On cyclic strong exceptional collections of line bundles on surfaces
aa r X i v : . [ m a t h . AG ] J u l ON CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLESON SURFACES
ALEXEY ELAGIN, JUNYAN XU, AND SHIZHUO ZHANG
Abstract.
We study exceptional collections of line bundles on surfaces. We prove that any full cyclicstrong exceptional collection of line bundles on a rational surface is an augmentation in the sense ofL. Hille and M. Perling. We find simple geometric criteria of exceptionality (strong exceptionality,cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, weclassify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles.Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptionalcollection of line bundles on it which does not come from augmentations, thus answering a questionby Hille and Perling.
Contents
1. Introduction 12. Divisors on surfaces and their properties 63. Toric systems and admissible sequences 124. Operations with toric systems 175. Toric systems of the first kind are augmentations 216. Surfaces with cyclic strong exceptional toric systems 267. An application to dimension of D b (coh X ) 298. A toric system which is not an augmentation 30Appendix A. Classification of weak del Pezzo surfaces 33References 341. Introduction
Our paper is devoted to the study of exceptional collections of line bundles on surfaces. By asurface we always mean a smooth connected projective surface over an algebraically closed field ofzero characteristic. Among the questions that are addressed in this paper are the following
Question 1.
Which surfaces admit exceptional/strong exceptional/cyclic strong exceptional col-lections of line bundles?
Question 2.
How to construct exceptional/strong exceptional/cyclic strong exceptional collec-tions of line bundles if they exist?
Question 3.
How to tell whether a given collection of line bundles on a surface is excep-tional/strong exceptional/cyclic strong exceptional?It is believed that any variety with a full exceptional collection in the bounded derived categoryof coherent sheaves is rational. Though, there is no proof yet even in the case when the collectionis formed by line bundles. On the other hand, on any rational surface one can construct a fullexceptional collection of line bundles, using a construction by Dmitry Orlov [Or93].
Mathematics Subject Classification.
Key words and phrases.
Weak del Pezzo surface, exceptional collection, line bundle.
For strong exceptional collections of line bundles the question is much more complicated. First,it is known that any surface with a full strong exceptional collection of line bundles is rational,see [BS17]. It was conjectured by Alastair King [Ki97] that any smooth toric variety has a strongexceptional collection formed by line bundles. In [HP06] Lutz Hille and Markus Perling constructeda surface which is a counterexample. Later in [HP11, Theorem 8.2] they established a criteriondetermining whether a toric surface admits a full strong exceptional collection of line bundles or not.It was demonstrated that such a collection exists if and only if the toric surface can be obtained fromsome Hirzebruch surface by two blow-up operations: on each step one can blow up several distinctpoints. Also it was explained that on any (not necessarily toric) rational surface obtained from aHirzebruch surface by two blow-up operations there exists a full strong exceptional collection of linebundles, see Theorem 5.9 in [HP11]. Therefore one arrives at a reasonable
Conjecture 1.1 ([HP11]) . A rational surface X has a full and strong exceptional collection of linebundles in the derived category if and only if X can be obtained from some Hirzebruch surface by atmost two steps of blowing up points (maybe several at each step). Of special interest are cyclic strong exceptional collections of line bundles. Recall that a strongexceptional collection ( O X ( D ) , . . . , O X ( D n )) is cyclic strong if it remains strong exceptional after“cyclic shifts”. That is, any segment ( O X ( D k +1 ) , . . . , O X ( D k + n )) in the infinite helix . . . , O X ( D i ) , . . . ,defined by the rule D i + n = D i − K X , is also strong exceptional. Cyclic strong exceptional collections ofline bundles and the tilting bundles formed by them have attracted much attention by specialists fromdifferent areas of mathematics and have been studied under different names (pull-back exceptionalcollections in [Br05], [BF12], [VdB04], first order approximation to pull-back exceptional collectionsin [BF12], 2-hereditary exceptional collections in [Ch16]). For example, let X be a smooth delPezzo surface and Y be the total space of the canonical line bundle on X . Then a full cyclic strongexceptional collection of line bundles on X will pull back to a tilting bundle T on Y . The algebraEnd( T ) gives a noncommutative crepant resolution of the anti-canonical cone of X in the sense ofMichel Van den Berg, see [VdB04].In Hille and Perling’s paper [HP11] the following results concerning full cyclic strong exceptionalcollections of line bundles were obtained: • if a rational surface X has a