COHOMOLOGY OF THE G-HILBERT SCHEME FOR 1 (1, 1, r 1) Oskar Kȩdzierski

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2 Sedica Math J 30 (2004), COHOMOLOGY OF THE G-HILBERT SCHEME FOR 1 (1, 1, 1) Oska Kȩdzieski Communicated by P Pagacz Abstact In this note we attempt to genealize a few statements dawn fom the 3-dimensional McKay coespondence to the case of a cyclic goup not in SL(3, C) We constuct a smooth, discepant esolution of the cyclic, teminal quotient singulaity of type 1 (1, 1, 1), which tuns out to be isomophic to Nakamua s G-Hilbet scheme Moeove we explicitly descibe tautological bundles and use them to constuct a dual basis to the integal cohomology on the esolution 1 Intoduction In the case of a finite, abelian goup G SL(3, C), Caw and Reid [2] constuct explicitly a smooth, cepant toic esolution of the quotient singulaity C 3 /G Moeove in [1] Caw shows that the integal cohomology of the esolution has ank equal to the ode of the goup G and 2000 Mathematics Subject Classification: Pimay 14E15; Seconday 14C05,14L30 Key wods: McKay coespondence; esolutions of teminal quotient singulaities;g-hilbet scheme

3 294 Oska Kȩdzieski constucts a dual basis using tautological bundles Fo finite G in GL(2, C) the cohomology of the minimal esolution has ank smalle than the ode of G (compae [7]) Caw and Reid calculated G-Hilb fo G = 1 (1,a, a), and fo most values of a it is vey discepant and still singula, with odinay double points xy = zt We show that in the case of a cyclic, teminal, quotient singulaity of type 1 (1,1, 1) the G-Hilbet scheme is a smooth, discepant esolution and its integal cohomology has ank 2 1 The dual basis to cohomology is constucted using tautological bundles intoduced by Gonzalez Spinbeg and Vedie We assume that the eade is familia with basic toic geomety ([4], [9]) Acknowledgments I would like to thank Pofesso Miles Reid fo the help duing my stay at the Univesity of Wawick and fo intoducing me into the subject Fo moal suppot I wish to thank Pofessos A Bia lynicki Biula, P Pagacz and my supeviso Jaos law Wiśniewski I am also gateful to Pofesso Dan Abamovich and the efeee fo numeous emaks concening the final vesion of this pape 2 Toic esolution Let us fix an intege 2 and the goup G geneated by the element diag(ε,ε,ε 1 ), whee ε = e 2πi The goup G has chaactes which may be identified with 1,ε,ε 2,,ε 1 To use toic geomety methods intoduce the lattice N = Z (1,1, 1)Z, and its dual M = Hom Z (N, Z) Conside the cone σ = R 0 e 1 + R 0 e 2 + R 0 e 3 geneated by non negative combinations of the standad basis vectos of Z 3 in N Z R and define X = C 3 /G Then it is easy to see that whee X = Spec C[x,y,z] G Spec C[σ M], σ = {u M : u,v 0 fo all v σ}, and the functions x,y,z ae identified with the dual elements e 1,e 2,e 3 (see [4] p 3 8 fo moe details) This identification will be used in the est of the pape Definition 21 Let p i = 1 ( i, i,i) fo i = 1,2,, be the points in the lattice N (note that p = e 3 ) Define Y as the toic vaiety given by the fan obtained fom the cone σ by the sequence of successive sta subdivisions along the ays R 0 p 1,, R 0 p 1 Denote by f : Y X the esulting pope, biational toic mophism given by the identity map on the lattice N, and let

