Deformations of rational surface singularities and reflexive modules with an application to flops

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Advances in Mathematics

www.elsevier.com/locate/aim

Deformations of rational surface singularities and reﬂexive modules with an application to ﬂops

TrondStølen Gustavsen^a,Runar Ile^b,∗

a UniversityofSoutheastNorway/DepartmentofMathematics,Universityof Bergen, Norway

b BINorwegianBusinessSchool/DepartmentofMathematics,Universityof Bergen, Norway

a r t i c l e i n f o a bs t r a c t

Article history:

Received14July2017

Receivedinrevisedform10October 2018

Accepted15October2018 Availableonline25October2018 CommunicatedbyKarenSmith

MSC:

primary14B07,14E30 secondary14D23,14E16

Keywords:

Flatifyingblowing-up

MaximalCohen–Macaulaymodule Simultaneouspartialresolution Smallresolution

Rationaldoublepoint Matrixfactorisation

Blowinguparationalsurfacesingularityinareﬂexivemodule gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of thepair(singularity, module)relates todeformationsof the correspondingpairofpartialresolutionandlocallyfreestrict transform,andtodeformationsoftheunderlyingspaces.The resultsimplysomerecentconjecturesonsmallresolutionsand ﬂops.

* Correspondingauthor.

E-mailaddresses:trond.gustavsen@math.uib.no(T.S. Gustavsen),runar.ile@bi.no(R. Ile).

https://doi.org/10.1016/j.aim.2018.10.023

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1. Introduction

We relate deformations of a rational surface singularity with a reﬂexive module to deformationsof apartial resolution of thesingularity withthelocally free stricttrans- form of the module. Our resultsimply three conjectures of C. Curto and D. Morrison abouthowafamilyofsmallresolutionsofa3-dimensionalindexoneterminalsingularity anditsﬂop areobtainedbyblowing upinamaximalCohen–Macaulaymodule andits syzygy.

Rational surfacesingularities were deﬁnedby M. Artin in[1]. Further foundational workwasdonebyE.Brieskorn [8] andJ.Lipman [41] and manystudieshavefollowed.

In the 1980s the geometrical McKay correspondence was establised by G. Gonzales- Sprinberg andJ.-L. Verdier[18] and generalisedin[4]. It givesabijection betweenthe isomorphismclassesof(non-projective)indecomposablereﬂexivemodules{Mi}andthe primecomponents{Ej}oftheexceptionaldivisorintheminimalresolution X˜→X of arationaldouble point(RDP),i.e.theA_n,D_n andE₆₋₈.Moreprecisely,if_F_i denotes the strict transform of M_i to X,˜ the Chern class of _F_i is dual to the prime divisor;

c1(Fi).Ej=δij,withrkMiequaltothemultiplicityofEiinthefundamentalcycle.For non-Gorensteinquotient surfacesingularitiesthere areingeneralmoreindecomposable reflexive modules than prime components as was shownby H. Esnault [17]. However, O. RiemenschneiderandhisstudentJ. Wunramgaveanaturalclassof‘special’reflexive modules(which we will call Wunrammodules) forwhich thecorrespondence holds for any rationalsurfacesingularity [48,58]. A. Ishiirefined Wunram’sresultbymeans ofa Fourier–Mukai transformin thecase of quotientsurface singularities[29]. M. Vanden Bergh’suse in[51] oftheendomorphism ringofahigher dimensionalWunrammodule toprovederivedequivalencesforflopsinducedalotofactivity,alsoattractingattention tothe2-dimensionalcasewithinterestingresultsbyM.Wemyssand collaborators,e.g.

O.IyamaandWemyss[30,31] and Wemyss[56].

TheMcKay–Wunram correspondence isfoundational forthis article: Weprovethat blowing up a rational surface singularity X in a reﬂexive module M (a special case of L. Gruson and M. Raynaud’s ﬂatifying blowing-up [46]) gives a partial resolution f: Y →X where Y inparticular is normal,dominated bytheminimal resolution,and the strict transform _M = f(M) is locally free. The partial resolution is determined by the Chern class c₁(_F) of the strict transform _F of M to X.˜ In particular, any partial resolution dominated by the minimal resolution is given by blowing up in a Wunram module. See Theorem 4.3 for more precise statements. As an example, the RDP-resolution(obtainedby contractingthe (−2)-curves intheminimal resolution) is givenbyblowingupinthecanonicalmoduleω_X.