full cyclic strong exceptional collection of line bundles thendeg( X ) := K X > • any del Pezzo surface X with deg( X ) > • a toric surface X has a full cyclic strong exceptional collection of line bundles if and only if − K X is numerically effective.In this paper we generalize the above results and give a complete classification of surfaces admittinga full cyclic strong exceptional collection of line bundles, answering Question 1 for such collections.First, any such surface is a weak del Pezzo surface. Recall that a weak del Pezzo surface is a smoothprojective surface X such that K X > − K X is numerically effective. Alternatively, X is theminimal resolution of singularities on a singular del Pezzo surface having only rational double pointsas singularities. Weak del Pezzo surfaces are rational, they are distinguished by their type (seeSection 2.2). Second, for any type of weak del Pezzo surfaces we determine whether they possess afull cyclic strong exceptional collection of line bundles. Theorem 1 (Propositions 6.2 and 6.3) . Let X be a smooth projective surface having a cyclic strongexceptional collection of line bundles of maximal length. Then X is a weak del Pezzo surface and X is one of the surfaces from Table 5. Moreover, any weak del Pezzo surface from Table 5 possesses afull cyclic strong exceptional collection of line bundles. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 3
Basing on Theorem 1, in a recent paper [Zh18] some new examples of non-commutative crepantresolutions in the sense of [VdB04] were constructed.For the study of exceptional collections of line bundles, we use two notions, proposed by Hille andPerling. First, it is convenient to pass from an exceptional collection( O X ( D ) , . . . , O X ( D n ))of line bundles (where D i are divisors on X ) to the sequence A , . . . , A n − of differences: A k = D k +1 − D k for k = 1 , . . . , n −
1. It is useful also to complete the sequence D , . . . , D n to an infinite helix D i , i ∈ Z , by the rule D k + n = D k − K X for all k , and to add the term A n = D n +1 − D n = D − K X − D n to the sequence A , . . . , A n − . Thus one gets a sequence A = ( A , . . . , A n ) , A i ∈ Pic( X ) , which contains essentially the same information as the original collection ( O X ( D ) , . . . , O X ( D n )).Elements A , . . . , A n generate Pic( X ) and have nice combinatorial properties (where indices aretreated modulo n ): • A i · A i +1 = 1; • A i · A j = 0 if j = i, i ± • P i A i = − K X .These properties comprise the definition of a toric system A , . . . , A n , see Definition 3.1.To answer Question 2, we use the second discovery of [HP11], the notion of augmentation. Thisoperation enables, starting from a toric system A = ( A , . . . , A n ) on some surface X , to obtain atoric system on the blow-up X ′ of X at a point. Namely, let 1 m n + 1 be an index and let E denote the exceptional divisor of the blow-up. Then the augmented toric system is the sequenceaugm m ( A ) = ( A ′ , . . . , A ′ m − , A ′ m − − E, E, A ′ m − E, A ′ m +1 , . . . , A ′ n )in Pic( X ′ ), where A ′ i is the pull-back of A i . One can start from some toric system on a Hirzebruchsurface and perform several blow-ups, augmenting the toric system at each step. The resulting toricsystem is called a standard augmentation . This construction gives us some (quite many) explicitexamples of toric systems on rational surfaces.It is natural to ask a question: does any full exceptional collection of line bundles correspondto a toric system which is a standard augmentation? The above cannot be literally true, but isclose to being true if we do not distinguish between exceptional collections which differ only by theordering of line bundles inside blocks of completely orthogonal bundles. Reordering of two mutuallyorthogonal line bundles in the exceptional collection corresponds to an operation with toric systemswhich we call transposition . By a permutation of a toric system we mean a composition of severaltranspositions, see Section 4 for details.A positive answer to the above question means that we are able to find all full exceptional collectionsof line bundles: any collection on a rational surface X is a standard augmentation for some minimalmodel X → F d and some exceptional collection on F d . For any fixed minimal model and exceptionalcollection on F d there exists only a finite number of such standard augmentations.Besides that, a positive answer to the above question helps to solve Conjecture 1.1. For standardaugmentations Hille and Perling proved the following: a toric system on a rational surface X whichis a standard augmentation along some sequence of blow-ups X → . . . → F d corresponds to a strongexceptional collection only if X was obtained from the Hirzebruch surface F d in at most two blow-ups(each time one can blow up several different points).Motivated by that result, Hille and Perling made the following Conjecture 1.2.
Any full strong exceptional collection of line bundles on a rational surface corre-sponds (up to permutation of completely orthogonal bundles) to a standard augmentation.
