL(2, ) case. Thus by [W, Lemma 3.3] the quiver of the endomorphism ring of the special M modules is precisely the Q defined above in this section. It is clear that the following composition of maps in the AR quiver do not factor through other special M modules R U N W U N 2 U W + W where the difference now (compared to Lemma 3.0) is that in the above picture the positions of R, W and U N are fixed; again the length of the distance between the positions of the other special M modules changes depending on n q (in the picture we ve drawn n q = 2) but this does not effect the fact that the maps are irreducible.

27 Reconstruction algebras of type D (II) 39 Remark 4.8 The picture in the above proof explains algebraically why in the case ν = N we don t have to directly connect R to the vertex W in = W, since any such map must necessarily factor through U N. The following result is the main result of this section: Theorem 4.9 For a group D n,q with parameter ν = N, denote R = [x, y] D n,q and let T n,q = R W + W N s= U s W in be the sum of the special M modules. Then D n,q = EndR (T n,q ) = D n,q. Proof. The proof is very similar to that of Theorem 3.. Setting m := n q then as before we may choose the following as representatives of the irreducible maps between the special M modules x m -y m «( -y m x m ) «x m y m ( y m x m ) «-y m 0 0 x m... «x m 0 0 y m «-y m 0 0 x m «x m 0 0 y m ( x y ) ` yx (xy) r 3 ` -y x (xy) r 3 ( x -y ) We note that with these choices we have f N,N G N, f,+ g +, F,N g N,N = (xy) r 3 w c 3 2 wd 3 3 = (xy)r 3 w 3 p NN = 2 f N,N G N, (f,+ g +, + f, g, )F,N g N,N = (xy) r 3 v c 3 2 vd 3 3 = (xy)r 3 v 3 f 0,N G N, f, g, F,N g N,0 = (xy) r 3 w c 3 2 wd 3 3 = (xy)r 3 w 3 p 00 = 2 f 0,N G N, (f,+ g +, + f, g, )F,N g N,0 = (xy) r 3 v c 3 2 vd 3 3 = (xy)r 3 v 3 { f0,n G N, f,+ g +, F,N g N,N = w 2 0N = 2 f 0,N G N, (f,+ g +, + f, g, )F,N g N,N = v 2

28 320 M. Wemyss and A 0N = xy. Now if α N > 2 there are extra arrows, and here in the moduli presentation we choose for representatives k t = (xy) r t+3 w c t+2 2 w d t+2 3 for all t α N 2. The proof that we can actually choose these is identical to before. Again the symmetric presentation is obtained by everywhere replacing w 2 by v 2 and w 3 by v 3. The relations for the reconstruction algebra are again induced by matrix multiplication such that x and y commute. By Lemma 2.3 and Lemma 2.4 it is very easy to verify that the algorithmic relations in Definition 4.3 are satisfied with these choices of polynomials. From here the proof is identical to that of Theorem 3. (actually all relations are now cycles, so in fact the proof of linear independence becomes a little easier) hence we suppress the details. 5. Moduli Examples In this section we justify some of the philosophy given in the introduction and also in the introduction of [W2]. Geometrically the key change in viewpoint is that we should not view the minimal resolution as G-Hilb (which we can do via a result of Ishii [I02]), but rather we should instead view the minimal resolution as being very similar to a space that we already understand. It is the reconstruction algebra which tells us which space to compare to, and it is the reconstruction algebra which encodes the difference. In this section (to ease notation) we shall use reconstruction algebras to compare the geometry of the following two examples from which the pattern is clear. The left example corresponds to the group D 3,2 whereas the right example corresponds to D 5,3. The first is an example inside SL(2, ) and so has been extensively studied by many people. Now by inspection of the quiver and relations defining the reconstruction algebra we have an obvious ring homomorphism D 3,2 D 5,3, and denoting by e the idempotent corresponding to the vertex this map restricts to

29 Reconstruction algebras of type D (II) 32 a ring homomorphism e D 3,2 e e D 5,3 e. But these are just [x, y] D 3,2 and [x, y] D 5,3 respectively, so we have D 3,2 D 5,3 [x, y] D 3,2 [x, y] D 5,3 We remark that the existence of the map [x, y] D 3,2 [x, y] D 5,3 is not clear using commutative algebra alone, since there are many choices for generators of the two invariant rings, and only the right choices of the generators on both sides will show the existence of such a map. We now choose the dimension vector and stability α = 2 θ = -5 for both reconstruction algebras and consider the resulting moduli spaces, denoted M 3,2 and M 5,3 respectively. Viewing this stability in the representation theoretic view of being -generated (see below) it is obvious that any θ-stable representation M of D 5,3 remains θ-stable if it is viewed as a D 3,2 -module, thus we get the following commutative diagram M 3,2 M 5,3 2 /D 3,2 2 /D 5,3 which allows us to compare the geometry of minimal resolutions which a priori have nothing to do with each other. It is instructive to view this explicitly, where we are able to see that M 5,3 is really built from M 3,2 by only a very minor modification. Below in Example 5. is an open cover of M 3,2 which we draw as follows:

30 c 322 M. Wemyss U U + U 2 U U 0 The curves are all ( 2)-curves. In Example 5.4 we see that by changing the reconstruction algebra from D 3,2 to D 5,3 the right hand ( 2)-curve in M 3,2 changes into a ( 3) curve in M 5,3 in a very efficient way. The open set U 0 changes, as does the glue between U 0 and U it is the change in glue which gives the change in self-intersection number. The rest of the open sets (and their glues) remain exactly the same and so we have the configuration of P s which forms the minimal resolution of 2 /D 5,3. This is geometrically the nicest solution since we are changing the least amount of information to obtain one space from the other. Note that if we use the G-Hilb description of the minimal resolution (or the McKay quiver) it is not obvious that this should be true. Example 5. onsider the group D 3,2 of order 8. Since this is inside SL(2, ) the reconstruction algebra D 3,2 is just the preprojective algebra b B D a A d aa = 0 c = 0 bb = 0 dd = 0 Aa Dd = Bb c. To simplify notation, below paths and their values in representations are represented by the same symbols. hoosing dimension vector and stability as above, to specify an open set in M 3,2 is equivalent to specifying, for each of the three vertices which are not either or the middle vertex, a non-zero path (which we can change basis to assume

31 Reconstruction algebras of type D (II) 323 to be the identity) from to that vertex. specifying paths (0 ) and ( 0) from to the middle vertex. Different choices in the above lead to different open sets. Note that we must be able to make such choices for any θ-stable module M since by definition M is -generated and so paths leaving the trivial vertex must generate the vector spaces at all other vertices. For a stable M, it must be true that a 0 and so after changing basis we can (and will) always assume that a = ( 0). Define the open sets U 0, U, U 2, U + and U by the following conditions: U 0 ab = a = abbd = a = ( 0) b = (0 ) U ab = a = ad = a = ( 0) b = (0 ) U 2 ab = a = ad = a = ( 0) d = (0 ) U + ab = add = ad = a = ( 0) d = (0 ) U addb = a = ad = a = ( 0) d = (0 ) Pictorially we draw this as follows: U 0 U U 2 U + U where the solid black lines correspond to the identity, and the dotted arrow corresponds to the choice of vector (0 ). Lemma 5.2 Proof. The open sets U 0, U, U 2, U + and U cover the moduli space. Take an arbitrary stable module M = ( b b 2 ) «( c c 2 ) 2 B B 2 «( 0 ) 2 D D 2 «( d d 2 ) «A A 2

32 324 M. Wemyss We must show that we can change basis so that M belongs to one of the above open sets. We first prove the following claim: if d is a linear multiple of ( 0) then we may choose b = (0 ). To see this, since we must generate the module M from, for every vertex +, and 2, one of the paths corresponding to a generator of the corresponding M module must be non-zero. By assumption on d we must be able to choose either b = (0 ) or c = (0 ) so that we generate the middle vertex. If we can choose b = (0 ) we are done, hence suppose b is a linear multiple of ( 0). In this case b = (µ 0) and d = (λ 0) for some λ, ν and so the preprojective relations force µb = 0 and λd = 0. onsequently the paths abb and add are zero. By inspection of the generators of the M modules at vertices +, and 2 this forces ab 0, a 0 and ad 0. In particular D 0 and B 0, but this in turn forces λ = µ = 0 which is a contradiction since the preprojective relation at the middle vertex cannot hold. With this claim, the proof is now easy: Suppose first d can be chosen as (0 ). If ab = 0 then by inspecting the generators of the M module at + we must have addb 0, which forces acb 0 by the preprojective relations. Thus M is in U. Hence we may suppose that ab 0. Now if a = 0 then we must reach vertex by the other generator, so add 0 and so M is in U +. Thus we may also assume that a 0. Now if ad = 0 then D = 0 so D 2 0 so that we generate. But D 2 = dd = 0, a contradiction. Hence necessarily ad 0 and so M is in U 2. The above covers the case when we can choose d = (0 ) so suppose now that this is not the case, i.e. d = (λ 0) for some λ. By the above claim we know we may choose b = (0 ). If ad 0 then D 0 from which dd = 0 gives us that λ = 0 and so d = 0. onsequently to generate at vertex + we need ab 0 and to generate at vertex we need a 0. Thus M is in U. Hence we may assume that ad = 0, and so to generate at vertex 2 requires that abbd 0 which forces acd 0. onsequently M is in U 0. Note that there are many choices of other open covers that we could take. We now compute the open sets above we do the U 0 calculation in full and just summarize the others. Any stable module in U 0 looks like

33 Reconstruction algebras of type D (II) ( 0 ) 2 B2 ( 0 ) ( c c 2 ) A A 2 «D ( d d 2 ) where the variables are scalars, subject only to the quiver relations. Now aa = 0 implies A = 0 bb = 0 implies B 2 = 0 c = 0 implies c = c 2 2 dd = 0 implies d 2 = d D and so plugging this in our module becomes 2 ( -c 2 2 c 2 ) ( 0 ) 2 0 ( 0 ) D ( d -d D ) 0 A2 But now there is only one relation left, namely Aa Dd = Bb c. This gives ( ) ( ) 0 0 d D d D 2 = A 2 0 d d D which yields the four conditions d D = c 2 2 d D 2 = c 2 A 2 = d + c d D = c 2 2 ( ) ( ) 0 c2 2 c c c 2 2