ConsiderthedeformationsDef_(Y,_M₎ofthepair(Y,_M) whichblowdowntodeforma- tionsofthe pair(X,M). Ourmain result(Theorem 5.1)saysthatinthecommutative diagramofdeformationfunctors

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Def_(Y,_M₎ ^β

α

Def_Y

δ

Def_(X,M₎ Def_X

the blowing down map α is injectiveand the forgetful map β is smooth and in many situationsanisomorphism.Theinjectivityofαissurprisingsincetheblowingdownmap δingeneralisnotinjective(cf.Remark5.8and[53,6.4]).OnspacesδisaGaloiscovering onto the Artin component A which for RDPs equals Def_X [9,50,45,2,55]. However, β is an isomorphism if M is Wunram (e.g. any reflexive on an RDP) implying that δ factors through aclosed embedding αβ⁻¹: Def_Y ⊆Def_(X,M) realising deformationsof thepartialresolutionas deformationsofthepairas conjecturedbyCurtoandMorrison intheRDPcase.AdeformationofXinthecomponentAliftsingeneraltoadeformation of (X,M) –and of Y –onlyafter a finite base change.However,adeformation of the pair(X,M) inthegeometricimageofDef_(Y,M)liftstoadeformationof(Y, M) without any basechange.NotethatDef_(X,M)ingeneralisnotdominatedbyDef_(Y,M),evenfor RDPs: in Example 5.11, M is the (rank two) fundamental module and Def_(X,M) has two componentswhileDef_Y hasone.A crucialingredient(firstprovedbyLipman [42]) in J. Wahl’s proof that the covering DefX˜ → A has Galois action by a product of Weyl groups was the injectivity of δ in the case Y is the RDP-resolution. This is an immediate consequenceof ourmain resultsince Def_{(X, ω}_X₎ ∼= Def_X; see Corollary 5.7.

While knowledge of Def_(X,M) would be interesting initself, these results also indicate thatthereareinterestingrelationsto Def_X,e.g.regardingthecomponentstructure.

In this article ourmain applicationof Theorem 5.1 is ageneralisation of three conjectures of Curto and Morrison [13] concerning the natureof small partial resolutions of 3-dimensionalindexoneterminalsingularitiesandtheirflops. Ifg:W →Z issucha smallpartialresolutionandX ⊆Z isasufficientlygenerichyperplanesectionwithstrict transform f:Y →X, a resultof M.Reid [47] says thatf is apartial resolution (normal, dominatedbytheminimalresolution)ofanRDP.Inparticular,gisa1-parameter deformation off and henceanelement inDef_Y. ByTheorem4.3,Y istheblowing-up of X inareflexivemoduleM.Thenαβ⁻¹ takesg toa1-parameterdeformation(Z,N) of thepair(X,M).Thebasicresultisthefollowing(cf.Theorem 6.3):

Corollary 1.1.There isamaximalCohen–Macaulay OZ-moduleN such that:

(i) The smallpartial resolutionW →Z isgivenby blowingup Z in N.

(ii) Blowing upZ in thesyzygymoduleN⁺ of N gives theunique ﬂopW⁺→Z.

(iii) The lengthof theﬂopequalstherankof N iftheﬂop issimple.

Theorem6.6isaversionofthisstatementforﬂatfamiliesofsuchsmallpartialresolu- tionsandﬂops.Thereisafamilyofpairs(X,M) inDef_(X,M)suchthattheblowingupof X inM andinthesyzygyM⁺givetwosimultaneouspartialresolutionsY →X←Y⁺

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whichinduceany local familyof flopsof g bypullback, forany g with hyperplanesec- tionf. ByaresultofS.Katzand Morrison,inthe simplecasethelength l ofthe flop determinesthegenerichyperplanesectionX[33],seealso[34].Moreprecisely,X equals A₁,D₄,E₆,E₇,E₈ or E₈ for l = 1,2,3,4,5 or 6, respectively. Byour result there is in eachcase aunique reflexive module M of rank l such that any simpleflop of length l isobtainedbypullbackfrom theY →X ←Y⁺ for thecorresponding(X,M).Hence Y →X ← Y⁺ gives the ‘universal’simple flop of length l realised as blowing-ups in familiesof reflexivemodulesas suggestedbyCurto andMorrison;seeRemark6.8.

Asanexampleconsider A1:x²+yzwhich hasaminimal versalfamilyx²+yz−u.

Afterthebasechangeu→t²itallowsasimultaneousdeformationoftheminimalreso- lutionandtheresultingfamilyisasmall resolutionofZ:x²+yz−t² withexceptional ﬁbreE∼=P¹;seeM.F. Atiyah[5,Thm. 2].Theonlynon-trivialindecomposablereﬂexive moduleMonA1extendstoamoduleN onZ withpresentationmatrixΦ=

x+t y

−z x−t

. Blowing up Z in N gives the simultaneous resolution W → Z of the family. Blowing upZ inthesyzygyN⁺ givesthesimpleﬂop W⁺ →Z oflengthone.Thepresentation matrixofN⁺ istheadjointΨ ofΦ andthepairmakesamatrixfactorisationofthehy- persurfaceZ.TheRDPs arehypersurfacesand any maximalCohen–Macaulay module is givenby amatrixfactorisation [15].Curto and Morrison phrasetheir conjecturesin termsofmatrixfactorisations(andforsimpleﬂops)andverifythemfortheAn andDn

byextensivecalculations.ThehigherranksoftheindecomposablemodulesfortheE₆₋₈ makes this approach diﬃcult, and for thenon-simple ﬂops practically impossible. Our argumentis conceptualanddoesnotrelyoncomputations.Thecoordinate-freeformu- lationofTheorems 6.3and 6.6makes theconjectures moretransparent andaccessible;

seeRemark6.7.ByaresultofO.VillamayorU.generatorsfortheblowing-upidealare readilyobtainedfromapresentationofthemodule[52],cf.commentsbelow(2.6.3).The singularitiesweworkwith arehenselisations ofﬁnite typealgebras andtheresultswill thereforehaveﬁnite typerepresentationslocallyintheétaletopology.

