DG QUIVERS OF SMOOTH RATIONAL SURFACES AGNIESZKABODZENTA Abstract. LetXbeasmoothrationalsurface. WecalculateaDGquiverofafullexceptionalcollection oflinebundlesonX obtainedbyanaugmentationfromastrongexceptionalcollectionontheminimal modelofX. Inparticular,wecalculatecanonicalDGalgebrasofsmoothtoricsurfaces. Introduction Derivedcategoriesofcoherentsheaveshavebecomeoneofthemainresearchareasinmodernalgebraic geometry. Animportanttoolallowingtoworkwithsuchcomplicatedcategoriesisgivenbyfullexceptional 3 collections. Let X be a smooth projective variety and let Db(X) denote the bounded derived category of 1 coherentsheavesonX. Itisprovedin[2]thatafullstrongexceptionalcollectionσleadstoanequivalence 0 between Db(X) and the bounded derived category of modules over a finite quiver with relations. By a 2 resultofBondalandKapranov,see[3],ifσ isnotstrongthenDb(X)isequivalenttothederivedcategory n a of modules over some DG category C . It is proved in [1] that in the latter case the DG category C is a σ σ J path algebra of a finite DG quiver with relations Qσ. 7 2 Calculatingthequiverofastrongexceptionalcollectionisequivalenttounderstandingendomorphisms of some sheaf. On the other hand, in order to calculate the DG quiver a priori one has to use injective ] G resolutions. In [1] more comprehensive methods are given for determining DG quivers for two types of A exceptionalcollections. Firstly,ifacollectionσ canbemutatedtoastrongoneτ thentheDGquiverQ σ . of σ can be calculated by means of the quiver of τ. On the other hand, if a collection σ =(cid:104)E ,...,E (cid:105) is h 1 n t almost strong, i.e. Exti(Ej,Ek)=0 for i(cid:54)=0,1, then one can construct a tilting object Eσ using universal a m extensions and coextensions defined in [6]. In this case endomorphisms of Eσ allow to calculate the DG [ quiver of σ. Many examples of almost strong exceptional collections are given by exceptional collections of line 1 v bundles on rational surfaces. Recall, that every rational surface X, not isomorphic to the projective 8 plane P2, is obtained from some Hirzebruch surface F by a sequence of blow-ups: 3 a 3 X =X −π−→n X →...→ X −π→1 X =F . 6 n n−1 1 0 a 1. In [5] Hille and Perling describe an augmentation process that allows to construct full exceptional 0 collections of line bundles on X starting from a full exceptional collection on X . Moreover, in [6] it 0 3 is proved that collections obtained by augmentation are almost strong. 1 : The main purpose of this note is to calculate the DG quiver of a full exceptional collection σ on a v i smoothrationalsurfaceobtainedviaaugmentationfromastrongfullexceptionalcollectiononX . Todo X 0 this we first present σ in the canonical form (see Proposition 2.1). Using this presentation we calculate r a thetiltingobjectE (seeProposition2.4),anditsendomorphisms. Thenusingtwistedcomplexeswecan σ calculate the DG quiver of σ and any of its mutations. InSection3weapplythesemethodstoasmoothtoricsurfaceY withT-invariantdivisorsD ,...,D . 1 n Recall,thatdivisorsDi correspondtotheraysinthefanΣY ⊂N⊗ZQofY. IftheorderofDi isinduced by an orientation of Q2 then the collection (cid:104)O ,O (D ),O (D +D ),...,O (D +...+D )(cid:105) Y Y 1 Y 1 2 Y 1 n−1 is full and exceptional on Y. For any k ∈{1,...,n} the same remains true for the collection (cid:104)O ,O (D ),O (D +D ),...,O (D +...+D )(cid:105) Y Y k Y k k+1 Y k k+n−2 TheauthorissupportedbyaPolishMNiSWgrant(contractnumberNN201420639). Key words and phrases. Derivedcategories,Exceptionalcollections,DGcategory,toricvarieties. 