4 Cohomology of the G-Hilbet scheme fo 1 (1,1, 1) 295 Ex(f) be the exceptional set of f (see [4] p 48 and pictue below showing the fan intesected with the hypeplane e 1 + e 2 + 2e 3 = 2) Lemma 21 Y is a smooth toic vaiety Poof Since the fan is simplicial it is enough to check that the pimitive vectos along geneating ays fo evey 3-dimensional cone in fom a Z-basis fo the lattice N This follows easily as det[e 1,e 2,p 1 ] = det[e j,p i,p i+1 ] = 1 fo j = 1,2, i = 1,, 1 Denote by τ i = R 0 p i the ay though p i fo i = 1,, 1 The ieducible components of exceptional set Ex(f) ae in one-to-one coespondence with the ays τ i Each component is a compact toic suface defined by the fan Sta(τ i ) in the quotient lattice N(τ i ) (details [4] p 52) It is also useful to have dual coodinates fo evey 3 dimensional cone in the fan They ae: σ e 1,e 2,p 1 = σ e 1 +(1 )e 3,e 2 +(1 )e 3,e 3, σ e 1,p i,p i+1 = σ e 1 e 2,ie 2 +(i )e 3,(i+1)e 2 +(i+1 )e 3, σ e 2,p i,p i+1 = σ e 1 +e 2,ie 1 +(i )e 3,(i+1)e 1 +(i+1 )e 3

5 296 Oska Kȩdzieski fo i = 1,, 1, whee fo example σ e1,e 2,p 1 denotes the cone geneated by R 0 e 1, R 0 e 2 and τ 1 Definition 22 Let S i be the i-th ieducible diviso in Ex(f) defined by the fan Sta(τ i ), that is S i = V(τ i ) Lemma 22 The exceptional ieducible divisos in Ex(f) ae S 1 P 2 and S i F i fo i = 2,, 1 whee F i is a Hizebuch suface (see [4] p 7) Poof Fo the suface S i pick two dual coodinates in an adjacent 3- dimensional cone in vanishing on τ i Evaluating them on pimitive vectos along ays geneating 2-dimensional cones containing τ i gives geneatos of ays in the fan Sta(τ i ) That is fo the suface S 1 choose the cone σ e1,e 2,p 1 and set X = e 1 + (1 )e 3 and Y = e 2 + (1 )e 3 Then (X(e 1 ),Y (e 1 )) = (1,0), (X(e 2 ),Y (e 2 )) = (0,1), (X(p 2 ),Y (p 2 )) = ( 1, 1), so S 1 P 2 Analogously fom σ e 2,p i,p i+1 pick X = ie 1 +(i )e 3 and Y = e 1 +e 2 Then (X(e 1 ),Y (e 1 )) = (i, 1), (X(p i 1 ),Y (p i 1 )) = (1,0), (X(e 2 ),Y (e 2 )) = (0,1), (X(p i+1 ),Y (p i+1 )) = ( 1,0), hence the lemma follows Fom the toic pictue it is easy to see that Ex(f) consists of a towe of P 2 and Hizebuch ational scolls, that is S i S i+1 = P 1 fo i = 1,, 2, whee P 1 coesponds to the cone spanned by τ i and τ i+1 Using homotopy x tx of C 3 we can contact X to a singula point The homotopy lifts via f to Y Since the exceptional set lies ove the singulaity on X one sees that Y is homotopic

6 Cohomology of the G-Hilbet scheme fo 1 (1,1, 1) 297 to a tubula neighbohood of Ex(f) so that H (Y, Z) H (Ex(f), Z) The basis of H 2 (F i, Z) consists of ational cuves L i and M i satisfying the elations L 2 i = 0, L i M i = 1, and M 2 i = i (see [10], Lemma 27) By induction on and using the Maye-Vietois sequence it is clea that the basis of H (Ex(f), Z) is given by the class of a point in degee 0, the classes of the cuves L i in degee 2 (L 1 stands fo P 1 in S 1 ) and by the classes of S i in degee 4 Definition 23 Nakamua The G-Hilbet scheme G- Hilb C 3 is the moduli space of G-clustes, that is 0-dimensional, G-invaiant subschemes Z C 3 such that H 0 (Z, O Z ) is the egula epesentation C[G] of the goup G Fo woking with G- Hilb C 3 schemes following Nakamua [8] it is convenient to intoduce the notion of a G-set Definiton 24 A subset Γ of monomials in C[x, y, z] is called a G-set if (1) it contains the constant monomial 1, (2) if pq Γ then p Γ and q Γ, (3) thee is a 1 to 1 coespondence between Γ and ieducible epesentations of G with espect to the induced action of G on C[x,y,z] We can identify G- Hilb C 3 with a moduli space fo ideals I in C[x,y,z] such that C[x,y,z]/I = C[G] The monomials in a basis of C[x,y,z]/I give elements of a G-set Lemma 23 The only possible G-sets in the case of 1 (1,1, 1) ae: Γ x i = {z i,z i 1,,1,x,x 2,,x i 1 } fo i = 0,, 2, Γ y i = {zi,z i 1,,1,y,y 2,,y i 1 } fo i = 0,, 2, Γ z = {z 1,z 2,,1} Poof If Γ is a G-set, then xz,yz / Γ since 1 aleady epesents tivial chaacte Moeove xy / Γ because x and y epesent the same chaacte ε, so Γ contains only monomials in one vaiable If z i is the maximal powe of z in Γ then eithe x i 1 o y i 1 must be in Γ, and the esult follows Lemma 24 The mophism f : Y X is a esolution of singulaities and Y G- Hilb C 3