ALEXEY ELAGIN, JUNYAN XU, AND SHIZHUO ZHANG
This conjecture can be viewed as a “homological version” of the minimal model program forrational surfaces.Note that Conjecture 1.2 implies Conjecture 1.1.Hille and Perling proved Conjecture 1.2 for toric surfaces, thus proving for toric surfaces Conjec-ture 1.1. Andreas Hochenegger and Nathan Ilten in [HI13, Main Theorem 3] proved Conjecture 1.2for toric surfaces of Picard rank Theorem 2 (See Theorem 5.3 and Corollary 5.4 for the precise statements) . Let X be a smooth ra-tional projective surface and A = ( A , . . . , A n ) be a toric system on X . Suppose that A is numericallycyclic strong: that is, the Euler characteristic χ ( X, A i ) > for all i . Then up to some permutations A is a standard augmentation. In particular, any full cyclic strong exceptional collection of line bundleson X corresponds (up to some permutations) to a toric system which is a standard augmentation. Therefore, one is able to list all full cyclic strong exceptional collections on a given surface. Thisgives an answer to Question 2 for full cyclic strong exceptional collections.Our initial idea was to investigate Conjecture 1.2 for weak del Pezzo surfaces. We have provedConjecture 1.2 for all weak del Pezzo surfaces of degree > Theorem 3.
Let X be a weak del Pezzo surface of degree > . Then any full strong exceptionalcollection of line bundles on X corresponds (up to some permutations) to a standard augmentation. The proof is heavily technical and uses some machine computations, we do not give it here. It canbe found in the extended version of this paper, see [EXZ17].What is more interesting, we found that in general Conjecture 1.2 is false . That is, we have foundcounterexamples to Conjecture 1.2 for some weak del Pezzo surfaces of degree 2.
Theorem 4 (See Section 8 for details) . Let X be a weak del Pezzo surface of degree of type A +2 A .There exists a full strong exceptional collection of line bundles on X such that the corresponding toricsystem is not a standard augmentation (up to any permutations). The example from Theorem 4 was found using a computer, as well as many other counterexamplesto Conjecture 1.2. It should be noted that any of the surfaces where a counterexample was foundcan be obtained from a Hirzebruch surface by two blow-ups. This gives some evidence supportingConjecture 1.1. It is interesting also that all surfaces where we found a counterexample to Con-jecture 1.2 have holes in the effective cone: such non-effective divisor classes which have a positivemultiple being effective.There are some examples known in the literature of strong exceptional collections of line bundleswhich are not standard augmentations, see [HI13] and [Ho13]. But some permutations in the citedcollections are standard augmentations, thus the cited examples agree with Conjecture 1.2. SeeRemark 4.15 for more details.In order to construct such a counterexample or to construct examples of cyclic strong exceptionaltoric systems (see Theorem 1) one needs to have a reasonable answer to Question 3: how to checkthat a given collection of line bundles on a surface is exceptional/strong exceptional? A priori oneneeds to check cohomology vanishing for lots of line bundles. To do this effectively, we propose asimple geometric criterion for exceptionality of toric systems, see Theorem 3.10. For cyclic strongexceptional toric systems it says:
Theorem 5.
Let X be a weak del Pezzo surface. Let A = ( A , . . . , A n ) be a toric system on X .Then A corresponds to a cyclic strong exceptional collection of line bundles if and only if the followingholds: N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 5 • A i > − for all i = 1 , . . . , n ; • for any cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] (see Definition 3.5) such that A k = A k +1 = . . . = A l − = A l = − the divisor A k + . . . + A l is neither effective nor anti-effective. Note that the divisors D = ± ( A k + . . . + A l ) in the above theorem are ( − D = − , D · K X = 0. It is a simple observation that such D is effective if and only ifit is a sum of some ( − X . For any type of weak del Pezzo surfaces the configurationof ( − D b (coh X ) of coherent sheaves on a smooth projectivevariety X and dimension of the category D b (coh X ) in the sense of Rouquier. They demonstrated,in particular, that a full strong exceptional collection ( E , . . . , E n ) in D b (coh X ) is cyclic strong if thegenerator ⊕ i E i has generation time equal to dim X . For collections of line bundles on surfaces weare able to provide a sort of converse statement. Theorem 6 (Proposition 7.4) . Let X be a smooth projective surface with a full cyclic strong excep-tional collection ( L , . . . , L n ) of line bundles. Then the generator ⊕ i L i of the category D b (coh X ) hasthe generation time two. In particular, for any surface X from Table 5 the category D b (coh X ) hasdimension two. The above theorem provides a new class of varieties X for which the dimension of D b (coh X ) equalsto the dimension of X . This supports a conjecture by Orlov saying that dim D b (coh X ) = dim X forall smooth projective varieties X .Let us briefly mention one other application of our results. Basing on Theorems 1 and 2, it isdemonstrated by the third author in [Zh18, Theorem 1.9] that a full exceptional collection of linebundles on a smooth projective surface is