1 2 A.BODZENTA if the indices are considered as elements of Z/nZ. Therefore we can consider all collections of such a form at once. Namely, let Z =Totω be the total space of the canonical bundle on Y and let p: Z →Y Y denote the canonical projection. As vector bundle E =O ⊕O (D )⊕...⊕O (D +...+D ) is a Y Y 1 Y 1 n−1 generator of Db(Y), we know that p∗(E) is a generator of Db(Z). Moreover, Hom (p∗(E),p∗(E))=Hom (E,p p∗(E))= Z Y ∗ (cid:77) =Hom (E,E ⊗p (O ))= Hom (E,E ⊗O (−nK )). Y ∗ Z Y Y Y n≥0 On Y we can consider an infinite sequence (A )∞ of line bundles defined by k k=0 A =O (sK +D +...+D ), k Y Y 1 r where k =sn+r for 0≤r <n. Denote by A =⊕A the sum of all elements in this sequence. By the Y k canonical DG algebra of Y we understand the DG algebra of endomorphisms of p∗(E) or equivalently of A . Methods described in Section 2 allow us to calculate the canonical DG algebra of any smooth toric Y surface. We give examples of canonical DG algebras of smooth toric surfaces. However, we do not investigate the connection between the combinatorial data of the fan Σ and the canonical DG algebra of Y. Y The structure of the paper is as follows. Section 1 contains definitions of quivers and DG quivers, twisted complexes and exceptional collections together with mutations, universal extensions and coextensions. We also recall basic facts about rational surfaces. In Section 2 we recall after [5] the construction of full exceptional collections on smooth rational surfaces. We present any such exceptional collection in the canonical form and describe its Ext-quiver. Then, using universal coextensions, we calculatetheassociatedtiltingobjectandwedescribeitsendomorphisms. Thisdataallowsustocalculate the DG quiver of the collection. In Section 3 we apply these methods to smooth toric surfaces. We start by recalling basic facts about toric surfaces and full exceptional collections on them. Then we define the canonical DG algebra of a toric surface and we show how to use the results of Section 2 to calculate it. We conclude with examples of the canonical DG algebras for the 1-st and 2-nd Hirzebruch surfaces and for surfaces obtained from the 1-st Hirzebruch surface by blowing-up one point. 1. Background 1.1. Quivers AquiverQconsistsoftwofinitesetsQ ,Q andtwomapsh,t:Q →Q .ElementsofQ arevertices 0 1 1 0 0 of Q and elements of Q are arrows of Q. The maps h and t indicate the head and the tail of an arrow 1 respectively. ApathinQisasequencep=a ...a ofarrowssuchthath(a )=t(a )for1≤i≤n−1; n 1 i i+1 we put h(p) = h(a ) and t(p) = t(a ). A path algebra CQ of a quiver Q is an algebra with basis n 1 consisting of paths in Q; the product p◦p(cid:48) of two basis elements is defined by means of concatenation of paths if t(p)=h(p(cid:48)) and is put to be zero otherwise. We also assume that for any vertex i∈Q there is 0 a trivial path e ∈Q with t(e )=i=h(e ). Then, the element (cid:80) e is the unit of CQ. i 1 i i i∈Q0 i A quiver with relations (Q,S) is a quiver Q together with a set S ⊂ CQ. Let I = (cid:104)S(cid:105) ⊂ CQ be an ideal generated by S. Then the path algebra C(Q,S) of a quiver with relations is defined to be CQ/I. If arrows in Q are Z - graded in such a way that deg(e ) = 0 for any i ∈ Q the path algebra CQ i 0 becomes a graded algebra; for a path p=a ...a we put deg(p)=deg(a )+...