7 298 Oska Kȩdzieski Poof Afte Lemma 21 it is enough to compute all G-sets (in the spiit of [8] o [1], Section 51) using dual coodinates fo evey cone in and check if all possible ae pesent Fo the cone σ e1,e 2,p 1 the dual coodinates on the coesponding affine open chat C 3 ae α = x z 1, β = y z 1, γ = z They give geneatos x αz 1,y βz 1,z γ of the ideal defining a G-cluste In this case the coesponding G-set is given by Γ z Similaly fo the cone σ e1,p i,p i+1 we get geneatos x αy,y i+1 βz i 1,z i γy i and the G-set Γ y i 1, and fo the cone σ e2,p i,p i+1 geneatos y αx,x i+1 βz i 1,z i γx i and the G-set Γ x i 1 3 Tautological bundles Tautological bundles on the esolutions of Kleinian singulaities wee defined by Gonzalez Spinbeg and Vedie [5] In the two dimensional case they define a basis of the K goup of the minimal esolution and have degee 1 on exactly one exceptional cuve of the minimal esolution In the toic case we adapt an equivalent definition (see [1] Def 47, [11] Section 4 and [5] p 417 fo oiginal teatment) Definiton 31 If ρ i : G GL(V i ) is an ieducible epesentation, let R i = Hom C[G] (V i, C[x,y,z]) be the O X -module geneated by monomials in the ε i -chaacte space Define tautological bundle R i as ie pullback modulo tosion R i = f R i /Tos OY Each R i is geneated by the monomials x i,y i,z i C[x,y,z] as an O X -module Multiplying by z i we see that it is isomophic to the ideal sheaf (x i z i,y i z i,z ) O X We claim that R i is an invetible sheaf Indeed on the toic pictue it is epesented as a Catie diviso by the piecewise linea function on the fan given by ie 1 + ie 3 on the cone σ e 2,e 3,p i, ie 2 + ie 3 on the cone σ e1,e 3,p i and by e 3 on σ p i,e 1,e 2 (see [11] p 5 8 and [2] Example 48) We note that this Catie diviso is equivalent to the Q-Catie diviso coesponding to (x i,y i,z i ) and it is moe convenient to expand it in tems of linea equivalence classes of exceptional sufaces:

8 Cohomology of the G-Hilbet scheme fo 1 (1,1, 1) 299 R 1 = 1 R 2 = 2 S 1 S 1 2 S 2 2 S 2 1 S 1, 2( 2) S 2 2( 3) S S 2 2 S 1, R i = i S 1 2( i) S 2 i( i) S i 2i S 2 i S 1, i( i 1) S i+1 R 1 = 1 S 1 2 S 2 2 S 2 1 S 1 Obseve that as a Q-Catie diviso R 1 is the discepancy diviso fo f (see [12] p ), that is f (K X ) = K Y + R 1 and the Catie diviso R 1 is linealy equivalent to K Y (the equivalence is given by linea function e 1 + e 2 + e 3 ) In fact K X is linealy tivial 4 Main esult Definition 41 Define vitual sheaves V i = (R 1 R i ) ((R 1 R i ) O Y ) These vitual sheaves will be used to constuct the dual basis to cohomology Fo any bundles F, G define c(f G) = c(f) c(g) Theoem 41 The tautological bundles R i fom the dual basis of H 2 (Y, Z), that is c 1 (R i ) L j = δ ij and the vitual sheaves V i fom the dual basis of H 4 (Y, Z), that is c 2 (V i ) S j = δ ij