cyclic strong exceptional iff it is a pull-back exceptionalcollection ([Br05], [BF12], [VdB04]), iff it is a first order approximation to a pull-back exceptionalcollection in the sense of [BF12] and iff it is a 2-hereditary exceptional collection ([Ch16]).The text is organized as follows. Sections 2,3, and 4 contain preliminary material. Here werecall, introduce and discuss necessary notions: divisors on surfaces, their orthogonality properties, r -classes, weak del Pezzo surfaces, exceptional collections and toric systems, admissible sequences,augmentations, transpositions. This material is partially known and/or contained in the paper [HP11]by Hille an Perling. Let us point out what is new.First, it is Proposition 2.9 providing a handy geometric tool for checking whether a given divisoron a weak del Pezzo surface is left-orthogonal or strong left-orthogonal. Further, a new results isTheorem 3.10. This theorem, in terms of the effective cone, gives an instrument for checking ((cyclic)strong) exceptionality of a given toric system on a weak del Pezzo surface.Also, in Section 4.2 we define and discuss different variants of augmentation. We distinguishbetween elementary augmentations, standard augmentations, augmentations in the weak sense, ex-ceptional augmentations, strong exceptional augmentations, and cyclic strong exceptional augmen-tations. Standard augmentations are the most natural but they do not exhaust all toric systems onmost surfaces if we do not allow permutations. Augmentations in the weak sense seem to be a goodnotion if we do not care about homological properties of collections like exceptionality. We arguethat strong exceptional augmentations are suitable for accurate formulation of Conjecture 1.2: anystrong exceptional toric system is a strong exceptional augmentation.Note here that terminology in [HP11] is different: they say that a strong exceptional collectionhas a normal form that is a standard augmentation. Normal form of a collection is obtained from ALEXEY ELAGIN, JUNYAN XU, AND SHIZHUO ZHANG the original collection by permutations. Unfortunately, the corresponding definitions in [HP11] arenot rigorous enough, this forced us to introduce the certain notions of augmentations.In Section 5 we treat toric systems of the first kind: such toric systems ( A , . . . , A n ) that χ ( A i ) > i . For example, toric systems corresponding to cyclic strong exceptional collections are of thefirst kind. Essentially, we prove that toric systems of the first kind are augmentations in certainsense. In particular, we prove Theorem 2.Section 6 is devoted to the proof of Theorem 1.In Section 7 we give an application of our results to the study of dimension of derived categoriesof coherent sheaves. In this section we prove Theorem 6.In Section 8 we present a counterexample to Conjecture 1.2.Finally, in Appendix we present classification of weak del Pezzo surfaces of degree >
3. It is neededfor Section 6.
Acknowledgements.
The authors would like to thank Valery Lunts, collaboration with whominitiated this project. We thank Michael Larsen for valuable remarks on the text. The first author isgrateful to Indiana University for its hospitality. The third author would like to thank Li Tang for hissupport. Finally, we are extremely grateful to the referees for the careful reading of the manuscriptand for many useful remarks and suggestions.2.
Divisors on surfaces and their properties
Throughout the paper we assume that the base field k is algebraically closed and has zero charac-teristic. All surfaces we consider are supposed to be smooth projective and connected.2.1. Divisors and r -classes. Let X be a smooth projective surface over k . Let K X be a canonicaldivisor on X . Let d = K X be the degree of X , further we always assume that d >
0. For a divisor D on X , we will use the following shorthand notations: H i ( D ) := H i ( X, O X ( D )) , h i ( D ) = dim H i ( D ) , χ ( D ) = h ( D ) − h ( D ) + h ( D ) . By the Riemann-Roch formula, one has χ ( D ) = χ ( O X ) + D · ( D − K X )2 . The following notions are introduced by Lutz Hille and Markus Perling in [HP11, Definition 3.1].
Definition 2.1. • A divisor D on X is numerically left-orthogonal if χ ( − D ) = 0. • A divisor D on X is left-orthogonal if h i ( − D ) = 0 for all i . • A divisor D on X is strong left-orthogonal if h i ( − D ) = 0 for all i and h i ( D ) = 0 for i = 0. Definition 2.2.
We call D an r -class if D is numerically left-orthogonal and D = r .Motivation: if C ⊂ X is a smooth rational irreducible curve, then the class of C in Pic( X ) is an r -class where r = C .If C is a smooth rational irreducible curve on X and r = C , it is said that C is an r -curve . An r -curve is negative if r < Proposition 2.3.
Let X be a surface with χ ( O X ) = 1 . Then a divisor D on X is numericallyleft-orthogonal if and only if D + 2 = − D · K X . In particular, D is an r -class if and only if D = r, D · K X = − r − . N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 7
For any numerically left-orthogonal divisor D on X one has χ ( D ) = D + 2 = − D · K X . Proposition 2.4.
Let X be a surface with χ ( O X ) = 1 . Suppose D , D are numerically left-orthogonal divisors on X . Then D + D is numerically left-orthogonal if and only if D D = 1 . Ifthat is the case, then χ ( D + D ) = χ ( D ) + χ ( D ) and ( D + D ) = D + D + 2 . Lemma 2.5.