+deg(a ). n 1 1 n A DG quiver is a quiver Q together with a Z grading on Q and a structure of a DG algebra on CQ 1 such that ∂(e ) = 0 for any i ∈ Q . The Leibniz rule guarantees that h(∂(p)) = h(p) and t(∂(p)) = t(p) i 0 if only ∂(p)(cid:54)=0. If the set S ⊂CQ consists of homogeneous elements one can analogously define a DG quiver with relations (Q,S). DG QUIVERS OF SMOOTH RATIONAL SURFACES 3 1.2. Twisted complexes RecallthataDGcategoryisapreadditivecategoryC inwhichabeliangroupsHom (A,B)areendowed C with a Z-grading and a differential ∂ of degree one. Moreover, the composition of morphisms Hom (A,B)⊗Hom (B,C)→Hom (A,C) C C C is a morphism of complexes and for any object C ∈C the identity morphism id is a closed morphism of C degree zero. ForaDGcategoryCBondalandKapranovin[3]definethecategoryCpre-troftwistedcomplexes. Todo thisfirstonehastointroducethecategoryC(cid:98)offormalshifts. TheobjectsofC(cid:98)areoftheformC[n]where C ∈ C and n ∈ N. For elements C1[k] and C2[n] of C(cid:98)we put HomlC(cid:98)(C1[k],C2[n]) = HomlC+n−k(C1,C2) and ∂ (f)=(−1)n∂ (f) for f ∈Hom (C [k],C [n]). C(cid:98) C C(cid:98) 1 2 A one-sided twisted complex over C is an expression ((cid:76)n C [r ],q ) where C ’s are objects of C, i=1 i i i,j i r ∈Z, n≥0 and q ∈Hom1(C [r ],C [r ]) are such that q =0 for i≥j and ∂q+q2 =0. i i,j C(cid:98) i i j j i,j LetC =((cid:76)C [r ],q)andC(cid:48) =((cid:76)C(cid:48)[r(cid:48)],q(cid:48))betwistedcomplexes. MorphismsinthecategoryCpre-tr i i j j from C to C(cid:48) are given the set of matrices f =(f ) for f ∈Hom (C [r ],C(cid:48)[r(cid:48)]). A map f =(f ) is i,j i,j C(cid:98) i i j j i,j homogeneous of degree k if f ∈Homk(C [r ],C(cid:48)[r(cid:48)]) for all pairs i,j. i,j C(cid:98) i i j j 1.3. Exceptional collections Let X be a smooth projective variety defined over C. Recall, that an object E ∈Db(X) is exceptional if Hom(E,E)=C and Exti(E,E)=0 for i(cid:54)=0. A sequence σ =(cid:104)E ,...,E (cid:105) of exceptional sheaves is an 1 n exceptional collection if Exti(E ,E ) = 0 for j > k and any i. An exceptional collection σ is full if the j k smallest strictly full subcategory of Db(X) containing E ,...,E equals Db(X). Finally, the collection σ 1 n is strong if Exti(E ,E )=0 for i(cid:54)=0 and any j, k. j k In [2] it is proved that a full strong exceptional collection σ leads to an equivalence of Db(X) with Db(mod-A ) for a finite dimensional algebra A . The algebra A is a path algebra of a quiver with σ σ σ relations obtained from sheaves E ,...,E . 1 n It is proved in [3] that when the collection σ is not strong the category Db(X) is equivalent to Db(C ) σ for some DG algebra C . It was proved in [1] that C can be chosen to be a path algebra of a finite DG σ σ quiver with relations Q . If the collection σ is strong the DG algebra C is quasi-isomorphic to A . σ σ σ In[2]BondaldefinesmutationsofexceptionalcollectionsonavarietyX. Ifapair(cid:104)E,F(cid:105)isexceptional thensoarethepairs(cid:104)L F,E(cid:105)and(cid:104)F,R E(cid:105)forL F andR E definedbydistinguishedtrianglesinDb(X) E F E F L F →E ⊗Hom(E,F)→F →L F[1] E E R E[−1]→E →Hom(E,F)∗⊗F →R E. F F For an exceptional collection σ = (cid:104)E ,...,E (cid:105) the i-th left mutation L σ and the i-th right mutation 1 n i R σ are exceptional collections defined by i L σ =(cid:104)E ,...,E ,L E ,E ,E ,...,E (cid:105), i 1 i−1 Ei i+1 i i+2 n R σ =(cid:104)E ,...,E ,E ,R E ,E ,...,E (cid:105). i 1 i−1 i+1 Ei+1 i i+2 n As described in [1] twisted