34 326 M. Wemyss The second and third conditions eliminate the variables c 2 and A 2, whereas the first and last conditions are the same. Substituting the second condition into the first we see that this open set is completely parameterized by d, D and 2 subject to the one relation d D = (d D 2 ) 2, so U 0 is a smooth hypersurface in 3. Similarly we have U U 2 U + U 2 ( -c 2 2 c 2 ) ( 0 ) 2 0 ( 0 ) ( -b 2 B 2 b 2 ) B2 D2 ( -d 2 D 2 d 2 ) 0 A2 2 ( -c 2 2 c 2 ) ( 0 ) 2 0 A2 0 ( 0 ) ( c -c ) ( -b 2 B 2 b 2 ) 0 2 B2 ( 0 ) ( 0 ) ( b -b B ) B 0 A2 2 ( -c 2 2 c 2 ) ( 0 ) 2 0 A2 0 ( 0 ) c 2 2 = d 2 D 2 c 2 = + d 2 A 2 = c d 2D 2 2 c 2 2 = d 2 D 2 b 2 B 2 = c 2 2 b 2 = c 2 A 2 = c b 2B 2 2 b 2 B 2 = c 2 2 b 2 B 2 = c b 2 = c 2 A 2 = b 2 B 2 2 c b 2 B 2 = c b B = c 2 2 c 2 = b B 2 A 2 = c b b B = c d 2,D 2, 2 /( + d 2 ) 2 = d 2 D 2 3 b 2,B 2, /( + b 2 ) 2 = b 2 B c,b 2, /( + c 2 )B 2 = c 3 b,b, 2 /(b B 2 ) 2 = b B

35 c Reconstruction algebras of type D (II) 327 Note in U 2 above the equation b 2 = c 2 really means that we have a choice of co-ordinate between b 2 and c 2 ; thus we could equally well parameterize U 2 as 3 c 2,B 2, 2 /c 2 2 = (c 2 )B 2. Hence we see that the space is covered by 5 open sets, each a smooth hypersurface in 3. By changing basis on the quiver it is also quite easy to write down the glues: U 0 (d, D, 2 ) (-d D 2, D, 2) U U (d 2, D 2, 2 ) (d 2, -d 2D 2, - 2 ) U 2 U 2 (c 2, B 2, 2 ) (-c 2 2 2, B 2, 2 ) U + U 2 (b 2, B 2, 2 ) (-b 2 B 2 2, B 2, 2) U from which we can just see the configuration of P s. The picture of the glues should (roughly) coincide with the picture drawn earlier. Remark 5.3 [N2]. Similar calculations to the above can be found in [L02] and We now illustrate how the reconstruction algebra changes this moduli picture. Example 5.4 onsider the group D 5,3. By Theorem 4.9 the moduli presentation of the reconstruction algebra D 5,3 is b D B d A k a aa = 0 c = 0 bb = 0 dd = 0 Aa Dd = Bb c k ad = dbbd adk = aca hoose dimension vector and stability as in the previous example. Notice that exactly the same conditions that defined an open cover in the previous example give an open cover here, since arrows pointing to don t add any more choices. Now the calculation from the previous example tells us almost everything, except now we have a new variable k inside every open set. The point is that the only open set which changes is U 0. The reason for this is quite simple: in the relations of the reconstruction algebra k ad = dbbd

36 328 M. Wemyss and notice that ad = in every open set except U 0. Thus k = dbbd in every open set except U 0 and consequently we can put k in terms of the other variables. Hence k isn t really an extra variable in these open sets, so they do not change. Thus the only open set that changes is U 0, and by the previous calculation (the relations above the line) and our new relations (shown below the line) we see that U 0 is given by ( 0 ) 0 2 ( -c 2 2 c 2 ) ( 0 ) 2 0 A2 D ( d -d D ) k d D = c 2 2 d D 2 = c 2 A 2 = d + c d D = c 2 2 k D = d D k = c 2 A 2 Since d = k D, instead of being given by d, D, 2 subject to d D = (d D 2 ) 2 the open set U 0 is now given by k, D, 2 subject to k D 2 = (k D 3 ) 2. Also, the gluing between U 0 and U has changed to U 0 (k, D, 2 ) D 0 (-(k D )D 2, D, 2) = (-k D 3, D, 2) U. Thus we see that the right hand curve has changed into a ( 3)-curve, nothing else in the open cover has changed and so the dual graph is now References [BR78] Behnke K. and Riemenschneider O., Diedersingularitäten. Abh. Math. Sem. Univ. Hamburg 47 (978), [BSW] Bocklandt R., Schedler T. and Wemyss M., Superpotentials and Higher Order Derivations. J. Pure Appl. Algebra 24 (9) (200), [B02] Bridgeland T., Flops and derived categories. Invent. Math. 47 (3) (2002),