In recent years there has been a lot of research linking properties of various non- commutative algebras and the flops, e.g.notably the description by W. Donovan and Wemyssof the Bridgeland–Chen autoequivalence interms of theuniversal familyof a non-commutativedeformationfunctor[14].J.Karmazyn[32] reconstructsthesmallpar- tialresolutionanditsflopbyaquiverGIT-constructionwheretheinputisendomorphism algebras.Wemyss[57] containsmanygeneralresultsdescribingflopsandminimalmodels ofsingularities (e.g.for cDVs) inhomologicalterms. Inparticular hedescribes flopsin terms of mutations, with applications to the GIT chamber structure. We offer on the otherhandadirectproofoftheoriginalCurto–Morrisonconjecturesusingdeformation theory where theblowing-up ideal forthe small, partial resolution is obtaineddirectly from the(parametrised)2-dimensional Wunrammodule.Moreover, any flopwith fixed RDPhyperplanesectionandDynkindiagramisapullbackfromapairofsuch‘universal’

blowing-ups. Wealsobelievethatthe geometric techniques usedinthis articlemaybe usefulinthestudy ofmoregeneralcontractions.SeeRemarks 6.5and6.9.

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Theinventoryofthearticleisasfollows.InSection2wegivepreliminaryresultscon- cerning rationalsurfacesingularities, blowing-up incoherentsheaves, strict transforms onpartial resolutionsandtheirChernclassesandacohomologyandbasechangeresult suited to our needs. In Section 3 we deﬁne the deformation functors. We also give a result which implies the compatibilityof blowing-up in afamily of modules with base change.InSection4weprovearesultconcerning thefractionalidealwhichdeﬁnes the blowing-up,normalityofblowing-up,andtheblowing-upversionoftheMcKay–Wunram correspondence. In Section5we prove themain theorem throughseveral intermediate steps. Existence of versal base spaces and a classical result of Lipman follows. There is also an example (the fundamental module). The article ends in Section6 with our treatmentoftheCurto–Morrisonconjectures.

Acknowledgement. Part of this work was doneduring the ﬁrst author’s most pleasant stayat NortheasternUniversity2013/14.

Theauthors thanktherefereeforadetailedandhelpfulreport.

2. Preliminaries

2.1. Partialresolutions ofrational surfacesingularities

Fix an algebraically closed ﬁeldk. All schemes and maps are assumed to be above Speck andallschemesareassumedto benoetherian.

Definition 2.1. A singularity is anaffine scheme X = SpecA where A is algebraic (the henselisationofafinitetypek-algebrainamaximalideal).Apartial resolutionof X is aproperbirationalmapf: Y →X withY normal.IfY isregular,f isaresolution.Let E(f)⊂Y denotethe(non-reduced)closedfibreoffandletΣ(f) denotetheexceptional set of f;theminimal closed subsetofY suchthatf restrictedto itscomplementis an isomorphism. A partial resolution f is small if Σ(f) does not contain any divisorial components.

If Afurthermoreis anormal domainofdimension two,X iscalled anormalsurface singularity. Moreover,X isarational surfacesingularity ifthere isaresolution f such thatR¹f_∗OY = 0;[1]. Arational surfacesingularitywhich isadouble point iscalled a rational double point(RDP).

AnormalsurfacesingularityisanRDPifandonlyifitisaGorensteinrationalsurface singularity;cf. [6,4.19].RDPis alsoequivalent to DuValasdefinedin[40, 4.4];cf.[6, 3.31,4.1].Afinite moduleonanormalsurfacesingularity isreflexiveifand onlyifitis maximal Cohen–Macaulay(MCM).

A fundamentalreferenceforthefollowingresultsisLipman[41]. Proposition2.2will be usedwithoutfurthermentioning.

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Proposition2.2([41,4.1,27.1]).LetX bearationalsurfacesingularityandf:Y →X a partialresolution.Let{Ei}i∈I denotetheprimecomponentsofE(f).Thereisaminimal resolutionofsingularitiesπ: ˜X →X (independentof f)suchthat:

(i) (Minimality) If f is a resolution of singularities then there exists a unique map g:Y →X˜ such thatf =πg.

(ii) (Singularities) Y has only rational surface singularities. If X is an RDP then Y hasonly RDP singularities.

(iii) (Contractingexceptionalcurves)Forany subsetJ⊆I thereexistsauniquepartial resolution g:Y_J → X and map h:Y → Y_J with f = gh such that g contracts exactlythecurves {Ei}i∈I\J.