complexes allow to define mutations of DG quivers in such a way that Q = L Q and Q = R Q . In particular, it is relatively easy to calculate a DG quiver of a Liσ i σ Riσ i σ collections that can be mutated to a strong one. 1.4. Universal extensions and coextensions Let σ = (cid:104)E ,...,E (cid:105) be an exceptional collection. We shall say that σ is almost strong if 1 n Exti(E ,E )=0 for i(cid:54)=0, 1 and for all j, k. j k In[6]HilleandPerlingdescribehowtoconstructatiltingobjectfromanalmoststrongfullexceptional collection. The main tool in their construction is universal extension and coextension. For a pair (E,F) 4 A.BODZENTA a universal extension E¯of E by F is defined by means of a distinguished triangle E[−1]−c−a→n F ⊗Ext1(E,F)∗ →E¯→E. Dually, a universal coextension of F by E is an object F¯ defined by a distinguished triangle F →F¯ →E ⊗Ext1(E,F)−c−a→n F. IfF isexceptionalandExti(F,E)=0 forallithenExt1(E,F)∗ isnaturallyisomorphictoHom(F,E¯) and thus E is the cone of the canonical map F ⊗Hom(F,E¯)−c−a→n E¯→E. Dually, if E is exceptional and Exti(F,E) = 0 for all i then Ext1(E,F) is naturally isomorphic to Hom(F¯,E) and up to a shift F is the cone of the canonical map F¯ −c−a→n E ⊗Hom(F¯,E)→F[1]. Theseobservationsareusedin[1]tocalculateaDGquiverofanyalmoststrongexceptionalcollection. Again, twisted complexes play an important part in the calculations. 1.5. Rational surfaces LetX beasmoothrationalsurface. ThenX isobtainedbyasequenceofblow-upsfromtheprojective plane P2 or a Hirzebruch surface F . We have a sequence of maps a X =X πn (cid:47)(cid:47) X πn−1 (cid:47)(cid:47) ... (cid:47)(cid:47) X π1 (cid:47)(cid:47) X , n n−1 1 0 where X =P2 or F . We can also assume that every π is a blow up of one point x ∈X . 0 a i i−1 i−1 Let E ⊂ X be the exceptional divisor of π . Denote by E the strict transform of E in X and by i i i i i R ⊂X its pullback under π ...π . i i+1 n The divisors R are mutually orthogonal and R2 = −1. Hille and Perling in [5] introduce a partial i i order on the set of indices {1,...,n}; i(cid:23)j if i>j and π ...π (x )=x . Then j−1 i−1 i−1 j−1 (cid:40) C if i(cid:23)j, Hom(O (R ),O (R ))=Ext1(O (R ),O (R ))= X i X j X i X j 0, otherwise. Moreover, H0(O (R ))=C, Hj(O (R ))=0 for j >0 and Hk(O (−R ))=0 for all k. X i X i X i 2. DG quivers of exceptional collections on rational surfaces 2.1. Exceptional collections on rational surfaces Again, let X be a smooth rational surface. We recall the augmentation procedure given in [5] by Hille andPerlingwhichallowstoconstructfullexceptionalcollectionsoflinebundlesonX fromanexceptional collection on X . To simplify the notation we identify a line bundle L on X with its pull back via π ’s 0 i j and denote them by the same letter. Let σ =(cid:104)L ,...,L (cid:105) be a full exceptional collection of line bundles on X . The augmentation of σ is 1 s i σ(cid:48) =(cid:104)L (R ),...,L (R ),L , L (R ),L ,...L (cid:105) – an exceptional collection on X . 1 i+1 k−1 i+1 k k i+1 k+1 s i+1 It follows from a result of Orlov, [7] that collections obtained via augmentation are full. It is proved in [6] that they are almost strong. Mutations allow to present each of the above described collections in the following form. Proposition 2.1. Any exceptional collection of line bundles on X obtained via augmentation can be mutated to (cid:104)O (R )[−1],...,O (R )[−1],O ,N ,...,N (cid:105), where (cid:104)O = N ,N ,...,N (cid:105) is an Rn n R1 1 X 1 t X0 0 1 t exceptional collection on X . 