Let D be an r -class on a surface X of degree d with χ ( O X ) = 1 . Then D ′ = − K X − D is an r ′ -class where r + r ′ = d − .Proof. Indeed, ( − K X − D ) + ( − K X − D ) · K X = D + D · K X = − D ′ is numerically left-orthogonal by Proposition 2.3. Also, using Proposition 2.3 we get( D ′ ) = K X + 2 D · K X + D = d + ( − r −
4) + r = d − − r . (cid:3) Denote the set of ( − X by I ( X ) and the set of ( − X by R ( X ). As wewill see, I ( X ) and R ( X ) depend only on deg X and do not depend on geometry of X .Suppose that X is a blow-up of P at n (maybe infinitesimal) points. Then the Picard groupPic( X ) of X is a finitely generated abelian group with the standard basis L, E , . . . , E n . We willoften use the following shorthand notation:(2.1) E i ...i k := E i + . . . + E i k , L i ...i k := L − E i ...i k . The intersection form in the basis
L, E , . . . , E n is given by L = 1 , E i = − , L · E i = 0 , E i · E j = 0for i = j . We have K X = − L + E + . . . + E n and deg( X ) = 9 − n . That is, the abelian groupPic( X ) with the intersection form and the distinguished element K X depend only on n and donot depend on the geometry of X . Since I ( X ) and R ( X ) are determined by numerical conditions D · K X = D = − D · K X = 0 , D = − X ) anddo not depend on geometry of X . If deg( X ) > n R ( X ) is a root system in somesubspace in N X = ( K X ) ⊥ ⊂ Pic( X ) ⊗ R (by [Ma74, Theorem 23.9] for 1 deg( X )
6, trivial fordeg( X ) = 7 , , X ) R ( X ) spans N X . Table 1.
Root systems R ( X ) for blow-ups X of P for 1 deg( X ) X ∅ ∅ A A + A A D E E E | R ( X ) | | I ( X ) | Definition 2.6.
Denote by R eff ( X ) ⊂ R ( X ) the subset of effective ( − R irr ( X ) ⊂ R eff ( X ) and I irr ( X ) ⊂ I ( X ) the subsets of classes of ( − − R slo ( X ) ⊂ R lo ( X ) ⊂ R ( X ) the subsets of strong left-orthogonal and left-orthogonal( − R eff ( X ) , R irr ( X ) , I irr ( X ) , R lo ( X ) and R slo ( X ) essentially depend on the surface X , incontrast with R ( X ). ALEXEY ELAGIN, JUNYAN XU, AND SHIZHUO ZHANG
Weak del Pezzo surfaces.
By definition, a weak del Pezzo surface is a smooth connectedprojective rational surface X such that K X > − K X is nef. A del Pezzo surface is a smoothconnected projective rational surface X such that − K X is ample. We refer to Igor Dolgachev [Do12,Chapter 8] or Ulrich Derenthal [De14] for the main properties of weak del Pezzo surfaces. A weakdel Pezzo surface is a del Pezzo surface if and only if it has no ( − X except for Hirzebruch surfaces F and F is a blow-up of P at several (maybe infinitesimal)points. That is, there exists a sequence X = X n p n −→ X n − → . . . → X p −→ X = P of n blow-ups, where p k is the blow-up of point P k ∈ X k − . Moreover, the surface X n as aboveis a weak del Pezzo surface if and only if n k the point P k does not belong to a( − X k − . Lemma 2.7.
Let C ⊂ X be a reduced irreducible curve with C < on a weak del Pezzo surface X .Then C is a smooth rational curve and C = − or − . If X is a del Pezzo surface then C = − .Proof. By the Riemann-Roch formula, one has χ ( O C ) = χ ( O X ) − χ ( O X ( − C )) = − C + C · K X . Since C is irreducible and reduced, one has h ( O C ) = 1. Hence0 h ( O C ) = 1 + C + C · K X . Note that C · K X − K X is nef and C < h ( O C ) = 0 and p a ( O C ) = 0. Hence C is rational and smooth by [Ha77, Exer. IV.1.8]. Moreover, C + C · K X = − C = − −
1. If X is a del Pezzo surface, then C · K X < C = − (cid:3) For the future use we formulate
Proposition 2.8 (See [Do12, Section 8.2.7]) . Let X be a weak del Pezzo surface. Then there existsa root subsystem Φ in the root system R ( X ) such that the set R irr ( X ) of classes of ( − -curves isthe set of simple roots in Φ and the set R eff ( X ) of effective ( − -classes is the set of positive rootsin Φ . Moreover, Φ is a disjoint union of root systems of types A, D, E . Two weak del Pezzo surfaces X and Y are said to have the same type if there exists an isomorphismPic( X ) → Pic( Y ) preserving the intersection form, the canonical class and identifying the sets ofnegative curves. Thus, for a weak del Pezzo surface it makes sense to consider the configuration(i.e., the incidence graph) of irreducible ( − A, D, E and will be denoted respectively. In mostcases, the configuration of irreducible ( − − − d with the configuration of ( − m irreducible ( − X d, Γ ,m . Number m is omitted if d and Γ determine the typeuniquely. For example, X ,A denotes a weak del Pezzo surface of degree 4 with ( − A and X ,A , denotes a weak del Pezzo surface of degree 4 with ( − A and with five ( − − N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 9
Now we formulate criteria for checking left-orthogonality and strong left-orthogonality of divisorson weak del Pezzo surfaces. Most of them are valid in greater generality: for all rational surfaces ofpositive degree.