9 300 Oska Kȩdzieski Poof The diviso R i has degee 1 on the fibe L i of ational scoll F i which coesponds to the line joining e 1 with p i in the toic pictue It has also degee 0 on L j fo i j This poves the fist pat of the theoem The second pat is poven by inspecting the following table of fist Chen classes computed on evey compact suface: c 1 (R 1 ) c 1 (R 2 ) c 1 (R 3 ) c 1 (R 1 ) P 2 L F 2 L 2 M 2 + 2L F 3 L 3 2L 3 M 3 + 3L 3 0 F 4 L 4 2L 4 3L 4 0 F 1 L 1 2L 1 3L 1 M 1 + ( 1)L 1 and by the equation c 2 (F F ) = c 1 (F)c 1 (F ), which holds fo any line bundles F, F The estiction of the bundle R i to the suface S j is computed by choosing fom the piecewise function fo R i a linea function on one of the 3-dimensional cones containing τ j and subtacting it fom the functions on all the othe cones Evaluating the esulting functions on pimitive vectos in ays geneating 2 dimensional cones containing τ j gives minus coefficients fo the desied tous invaiant Catie diviso on the fan Sta(τ j ) (see [9] fo moe details) Obseve also that c 1 (V i ) = 0, so the second Chen class of V i is integal This esult computes also ank H (Y, Z) = 2 1 ( 1 fo the second and fouth cohomology and 1 fo the zeoth) It would be also inteesting to obtain simila esults in the geneal case of 1 (1,a, a) fo the economic, smooth esolution (see [12], Section 5) We note also that this economic esolution is isomophic to the G-Hilbet scheme only fo a = 1

10 Cohomology of the G-Hilbet scheme fo 1 (1,1, 1) 301 REFERENCES [1] A Caw The McKay coespondence and epesentations of the McKay quive Univesity of Wawick PhD thesis, caw/thesisps [2] A Caw, M Reid How to calculate A-Hilb C 3 Séminaies et Congès 6 (2002), [3] W Fulton Intesection theoy Egebnisse de Mathematik und ihe Genzgebiete (3), Folge A, 2 Spinge Velag, Belin, 1998 [4] W Fulton Intoduction to toic vaieties Annals of Mathematics Studies, vol 131 The William H Roeve Lectues in Geomety Pinceton Univesity Pess, Pinceton, 1993 [5] G Gonzalez-Spinbeg, J-L Vedie Constuction géométique de la coespondance de McKay Ann Sci École Nom Sup (4) 16, 3 (1983), [6] R Hatshone Algebaic geomety Gaduate Texts in Mathematics, No 52 Spinge-Velag, New Yok-Heidelbeg, 1977 [7] A Ishii On the McKay coespondence fo a finite small subgoup of GL(2, C) J Reine Angew Math 549 (2002), [8] I Nakamua Hilbet schemes of abelian goup obits J Algebaic Geom 10, 4 (2001), [9] T Oda Convex bodies and algebaic geomety An intoduction to the theoy of toic vaieties Egebnisse de Mathematik und ihe Genzgebiete (3), vol 15, Spinge-Velag, Belin, 1988 [10] M Reid Chaptes on algebaic sufaces Complex algebaic geomety (Pak City, UT, 1993), 3 159, IAS/Pak City Math Se, 3, Ame Math Soc, Povidence, RI, 1997 [11] M Reid McKay coespondence In: Poc of algebaic geomety symposium(kinosaki, Nov 1996) (Ed T Katsua), 1997, 14 41

11 302 Oska Kȩdzieski [12] M Reid Young peson s guide to canonical singulaities Algebaic geomety, Bowdoin, 1985 (Bunswick, Maine, 1985), , Poc Sympos Pue Math, 46, Pat 1, Ame Math Soc, Povidence, RI, 1987 Institute of Mathematics Polish Academy of Sciences ul Śniadeckich 8 POBox Waszawa 10 Poland oska@impangovpl Received Mach 31, 2004 Revised June 17, 2004