0 DG QUIVERS OF SMOOTH RATIONAL SURFACES 5 Proof. The collection on X obtained via augmentation is of the form (cid:104)L (R ),...,L (R ),L ,L (R ),L ,...L (cid:105), 1 n i−1 n i i n i+1 s where L ’s are pull backs of line bundles on X . j n−1 Equality Hom(L ,L (R ))=Hom(O ,O (R ))=C i i n X X n and the short exact sequence 0→L →L (R )→O (R )→0 i i t Ri i show that this collection can be mutated to (cid:104)L (R ),...,L (R ),O (R )[−1],L ,L ,...L (cid:105). 1 n i−1 n Rn n i i+1 s Then Hom(L (R ),O (R ))=Hom(O (R ),O (R ))=Hom(O ,O )=C i n Rn n X n Rn n X Rn and exact sequences 0→L →L (R )→O (R )→0 i i n Rn n provide further mutations to (cid:104)O (R )[−1],L ,...,L (cid:105). Rn n 1 s The collection (cid:104)L ,...,L (cid:105) is a pull back of a collection on X and it again has 1 s n−1 the form (cid:104)L(cid:48)(R ),...,L(cid:48) (R ),L(cid:48),L(cid:48)(R ),L(cid:48) ,...,L(cid:48) (cid:105) for some k. As before, it 1 n−1 k−1 n−1 k k n−1 k+1 s−1 can be mutated to (cid:104)L(cid:48)(R ),...,L(cid:48) (R ), O (R )[−1],L(cid:48),...,L(cid:48) (cid:105) and then to 1 n−1 k−1 n−1 Rn−1 n−1 k s−1 (cid:104)O (R )[−1],L(cid:48),...,L(cid:48) (cid:105). Rn−1 n−1 1 s−1 Continuing, we can mutate the collection on X to (cid:104)O (R )[−1],...,O (R )[−1], O ,N ,...,N (cid:105). Rn n R1 1 X 1 t (cid:3) From now on we will assume that the collection (cid:104)O ,N ,...,N (cid:105) on X is strong. X0 1 t 0 2.2. Ext-quiver of (cid:104)O (R )[−1],...,O (R )[−1],O ,N ,...,N (cid:105). Rn n R1 1 X 1 t To draw the Ext-quiver of this collection in particular we need to understand the compositions Ext1(O (R ),O (R ))⊗Hom(O (R ),O (R ))→Ext1(O (R ),O (R )), Rj j Rk k Ri i Rj j Ri i Rk k Hom(O (R ),O (R ))⊗Ext1(O (R ),O (R ))→Ext1(O (R ),O (R )) Rj j Rk k Ri i Rj j Ri i Rk k for i(cid:23)j (cid:23)k. Denote by C the subcategory of Db(X) generated by objects O (R ),...,O (R ) and by C(cid:48) the Rn n R1 1 subcategory of Db(X) generated by O(R ),...,O(R ). Then C is a mutation of C(cid:48) over O and hence n 1 X understanding morphisms between generators of C is equivalent to understanding morphisms between generators of C(cid:48). Lemma 2.2. Let i(cid:23)j (cid:23)k. The composition Ext1(O (R ),O (R ))⊗Hom(O (R ),O (R ))→Ext1(O (R ),O (R )) X j X k X i X j X i X k is an isomorphism. Proof. The exact sequence 0→O (R )→O (R )→O (R )→0 X i X j Rj−Ri j gives α 0→Hom(O (R ),O (R ))→Hom(O (R ),O (R ))−→Hom(O (R ),O (R ))→ Rj−Ri j X k X j X k X i X k →Ext1(O (R ),O (R ))→Ext1(O (R ),O (R ))−→β Ext1(O (R ),O (R ))→ Rj−Ri j X k X j X k X i X k →Ext2(O (R ),O (R ))→0. Rj−Ri j X k The morphism α: Hom(O (R ),O (R )) → Hom(O (R ),O (R )) is an isomorphism because its X j X k X i X k kernel is zero and both spaces are one dimensional. 6 A.BODZENTA β is an isomorphism if and only if Ext1(O (R ),O (R )) is zero. Rj−Ri j X k We have a short exact sequence 0→O (R )→O (R +R )→O (R +R )(cid:39)O (R )→0. Rj−Ri j Rj j i Ri j i Ri i It is easy to check that Ext1(O (R ),O (R ))=C. From short exact sequences Ri i Rk k 0→O (R )→O (R +R )→O (R +R )→0, X i X i j Rj i j 0→O (R )→O (R +R )→O (R +R )(cid:39)O (R )→0 X j X i j Ri i j Ri i we deduce that Ext1(O (R +R ),O (R ))(cid:39)Ext1(O (R +R ),O )(cid:39)Ext1(O (R +R ),O )=C. Rj j i X k Rj j i X X j i X It follows that Ext1(O (R ),O (R ))=0. (cid:3) Rj−Ri j X k Remark 2.3. If i(cid:23)j (cid:23)k the composition Hom(O (R ),O (R ))⊗Ext1(O (R ),O (R ))→Ext1(O (R ),O (R )) X j X k X i X j X i X k does not have to be an isomorphism. Indeed, consider a surface X obtained from its minimal model by three blow-ups such that E2 = −3, E2 = −2, E2 = −1, E E = 0, E E = 1 and E E = 1. 