Proposition 2.9.
Let D be an r -class on a weak del Pezzo surface X of degree d . Then Table 2gives necessary and sufficient conditions for D being left-orthogonal and strong left-orthogonal. Inparticular, if D > − then D is left-orthogonal if and only if D is strong left-orthogonal.More generally, let X be a rational surface of degree d > and D be an r -class on X . Thencriteria of left-orthogonality from Table 2 hold, and for r − criteria of strong left-orthogonalityhold. Table 2.
Criteria of left-orthogonality and strong left-orthogonality of divisors onrational surfaces of positive degree. Criteria marked with ∗ are valid only for weak delPezzo surfaces. r D is left-orthogonal? D is strong left-orthogonal? − iff h ( − D ) = 0 no − iff h ( − D ) = 0 iff h ( D ) = h ( − D ) = 0 − r d − yes yes ∗ > d − iff h ( K X + D ) = 0 iff h ( K X + D ) = 0 ∗ For the proof we will need some lemmas.
Lemma 2.10.
Let X be a rational surface with deg( X ) = K X > . Then − K X is effective.Proof. We have h ( − K X ) = h (2 K X ) = 0 by Serre duality since X is rational. By Riemann-Rochformula we have χ ( − K X ) = h ( − K X ) − h ( − K X ) = K X + 1 >
2. It follows that h ( − K X ) > − K X is effective. (cid:3) Lemma 2.11.
Let X be a rational surface with − K X effective. Let D be a divisor on X with h ( − D ) = 0 . Then h ( D ) = 0 .Proof. We have h ( D ) = h ( K X − D ) by Serre duality. Since − K X is effective, we have an embedding H ( X, O X ( K X − D )) ⊂ H ( X, O X ( − D )) = 0 , hence h ( K X − D ) = 0. (cid:3) Lemma 2.12.
Let X be a rational surface with deg( X ) > and D be an r -class on X with r > − .Then (1) h ( − D ) = 0 , (2) h ( D ) > .Proof. (1) First we prove the statement for X being a blow-up of P at several (maybe infinitesimal)points. Let L, E , . . . , E n be the standard basis in Pic( X ). Since deg( X ) >
0, we have n
8. Let D = aL + P ni =1 b i E i . Claim.
We have D = E i for some i or a > , a + b i > i .To prove the claim, assume that r = −
1. Then all ( − X ) are listed in [Ma74,Prop. 26.1] and the claim follows. Now assume r >
0. Then by Proposition 2.3 we have(2.2) D = a − X b i > D · ( − K X ) = 3 a + X b i > . It follows that (using an inequality between arithmetic and quadratic means) | a | > vuut n X i =1 b i > P ni =1 b i √ n > − a √ n > − a √ √ | a | + 3 a > . Therefore a >
0. Finally, a + b i > − D cannot be effective: otherwise the image − aL of − D on P is effective.Now we consider the general case: assume X is a blow-up of some Hirzebruch surface F s at severalpoints. Moreover, unless X = F s we can assume that s = 2 m + 1 is odd. Let B, F ⊂ X be thepreimages of the ( − s )-curve and the fiber respectively, we have B = − s, B · F = 1 , F = 0. Denoteby B, F, E , . . . , E n the standard basis in Pic( X ). Since deg( X ) > n
7. Consider ablow-up Y of P at n + 1 points, denote by L, E ′ , E ′ , . . . , E ′ n the standard basis in Pic( Y ). Definethe linear map φ : Pic( Y ) → Pic( X )be the rule φ ( L ) = B + ( m + 1) F, φ ( E ′ ) = B + mF, φ ( E ′ i ) = E i for i = 1 , . . . , n. The reader is welcome to check that φ is an isometry and maps K Y to K X . Thus φ identifies r -classeson Y and X . Let D ′ := φ − ( D ) be the r -class on Y , we can write D ′ = aL + P ni =0 b i E ′ i . By theabove Claim, we have either D ′ = E ′ i for some i or a > a + b >
0. Hence, D = φ ( D ′ ) = E i , D = B + mF or D = a ( B + ( m + 1) F ) + b ( B + mF ) + n X i =1 b i E i . In the first two cases we readily have h ( − D ) = 0, in the latter case we have D = ( a + b ) B + ( a ( m + 1) + b m ) F + n X i =1 b i E i . The coefficient at F equals to ( a + b ) m + a and is positive by our assumptions. The divisor S = B + sF is the class of a positive s -curve, hence S is nef. We have S · ( − D ) = − ( a ( m + 1) + b m ) <
0, hence − D is not effective.It remains to consider the case when X is a Hirzebruch surface F s with even s , we leave it to thereader.(2) By Lemma 2.10, − K X is effective. By part (1) we get h ( − D ) = 0 and by Lemma 2.11 we get h ( D ) = 0. Now using Proposition 2.3 we get h ( D ) > h ( D ) − h ( D ) = χ ( D ) = D + 2 > . (cid:3) Lemma 2.13.