Proof. Fortheminimalresolution,(i)and(ii)see[41,4.1and1.2].For(iii)see27.1and Remarks p.275in[41]. 2

Proposition 2.3. Let X be a normal singularity of dimension at least two and suppose f: Y → X is a partial resolution. Let {E_i}i∈I denote the prime components of E(f).

AssumedimEi= 1 foralli∈I andR¹f_∗OY = 0.Then:

(i) Ej∼=P¹ forallj,theintersections aretransversalandE(f)containsnoembedded components.

(ii) (Intersection numbers) Let L be an invertible sheaf on Y and C ∈ {Ei}i∈I.Put L.C = deg_C(L⊗ OC);cf. [41,§10-11].Then:

(a) L ∼=OY ifandonly if L.C= 0forallC∈ {Ei}i∈I.

(b) _L isgeneratedbyitsglobalsectionsifandonlyif_L.C0forallC∈ {E_i}i∈I. In thatcase R¹f_∗_L = 0.

(c) L isampleifandonlyifL.C >0forallC∈ {Ei}i∈I.InthatcaseL isvery ample forf.

(iii) (ThePicard group) For each i∈ I there is an eﬀective prime Cartierdivisor Di

whichintersects∪i∈IE_i transversallyinapointcontainedinE_i.Moreover,{D_i}i∈I

givesaZ-basis forPic(Y).

(iv) (Hyperplane sections) Assume f is small and dimX 3. Let g: H → H de- note the strict transform along f of a hyperplane section H ⊂ X deﬁned by a non-zero-divisor u. Assume that H and H are normal. Then the restriction map Pic(Y)→Pic(H) isanisomorphism. Moreover,

O(Di).E(f) =O(Di∩H).E(g).

Proof. (i) Note that 0 = R¹f_∗OY R¹f_∗OC for all subschemes C with support in

∪E_j. Itfollows thatp_a(E_j)= 0 (whichimpliesE_j ∼=P¹)and thattheintersections are transversal. Since f_∗OY f_∗OE(f) and f_∗OY =OX by[49, Lemma 0AY8],itfollows thatH⁰(OE(f))∼=k andE(f) cannothaveembeddedcomponents.(ii)is[41,12.1].

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(iii)Weimitatetheproofof[41,14.3].Lety ∈Ei∪j=iEj beaclosedpointand¯ta generatorforthemaximalidealinOEi,y.Lett∈ OY,y bealiftingof¯t.Onemayassume thatnoEj isacomponentoftheprincipalCartierdivisor(t).Put(t)=Di+D_i where D_i∩(∪E_j)={y}and y /∈D_i (usethatX is henselian). There isa mapθ: Pic(Y)→ Hom_Z(⊕iZEi,Z) givenbyL →(L.−).TheexistenceofDi showssurjectivityofθand (ii)shows injectivity.

(iv)Note thatthestrict transformequals thetotaltransform.Inparticular,{Ei}i∈I

are the prime components of g⁻¹(x). The sequence (u,t) is OY,y-regular. It implies that the standard Cartier divisor in Pic(H) given in(iii) corresponding to the prime component E_i canbe taken to be D_i∩H. Since OE(f),y ∼=OE(g),y the moreoverpart follows. 2

Remark2.4.Notein(iii)thataCartierdivisorD whichintersects∪i∈IE_i transversally is containedinanyopen U ⊆Y whichcontainestheintersectionpoints.

2.2. Blowingupincoherent sheaves

Let X be a scheme, i: U → X a non-empty open subscheme with complement Z, and _F a quasi-coherent OX-module. Suppose f:Y → X is ascheme map such that the restrictionfU of f to f⁻¹(U) is an isomorphismf⁻¹(U)∼=U.Let j: f⁻¹(U)→Y denotetheopeninclusion.DeﬁnetheZ-stricttransformof_F along f tobetheimageof the naturalrestrictionmap f^∗F →j_∗f_U^∗(F_|U) –aquasi-coherent OY-module denoted f_Z_F. Thekernel of therestriction map is the subsheaf_H_f⁰₋₁_Z(f^∗_F) ofsections with support in f⁻¹(Z). Let U ⊆ X be another open subscheme with f⁻¹(U) ∼= U and suppose _F_|_U_∪_U is locally free and both f⁻¹(U) and f⁻¹(U) are dense in Y. Then f_ZF ∼= f_ZF. We use the simpliﬁednotation fF for the maximal such U and call it the strict transform. If Y is integral then f⁻¹Z does not contain the generic point of Y andalllocal sections of H_f⁰−1Z(f^∗F) aretorsion. If F_|U is locally free(asinthe applications below),thenalltorsionlocalsectionsinf^∗F havesupportinf⁻¹Zsincea locally freesheafhasnotorsion; i.e. H_f⁰−1Z(f^∗F)= (f^∗F)tors.