1 2 3 1 2 1 3 2 3 Then R = E + E + 2E , R = E + E and R = E . Let α¯ ∈ Ext1(O (R ),O (R )) and 1 1 2 3 2 2 3 3 3 X 3 X 2 β ∈Hom(O (R ),O (R )) be non-zero elements. We have a short exact sequence X 2 X 1 β 0→O (E +E )−→O (E +E +2E )→O (E +E +2E )→0. X 2 3 X 1 2 3 E1+E3 1 2 3 Asintheproofofthepreviouslemmaβ◦α¯ =0ifandonlyifHom(O (E ),O (E +E +2E ))(cid:54)=0. X 3 E1+E3 1 2 3 We have Hom(O (E ),O (E +E +2E ))=H0(X,O (E +E +E )). The latter sheaf fits X 3 E1+E3 1 2 3 E1+E3 1 2 3 into a short exact sequence 0→O (cid:39)O (E +E )→O (E +E +E )→O (E +E +E )(cid:39)O (−2)→0 E3 E3 2 3 E1+E3 1 2 3 E1 1 2 3 E1 from which it follows that H0(X,O (E +E +E ))=C. E1+E3 1 2 3 Thus, we know that between O (R ) and O (R ) there is either no arrow or two arrows, one in Ri i Rj j degree zero and one in degree one. Moreover, β¯◦α(cid:54)=0 and β◦α(cid:54)=0 for α (cid:47)(cid:47) β (cid:47)(cid:47) O (R ) (cid:47)(cid:47) O (R ) (cid:47)(cid:47) O (R ), Ri i Rk k Rj j α¯ β¯ where α,β are non-zero morphisms and α¯,β¯ are non-zero elements of the first Ext groups. ItremainstounderstandwhatarethemapsfromO (R )toN andthecompositionsbetweenthem. Rk k i As N are torsion-free we know that Hom(O (R ),N )=0. From the short exact sequence i Rk k i (1) 0→N →N ⊗O (R )→O (R )→0. i i X k Rk k wededucethatExt1(O (R ),N )(cid:39)Hom(O (R ),O (R ))=C. Letζi denotethenon-zeroelement Rk k i Rk k Rk k k of the group Ext1(O (R ),N ). Rk k i DG QUIVERS OF SMOOTH RATIONAL SURFACES 7 Moreover, the diagram 0 0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) 0 N N ⊗O (R ) O (R ) 0 i i X j Rj j = (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) 0 N N ⊗O (R ) O (R ) 0 i i X k Rk k (cid:15)(cid:15) (cid:15)(cid:15) = (cid:47)(cid:47) O (R ) O (R ) Rk−Rj k Rk−Rj k (cid:15)(cid:15) (cid:15)(cid:15) 0 0 shows that the composition Hom(O (R ),O (R ))⊗Ext1(O (R ),N )→Ext1(O (R ),N ) Rj j Rk k Rk k i Rj j i is an isomorphism. To understand the composition Ext1(O (R ),N )⊗Hom(N ,N )→Ext1(O (R ),N ) Rk k i i l Rk k l we apply the functor Hom(−,N ) to the short exact sequence (1). It follows that for φ ∈ Hom(N ,N ) l i l the composition φ◦ζi is zero if and only if φ factors through N (−R ). k l k 2.3. DG quiver of (cid:104)O (R )[−1],...,O (R )[−1],O ,N ,...,N (cid:105) Rn n R1 1 X 1 t Now, we will present calculations allowing to determine the DG quiver of the collection (cid:104)O (R )[−1],...,O (R )[−1],O,N ,...,N (cid:105). Recall, that we work under the assumption that the Rn n R1 1 1 t collection (cid:104)O ,N ,...,N (cid:105) on X is strong. X0 1 t 0 To calculate the DG category of the collection (cid:104)O (R )[−1],...,O (R )[−1],O , N ,...,N (cid:105) we Rn n R1 1 X 1 t substitute some objects with universal coextensions. 2.3.1. Tilting object Note that if 2(cid:23)1 then we have a unique nontrivial extension 0→O (R )→O (R +R )→O (R )→0. R1 1 R1+R2 1 2 R2 2 Hence O (R +R ) is the universal coextension of O (R ) by O (R ). R1+R2 1 2 R1 1 R2 2 We will show that for i (cid:23)...(cid:23)i (cid:23)s the universal coextension of O (R +...+R ) by k 1 Ri1+...+Rik i1 ik O (R ) is O (R +R +...+R ). Rs s Rs+Ri1+...