Let X be a weak del Pezzo surface of degree d . Then a general anticanonical divisor Z ∈ | − K X | is irreducible.Proof. Recall that by [Do12, Theorem 8.3.2], the linear system | − K X | has no fixed components.If d > | − K X | = d > Z is irreducible by Bertini’s theorem. For d = 1 we havedim | − K X | = 1 and argue as follows. Let X f −→ P be the blow-up of points P , . . . , P . To prove thata general divisor Z ∈ | − K X | = | L − P E i | is irreducible, it suffices to check that a general divisorin the linear system | L − P P i | on P is irreducible. Note that any reducible divisor Z ∈ | L − P P i | is a sum of a line l Z ∈ | L − P j ∈ J P j | and a conic q Z ∈ | L − P j ∈{ ,..., }\ J P j | for some subset N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 11 J ⊂ { , . . . , } . If l Z or q Z is movable in the corresponding linear system then (as dim | − K X | = 1)we get | − K X | = | L − X j ∈ J P j | × | L − X j ∈{ ,..., }\ J P j | . It follows that | − K X | has a fixed part, what contradicts to [Do12, Theorem 8.3.2]. Otherwise Z isuniquely defined by J , and there are only finitely many such reducible divisors. So the statementfollows. (cid:3) Proof of Proposition 2.9.
First, we find criteria of left-orthogonality. We assume that X is a rationalsurface with deg( X ) >
0. Since D is numerically left-orthogonal, we have h ( − D ) − h ( − D ) + h ( − D ) = 0. Therefore D is left-orthogonal if and only if h ( − D ) = h ( − D ) = 0. Suppose r > − h ( − D ) = 0 by Lemma 2.12(1). Suppose r d −
3, denote D ′ = − K X − D . Then by Lemma 2.5 D ′ is an r ′ -class where r ′ = d − − r > −
1. By Lemma 2.12(1) we have h ( − D ′ ) = 0 and by Serreduality we have h ( − D ) = h ( K X + D ) = h ( − D ′ ) = 0. Now the statement follows.Now we prove criteria of strong left-orthogonality. As before, X is any rational surface withdeg X >
0. For a strong left-orthogonal divisor D one has D = χ ( D ) − h ( D ) − > − r − r = −
2. Suppose D is left-orthogonal, then h ( D ) = 0 by Lemmas 2.10 and 2.11. Since D isnumerically left-orthogonal and χ ( D ) = D + 2 = 0, D is strong left-orthogonal iff h ( D ) = 0. Thestatement for r = − X is a weak del Pezzo surface. It remains to demonstrate that aleft-orthogonal divisor D with D > − h ( D ) = 0.By Lemma 2.13, we can choose an irreducible divisor Z ∈ | − K X | . Consider the standard exactsequence 0 → O X ( K X ) → O X → O Z → . Twisting it by O X ( − D ), we get the exact sequence(2.3) 0 → O X ( K X − D ) → O X ( − D ) → O Z ( − D ) → . The long exact sequence of cohomology associated with (2.3) yields that h ( O X ( K X − D )) = h ( O Z ( − D )). Since ( − D ) · Z = D · K X = − − D < Z is irreducible, we have h ( O Z ( − D )) = 0and by Serre duality h ( D ) = h ( K X − D ) = 0. (cid:3) The following consequences and variations of Proposition 2.9 will be needed below.
Lemma 2.14. (1) Let X be a rational surface with − K X effective. Then R ( X ) = R eff ( X ) ⊔ ( − R eff ( X )) ⊔ R slo ( X ) . (2) Let X be a rational surface with deg( X ) > . Then R lo ( X ) = R slo ( X ) ⊔ R eff ( X ) . Proof. (1) Note that D ∈ R ( X ) ⇐⇒ − D ∈ R ( X ), also note that χ ( D ) = χ ( − D ) = 0 for any( − D . Hence D is strong left-orthogonal iff h i ( D ) = h i ( − D ) = 0 for i = 0 ,
2. Now assume D ∈ R ( X ) and h ( D ) = h ( − D ) = 0. By Lemma 2.11 applied to D and − D we get that h ( D ) = h ( − D ) = 0. It follows that R eff ( X ) = { D ∈ R ( X ) | h ( D ) > } , − R eff ( X ) = { D ∈ R ( X ) | h ( − D ) > } ,R slo ( X ) = { D ∈ R ( X ) | h ( D ) = h ( − D ) = 0 } , what concludes the statement.(2) Follows from Lemma 2.10, Proposition 2.9 and part (1). (cid:3) Corollary 2.15.