ThefollowingisaspecialcaseofGrusonandRaynaud’stheoremonﬂatteningblowing- up(withtheuniversalproperty); cf.[46, 5.2.2].

Proposition 2.5. SupposeX isascheme, U anopen subschemeof X and F acoherent OX-module such that _F_|_U is locally free. Put Z = X\U. Then there is a projective scheme mapf:Y →X whichis universalwith respecttothefollowing propertiesfora scheme mapf: Y→X.

(i) Therestriction f_U is anisomorphismand f⁻¹(U)is denseinY. (ii) TheZ-stricttransformf_ZF islocallyfree on Y.

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The proof realises Y as the scheme-theoretic closed image (so possibly with non- reduced structure;[21, 9.5]) of amap from U to the scheme of quotients Quot_F/X/X; see[46,§5.2].DenoteY byBlZ,F(X).LetU⊆X beanotheropensubschemeofX with F_|U locally freeand suchthat both U and U aredense inU ∪U. Put Z =X\U. Then BlZ,F(X) equalsBl_Z,F(X). Thesimpliﬁednotation f: Bl_F(X)→X is used if U ismaximalwith F_|U locallyfreeandf iscalledtheblowing-upofX in F.Notethat A. OnetoandE. Zatini[44] deﬁnedtheblowing-upastheclosureoftheimageofU with reducedstructure.Manyoftheirresultsextendtothenon-reducedcontext.

Asweshallconsiderbasechangesofblowing-ups,thefollowingcorollarywillbeuseful.

Corollary2.6. Given acommutativediagram of schememaps

Y2 f2

g Y1 f1

X₂ ^p X₁

andan open subschemeU1 ⊆X1. Put U2 =p⁻¹(U1)andZi=Xi\Ui. Assumethat fi

isan isomorphism above Ui and that f_i⁻¹(Ui)is dense inYi fori= 1,2.Suppose F is acoherent OX1-modulesuchthat _F_|_U₁ and(f₁)_Z₁_F are locallyfree.

(i) The naturalmapg^∗((f1)_Z₁F)→(f2)_Z₂(p^∗F)isan isomorphism.

(ii) Iff1equalsBlZ1,F(X1)→X1andY2= BlZ1,F(X1)×X2,thenY2 isisomorphicto Bl_Z₂_,p∗F(X₂) overX₂.

Proof. (i)Thereisanaturalmap g^∗H_f⁰−1

1 Z1(f₁^∗F)−→H_f⁰−1

2 Z2((pf2)^∗F) (2.6.1) inducing asurjection ϕ:g^∗((f1)_Z₁F)→ (f2)_Z₂(p^∗F). Since ϕ restricted to the dense (f₁g)⁻¹(U₁) isanisomorphismandg^∗((f₁)_Z₁_F) islocallyfree,ϕisanisomorphism.

(ii) By (i), (f2)_Z₂(p^∗F) is locally free. By the universal property in Proposi- tion 2.5 there is an X2-map r: Y2 → BlZ2,p^∗F(X2). Similarly, there is an X1-map BlZ2,p^∗F(X2) → Y1, i.e. an X2-map s: BlZ2,p^∗F(X2) → Y2. By universality r and s areinverseisomorphisms. 2

Assume(forsimplicity)that_F hasaconstantrankrandletK(X) denotethesheaf ofmeromorphicfunctions;cf.[35],[49,Deﬁnition01X2] and[49,Lemma02OV].Ifr= 1 let Fⁿ betheimage ofthenaturalmap F^⊗ⁿ →i_∗(F_|_U^⊗ⁿ).Then

Bl_F(X)∼= Proj

n0

Fⁿ

(2.6.2)

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is thescheme-theoreticclosedimageofU inP(F).Oneto andZatiniobservedthatthe Plücker embeddingoftheGrassmanngivesthefractionalidealsheaf

JFK= im_r

F →rF⊗OXK(X)∼=K(X) (2.6.3) for the blowing-up f: Bl_F(X) → X; cf. [44, 1.4, 3.1], [52, 3.3].Villamayor has given an explicit description of an equivalent ideal. Suppose X = SpecA for a ring A and F is givenbyanA-module M.Choose ngeneratorsforM andletSyz(M) denote the kerneloftheresultingmapA^⊕ⁿ→M.Thenrk Syz(M)=n−randanychoiceofn−r elements inSyz(M) whichinducesgeneratorsforK(A)⊗Syz(M)∼=K(A)^⊕n−r deﬁnes alinearmapψ:A^⊕n−r→A^⊕nsuchthattheidealofmaximalminorsofψisisomorphic to JMK.See[52,3.3].

Curto and Morrisondefines a‘Grassmann blowup’as theclosureinC^N×Grass(n− r,n) of a set defined interms of the smooth locus and the presentation matrix ϕ. In the case of a matrix factorisation of ahypersurface they state in [13, 2.1] a universal propertyforthenormalizationoftheGrassmannblowupfor‘birational’mapsh:Y →X suchthathM islocallyfree.ByourdiscussionandProposition2.2itfollowsthattheir normalizedGrassmannblowupequalsBlM(X) forRDPsonceweknow thatBlM(X) is normal.Normalityisnotobviousandwillbeprovedforareflexivemoduleonarational surfacesingularityinProposition4.2.