+Rik s i1 ik Proposition 2.4. Let (cid:104)O (R )[−1],...,O (R )[−1],O ,N ,...,N (cid:105) be an exceptional collection on Rn n R1 1 X 1 t X such that (cid:104)O ,N ,...,N (cid:105) is a strong exceptional collection on X . Then X0 1 t 0 O (S )[−1] ⊕ O (S )[−1]⊕...⊕O (S )[−1] ⊕ O ⊕ N ⊕...⊕N Sn n Sn−1 n−1 S1 1 X 1 t is tilting on X, where S are defined as k (cid:88) S = R . k j j(cid:23)k To prove Proposition 2.4 we shall need the following Lemma. 8 A.BODZENTA Lemma 2.5. For i(cid:23)k (cid:23)l we have Hom(O (R ),O (R +...+R ))(cid:39) Ri i Rl+...+Rk l k Hom(O (R ),O (R ))⊗Hom(O (R ),O (R +...+R ))=C, Ri i Rk+1 k+1 Rk+1 k+1 Rl+...+Rk l k Hom(O (R ),O (R +R +...+R ))(cid:39) Ri i Rl+Rl+1+...+Rk l l+1 k Ext1(O (R ),O (R ))⊗Ext1(O (R ),O (R +...+R ))=C, Ri i Rk+1 k+1 Rk+1 k+1 Rl+...+Rk l k where the sum R +...+R is taken over all divisors R such that k (cid:23)j (cid:23)l. l k j Proof. We proceed by induction. The basis case, for k =l follows from Lemma 2.2. The induction step follows from applying the functor Hom(O (R ),−) to the short exact sequence Ri i (2) 0→O (R +...+R )→O (R +...+R )→O (R )→0. Rl+...+Rk−1 l k−1 Rl+...+Rk l k Rk k (cid:3) Proof of Proposition 2.4. From the above lemma and the short exact sequence (2) it follows that if i(cid:23)k (cid:23)lthesheafO (R +...+R )istheuniversalcoextensionofO (R +...+R ) Rl+...+Rk l k Rl+...+Rk−1 l k−1 by O (R ). Hence, by the construction described in [6] the object Ri i O (S )[−1] ⊕ O (S )[−1]⊕...⊕O (S )[−1] ⊕ O ⊕ N ⊕...⊕N Sn n Sn−1 n−1 S1 1 X 1 t is tilting on X. (cid:3) Endomorphisms of the tilting object depend not only on the order of the divisors but also on the mutual position of the exceptional curves. Consider the blow-up with the following mutual position of exceptional divisors. E3 E2 E1 Then E2 =−2, E2 =−2, E2 =−1, 1 2 3 E E =1, E E =0, E E =1 R =E +E +E , R =E +E , R =E , 1 2 1 3 2 3 1 1 2 3 2 2 3 3 3 and the order is 3(cid:23)2(cid:23)1. The endomorphisms of O (R )⊕O (R +R )⊕O (R +R +R ) are R3 3 R2+R3 2 3 R1+R2+R3 1 2 3 α3(cid:47)(cid:47) α2(cid:47)(cid:47) O (R ) (cid:111)(cid:111) O (R +R ) (cid:111)(cid:111) O (R +R +R ) R3 3 R2+R3 2 3 R1+R2+R3 1 2 3 β3 β2 with β ◦α =0, α ◦β =β ◦α . 3 3 3 3 2 2 However, if the picture is E2 E3 E1 DG QUIVERS OF SMOOTH RATIONAL SURFACES 9 then E2 =−3, E2 =−2, E2 =−1, 1 2 3 E E =0, E E =1, E E =1, 1 2 1 3 2 3 R =E +E +2E , R =E +E , R =E , 1 1 2 3 2 2 3 3 3 the order is still 3(cid:23)2(cid:23)1. and the endomorphisms of the tilting object are α3 (cid:47)(cid:47) α2 (cid:47)(cid:47) O (R ) (cid:111)(cid:111) O (R +R ) (cid:111)(cid:111) O (R +R +R ) R3 3 R2+R3 2 3 R1+R2+R3 1 2 3 β3 β2 with β ◦α =0, β ◦α =0. 3 3 2 2 2.3.2. Ext1(O (S ),N ) Sk k i Lemma 2.6. Let i (cid:23)i (cid:23)...(cid:23)i . Then k k−1 1 Ext1(O (R +...+R ),N )=Ck Ri1+...+Rik i1 ik i and the remaining Ext groups are zero. Proof. We proceed by induction. The short exact sequence 0→O (R +...+R )→O (R +...+R )→O (R )→0 Ri1+...+Rik−1 i1 ik−1 Ri1+...+Rik i1 ik Rik ik together with an equality Exti(O (R ),N )=Exti(O (E ),N ) Rik ik i Eik ik i completes the proof. (cid:3) If we apply the functor Hom(O (S ),−) to the short exact sequence Sk k 0→N →N ⊗O (S )→O (S )→0 i i X k Sk k we get an isomorphism (3) Ext1(O (S ),N )(cid:39)Hom(O (S ),O (S )). Sk k i Sk k Sk k The identity morphism in the latter space corresponds to an element ζi ∈Ext1(O (S ),N ). k Sk k i The diagram 0 0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) 0 N N ⊗O (S ) O (S ) 0 i i X k Sk k = ι (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) 0 