2.3. Stricttransformsand Chernclasses

The strict transform ofareflexive sheaf alonga resolution ofarational surfacesin- gularityislocallyfree;see[18,2.10] for quotientsingularties,thegeneralcaseiscitedin [4, 1.1].Esnault proves a characterisationof sheaves onthe resolution which arestrict transformsofreflexivemodulesin[17,2.2].Wegivethefollowingnaturalgeneralisation of Esnault’sresultwhichneedsaslightlydifferentproof.

Proposition 2.7. Letf:Y →X beapartialresolution ofarational surfacesingularity.

(i) SupposeM isareﬂexiveOX-module.ThenthestricttransformfM isareﬂexive OY-module generated by global sections, the natural map M → f_∗fM is an iso- morphism, andR¹f_∗Hom_Y(fM ,ωY)= 0.In particular,fM islocally free ifY isregular.

(ii) If_F isareﬂexiveOY-modulewithR¹f_∗_Hom_Y(_F,ω_Y)= 0thenf_∗_F isareﬂexive OX-module. Moreover, if F is generated by global sections then the natural map ff_∗F →F isan isomorphism.

Proof. (i)Put _M =fM. As a quotient of f^∗M, _M is generated by global sections.

LetU denotethenon-singularlocusinX.Sincef isanisomorphismaboveU andf_∗_M istorsionfree,thenaturalmapα:M →f_∗M isanisomorphismby[49,Lemma0AVS].

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Alsonote thatM islocallyfreeonthecomplementofa0-dimensionallocussinceM is torsionfreeandY isnormal;cf.[7,Chap.VII,§4.9,Thm.6].

Thedualitytheorem [24,VII3.4] (cf.[12, 3.4.4])givesanisomorphism:

Rf_∗RHom_Y(M, ωY)−−−−−→^∼ RHom_X(Rf_∗M, ωX) (2.7.1) Rationalitygives Rf_∗_M f_∗_M and the resulting spectral sequence gives short exact sequences:

0→R¹f_∗Ext^p−1_Y (M, ωY)−→Ext^p_X(M, ωX)−→f_∗Ext^p_Y(M, ωY)→0 (2.7.2) Since M is maximal Cohen–Macaulay, _Ext^p_X(M,ω_X) = 0 for all p > 0 which implies Ext^p_Y(_M,ω_Y) = 0 because _Ext^p_Y(_M,ω_Y) has zero dimensional support for p > 0. It followsthat_M ismaximal Cohen–Macaulay,i.e.reﬂexivesinceY is normal.Moreover, R¹f_∗Hom_Y(M,ωY)= 0 by(2.7.2).Forlocalcohomology; cf.[10,Chap. 3].

(ii) Since Y is normal, F is maximal Cohen–Macaulay, so (2.7.1) gives (with F replacing M)anisomorphismRf_∗Hom_Y(F,ωY)RHom_X(Rf_∗F,ωX).Theassoci- atedsecond quadrantcohomologicalspectralsequencegivesanexactsequence:

0→Ext¹_X(R¹f_∗_F, ω_X)→f_∗_Hom_Y(_F, ω_Y)→Hom_X(f_∗_F, ω_X)

→Ext²_X(R¹f_∗_F, ω_X)→R¹f_∗_Hom_Y(_F, ω_Y)→Ext¹_X(f_∗_F, ω_X)→. . . (2.7.3) SinceR^qf_∗_Hom_Y(_F,ω_Y)= 0 forq >0,(2.7.3) gives

Ext^q_X(f_∗F, ωX)∼=Ext^q+2_X (R¹f_∗F, ωX) (q >0) (2.7.4) and the latter is zero by [10, 3.5.11], i.e. f_∗F is maximal Cohen–Macaulay. Any map O^⊕n_Y →F factorsas

O^⊕Yⁿ∼=ff_∗O^⊕Yⁿ−→ff_∗_F −−→^ρ F (2.7.5) henceiftheformerissurjectivesoisρ.Butsinceff_∗F istorsionfree,ρisanisomor- phism. 2

Remark2.8.Theargumentin(ii)worksforanynormalsurfacesingularity.Seealso[39, 2.74].

Lemma 2.9.Suppose f:Y → X is apartial resolution of a rational surface singularity and F is a locally free OY-module of rank r generated by global sections. A generic choiceof rglobal sectionsgives ashortexact sequenceof coherentOY-modules

α: 0→ OY^⊕r

(s1,...,sr)

−−−−−−→F −→ OD→0

whereD isan eﬀective, aﬃne,smoothdivisorintersecting E(f)red transversally.

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Moreover, ther−1sections s2,. . . ,sr givea shortexactsequence β: 0→ O^⊕r−1_Y −−−−−−→^(s²^,...,s^r⁾ F −−→^w rF →0 where w(m)=m∧s2∧ · · · ∧sr andrF ∼=OY(D).