N N ⊗O (S ) O (S ) 0 i i X l Sl l (cid:15)(cid:15) (cid:15)(cid:15) = (cid:47)(cid:47) O (S ) O (S ) Sl−Sk l Sl−Sk l (cid:15)(cid:15) (cid:15)(cid:15) 0 0 shows that for an inclusion ι: O (S )→O (S ) we have ζi◦ι=ζi. Sk k Sl l l k 10 A.BODZENTA The isomorphism (3) allows also to calculate the Yoneda composition Hom(N ,N )⊗Ext1(O (S ),N )→Ext1(O (S ),N ). i k Sk k i Sk k k Thus, if the collection (cid:104)O ,N ,...,N (cid:105) on X is strong we know the endomorphism algebra of the X0 1 t 0 tilting object O (S )[−1] ⊕ O (S )[−1]⊕...⊕O (S )[−1] ⊕ O ⊕ N ⊕...⊕N . Sn n Sn−1 n−1 S1 1 X 1 t UsingtwistedcomplexesonecanthencalculatetheDGquiverofthecollection(cid:104)O (R )[−1],...,O (R )[−1], Rn n R1 1 O ,N ,...,N (cid:105) and of any of its mutations. X 1 t 3. Canonical DG algebras of toric surfaces 3.1. Toric surfaces We recall some information about toric surfaces. More details can be found for example in [4]. A smooth projective toric surface Y is determined by its fan, spanned by a collection of elements ρ ,...,ρ inalatticeN =Hom(C∗,T)∼=Z2,whereT =(C∗)2 isatwo-dimensionaltorus. Weenumerate 1 n ρ ’s clockwise and consider their indexes, i’s, to be elements of Z/nZ. Then, for every i∈Z/nZ, vectors i ρ and ρ form an oriented basis of N. Moreover, for every such pair there is no other ρ lying in the i i+1 k rational polyhedral cone generated by ρi and ρi+1 in NQ =N ⊗Q. There is a one-to-one correspondence between one-dimensional orbits of the T-action on Y and the rays in the fan generated by ρ ’s. For every i we denote by D the closure of this orbit. Then D ’s are i i i T-invariant divisors on X. Every D is isomorphic to P1 and the intersection form is given by i  a if i=j,  i D D = 1 if j ∈{i−1,i+1} i j  0 otherwise, where a ∈ Z are such that ρ +a ρ +ρ = 0. Conversely, the numbers (a ,...,a ) determine the i i−1 i i i+1 1 n toric surface Y. Divisors D and D intersect transversely in a T-fixed point p corresponding to the cone spanned i i+1 i by vectors ρ and ρ . i i+1 A surface Y obtained from Y by a blow-up of a torus-fixed point p is again a toric surface. The fan 1 i of Y is determined by vectors ρ ,...,ρ ,ρ +ρ ,ρ ,...,ρ . Moreover, every toric surface different 1 1 i i i+1 i+1 n from P2 can be obtained from some Hirzebruch surface F by a finite sequence of blow-ups of T-fixed a points. A canonical divisor of a toric surface is given by K = −(cid:80)n D . The Picard group of Y is Y i=1 i Pic(Y)=Zn−2. 3.2. Exceptional collections on toric surfaces The a-th Hirzebruch surface F has a fan with four vectors and we can assume that w =(1,0), w = a 1 2 (0,−1),w3 =(−1,a)andw4 =(0,1). Thecollection(cid:104)OFa,OFa(D1),OFa(D1+D2),OFa(D1+D2+D3)(cid:105) is a full strong exceptional collection on F . a IfY isobtainedfromF byasequenceofT-equivariantblow-upsthenwecanassumethatthevectors a ρ ,...,ρ determining Y are numbered in such a way that ρ = w = (0,1). Then the collection 1 n n 4 (cid:104)O ,O (D ),O (D +D ),...,O (D +...+D )(cid:105) on Y is obtained by augmentation from the Y Y 1 Y 1 2 Y 1 n−1 strong collection on F and hence it is full. The following lemma tells us that in fact the numeration of a T-invariant divisors is not important. Lemma 3.1 (cf. Theorem 4.1 of [2]). Let (cid:104)E ,...,E (cid:105) be a full exceptional collection on a smooth 1 n projective variety Z of dimension m. Then the n-fold mutation of E to the left, LnE =E ⊗ω [m−n], n n n Z where ω is the canonical line bundle on Z. Z