Proof. ByProposition2.3theprimecomponentsofE(f)redaresmooth.Thenαfollows as in [4, 1.2]. Pushout of O^⊕rY → F along the ﬁrst projection O^⊕rY → OY gives a s.e.s. 0→ O^⊕Y^r⁻¹ →F −→^p E → 0 where_E is an invertible sheaf by[51, 3.5.1]. Since im(s2,. . . ,sr)⊆kerwthereisaninducedmapi:E →rF withw=ip.Themapwis surjective sincepsplits locally. Theni is anisomorphism.Applying Hom_Y(−,OY) to the induceds.e.s.0→ OY →E → OD→0 givesthe s.e.s.0→E^∨→ OY → OD →0 whichimpliesthat_E ∼=OY(D). 2

For a locally free sheaf _F of rankr we use the notation c₁(_F) =rF. Note that Wunramin[58,A2] gavetwonon-isomorphicindecomposablereﬂexivemodulesofrank 3 onI7 withequalChernclasses.

2.4. Basechange andcohomology

We will need a base change result for Ext which is not covered by [20, 7.7.5]. Let f: Y → X = SpecR and g: X → S = SpecA be maps of schemes and _E and _F coherent OY-modules such that Y, E and F are S-ﬂat. Assume that g is local (i.e.

given byalocal mapof localk-algebrasA→R)and f is proper.Put π=gf. Forany quasi-coherent OY-module G and any n, Extⁿ_Y(E, G) is a quasi-coherent OY-module, moreover,π_∗_Extⁿ_Y(_E,_G) isquasi-coherentsinceπisproper;[49,Lemma01XJ].Alsonote thatExtⁿ_Y(E,G) isnaturallyanR-modulewhichisﬁnitelygeneratedifG iscoherentby thelocal-to-globalspectralsequenceE^p,q₂ = H^q(Ext^p_Y(E, G))⇒Extⁿ_Y(E,G) andproper- ness ([19,3.2.1]).Thenaturalisomorphismof functorsf_∗Hom_Y(E,−)∼= Hom_Y(E,−)^˜ extendstoanisomorphismoftherightderiveduniversalδ-functors:

Extⁿ_f(_E,−)∼= Extⁿ_Y(_E,−)^˜

n∈Z:QCoh(Y)−→QCoh(X) (2.9.1) whichrestrictsto functorsofcoherentsheavesCoh(Y)→Coh(X).

Foreveryintegernwedeﬁneafunctorofquasi-coherentsheaves

Fⁿ:QCoh(S)−→QCoh(X) by Fⁿ(I) = Extⁿ_Y(E,F⊗π^∗I)^˜. (2.9.2) ThefunctorgivenbyI→F⊗π^∗I isexactsince F isS-flatand{Fⁿ}n∈Zisacohomo- logicalδ-functor.Moreover,Fⁿ(I) isacoherentOX-moduleifIisacoherentOS-module, and Fⁿ commuteswithfiltereddirectlimits.Hencetheconditionsin[43,5.1-2] aresat- isfied andtheconclusionsapply totheexchangemaps

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eⁿ_I:Fⁿ(OS)⊗OXg^∗I−→Fⁿ(I) (2.9.3) whicharedeﬁnedessentiallybyapplyingFⁿ tothemultiplicationmaps·u:OS →I for u∈I,seethebeginningofSection4in[43] or[20,7.2.2].

Weﬁrstextendtheexchange mapto ordinaryﬁbreproductsby alocal scheme map p:T = SpecB→S.PutX:=X×ST andY:=Y×ST.Letpr_X:X→X,q:Y→Y, g: X → T, f: Y → X and π = g◦f denote the projections. Suppose _G is a quasi-coherentOY-module.ApplyingRf_∗tothenatural,functorialisomorphismin[24, II5.10] gives

R(pr_X)_∗RHom_f(Lq^∗E,G)RHom_f(E,Rq_∗G). (2.9.4) NotethatLq^∗E q^∗E since E isS-ﬂat.Moreover,qandpr_X areaﬃne,so (2.9.4) gives isomorphisms

η: Extⁿ_Y(E, q_∗G)^˜∼= (pr_X)_∗Extⁿ_Y(q^∗E,G)^˜. (2.9.5) Suppose now that I is a quasi-coherent OT-module and let eⁿ_I denote the exchange map eⁿ_I: Extⁿ_Y(q^∗E,q^∗F)^˜⊗OX(g)^∗I →Extⁿ_Y(q^∗E,q^∗F⊗OY(π)^∗I)^˜. We deﬁne the (ordinary)basechangemapbⁿ_I bythefollowingcommutativediagram

pr^∗_XExtⁿ_Y(E,F)^˜⊗OX(g)^∗I ^b

n I

a

Extⁿ_Y(q^∗E, q^∗F⊗OY(π)^∗I)^˜

pr^∗_XExtⁿ_Y(E, q_∗q^∗F)^˜⊗OX(g)^∗I ^η

ad⊗id

Extⁿ_Y(q^∗E, q^∗F)^˜⊗OX(g)^∗I

eⁿ_I

(2.9.6)

whereaisinducedbythecanonicalmap F →q_∗q^∗F.

Toﬁt ourapplicationweassumeR ishenselian. LetgT:XT = Spec(R⊗AB)^h→T denote the projection where the h denotes henselisation in the canonical k-point. Let f_T: Y_T → X_T denote the (ordinary) pullback of f to X_T and let p_X: X_T → X and pY: YT → Y denote the induced projections. Put πT = gTfT, FT = p^∗_YF, and so on. Let h: XT → X denote the henselisation map and hY: YT →Y the pullback of h. Flat base changeby hgivesacanonicalisomorphism (e.g.by Lazard’s theorem [49, Theorem 058G],[49,Lemma 07TB] and thelocal toglobalspectral sequence):

h^∗Extⁿ_Y(q^∗E, q^∗F⊗OY(π)^∗I)^˜∼= Extⁿ_Y_T(ET,FT⊗O_YTπ^∗_TI)^˜ (2.9.7) ThereisalsoanisomorphismofOXT-modules

s: Extⁿ_Y(E,F)^˜_T⊗OXTg^∗_TI−−−→ h^∗

pr^∗_XExtⁿ_Y(E,F)^˜⊗OX(g)^∗I

. (2.9.8) DeﬁnetheOXT-linear(henselian)base change map

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cⁿ_I: Extⁿ_Y(E,F)^˜_T⊗O_XTg^∗_TI−→Extⁿ_Y_T(ET,FT⊗O_YTπ_T^∗I)^˜ (2.9.9) as thecompositionof h^∗(bⁿ_I)◦swith (2.9.7). PutX0=X×SSpeck,Y0 =Y×XX0, let E0 denotethepullbackof E toY0,andso on.

Proposition 2.10.Assumethebase changemap

cⁿ_k: Extⁿ_Y(_E,_F)^˜₀−→Extⁿ_Y₀(_E₀,_F₀)^˜ is surjective. Then:

(i) Forall local maps T →S and quasi-coherent OT-modules I, the basechange map cⁿ_I isanisomorphism.

(ii) Thefollowingstatements are equivalent:

(a) cⁿ⁻¹_k issurjective.

(b) The OX-module Extⁿ_Y(_E,_F)^˜isS-ﬂat.

Proof. We ﬁrst establish a compatibility of eⁿ_p_∗_I with (pr_X)_∗(eⁿ_I). There is a natural isomorphism τ: F⊗OYπ^∗p_∗I ∼=q_∗(q^∗F⊗OY(π)^∗I) with adjoint τ^ad. Note that the canonicalmap _F⊗π^∗p_∗I→q_∗q^∗(_F⊗π^∗p_∗I) composedwith

q_∗τ^ad:q_∗q^∗(_F⊗π^∗p_∗I)−→q_∗(q^∗_F⊗(π)^∗I) (2.10.1) equals τ.LetubeanelementinIandlet·udenotethemapOS→p_∗I.Tosimplifythe notationwealsowrite·uforsomeoftheinducedmapslikeid⊗π^∗(·u) :F →F⊗π^∗p_∗I.

There isadiagram ofOX-linearmaps:

Extⁿ_Y(E,F)

(·u)∗

a Extⁿ_Y(E, q_∗q^∗F)

(q_∗q^∗(·u))_∗

η Extⁿ_Y(q^∗E, q^∗F)

(q^∗(·u))_∗

Extⁿ_Y(E,F⊗π^∗p_∗I)

τ_∗

a Extⁿ_Y(E, q_∗q^∗(F⊗π^∗p_∗I))

(q_∗τ^ad)_∗

η Extⁿ_Y(q^∗E, q^∗(F⊗π^∗p_∗I))

(τ^ad)_∗

Extⁿ_Y(E, q_∗(q^∗F⊗(π)^∗I)) ^η Extⁿ_Y(q^∗E, q^∗F⊗(π)^∗I) (2.10.2) Since η isfunctorial thediagram commutes.The compositionτ^ad◦q^∗(·u) isthe multi- plicationmap ·u:q^∗_F →q^∗_F⊗(π)^∗I.Thereisanaturalisomorphism

γ: (pr_X)_∗

pr^∗_XExtⁿ_Y(_E,_F)⊗OX(g)^∗I∼= Extⁿ_Y(_E,_F)⊗OXg^∗p_∗I. (2.10.3) With the compatibility in(2.10.2) one shows that (pr_X)_∗bⁿ_I = (pr_X)_∗[eⁿ_I◦(η^ad⊗id)◦a]

equals η ◦τ_{∗ ◦}eⁿ_p_∗_I ◦γ where γ, τ_∗: Extⁿ_Y(E,F⊗π^∗p_∗I) → Extⁿ_Y(E,q_∗(q^∗F⊗(π)^∗I)