Any ( − -class on a del Pezzo surface is strong left-orthogonal.Proof. Use Lemma 2.14(1). Indeed, any effective ( − − (cid:3) We finish this section with an example demonstrating that (some part of) Proposition 2.9 failsif X is only assumed to be of positive degree. Example 2.16.
Let X = F be a Hirzebruch surface with a ( − B and a fiber F . We have K X = 8 > X is not a weak del Pezzo surface. Consider the divisor D = B + F on X , it is a( − D is left-orthogonal by Proposition 2.9, but not strong left-orthogonal: χ ( D ) = 1while h ( D ) = h ( F ) = 2.3. Toric systems and admissible sequences
Exceptional objects.
We start with recalling some concepts related with exceptional collec-tions in triangulated categories. An object E in the derived category D b (coh( X )) of coherent sheaveson X is said to be exceptional if Hom i ( E , E ) = 0 for all i = 0 and Hom( E , E ) = k . An orderedcollection ( E , . . . , E n ) of exceptional objects is said to be exceptional if Hom i ( E l , E k ) = 0 for all i and k < l . An exceptional collection ( E , . . . , E n ) is said to be strong exceptional if Hom i ( E l , E k ) = 0for all i = 0 and all k, l . An ordered collection ( E , . . . , E n ) of objects in D b (coh( X )) is said to be numerically exceptional if for the Euler form χ ( E , F ) := P i ( − i dim Hom i ( E , F ) one has: χ ( E k , E k ) = 1 , χ ( E l , E k ) = 0 for k < l. A collection ( E , . . . , E n ) is said to be full if objects E , . . . , E n generate D b (coh X ) as triangulatedcategory. A collection ( E , . . . , E n ) is said to be of maximal length if the classes of objects E , . . . , E n generate K ( X ) num , the Grothendieck group of X modulo numeric equivalence. Recall that for arational surface X one hasrank K ( X ) = rank Pic( X ) + 2 = 12 − deg( X ) . Any line bundle on a rational surface is exceptional. Therefore an ordered collection of line bundles( O X ( D ) , . . . , O X ( D n )) on a rational surface X is exceptional (resp. strong exceptional) if and onlyif the divisor D l − D k is left-orthogonal (resp. strong left-orthogonal) for all k < l .Clearly, any full exceptional collection has maximal length. In general, the converse in not true:there are examples of surfaces of general type (classical Godeaux surface, [BBS13] or Barlow surface,[BBKS15]) possessing an exceptional collection of maximal length which is not full. But for rationalsurfaces there are no such examples known. For weak del Pezzo surfaces of degree >
2, it followsfrom a result by Sergey Kuleshov [Ku97, Theorem 3.1.8] that any exceptional collection of vectorbundles of maximal length is full.3.2.
Toric systems.
Next we recall the important notion of a toric system , introduced by Hille andPerling in [HP11].For a sequence ( O X ( D ) , . . . , O X ( D n )) of line bundles one can consider the infinite sequence (calleda helix ) ( O X ( D i )) , i ∈ Z , defined by the rule D k + n = D k − K X . From Serre duality it follows thatthe collection ( O X ( D ) , . . . , O X ( D n )) is exceptional (resp. numerically exceptional) if and only ifany collection of the form ( O X ( D k +1 ) , . . . , O X ( D k + n )) is exceptional (resp. numerically exceptional).One can consider the n -periodic sequence A k = D k +1 − D k N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 13 of divisors on X . Following Hille and Perling, we will consider the finite sequence ( A , . . . , A n )with the cyclic order and will treat the index k in A k as a residue modulo n . Vice versa, for anysequence ( A , . . . , A n ) one can construct the infinite sequence ( O X ( D i )) , i ∈ Z , with the property D k +1 − D k = A k mod n . Then the corresponding collection of line bundles would be(3.1) ( O X ( D ) , O X ( D ) , . . . , O X ( D n )) == ( O X ( D ) , O X ( D + A ) , O X ( D + A + A ) , . . . , O X ( D + A + . . . + A n − )) . Definition 3.1 (See [HP11, Definitions 3.4 and 2.6]) . Let X be a smooth projective surface. Asequence ( A , . . . , A n ) in Pic( X ) is called a toric system if n = rank K ( X ) num and the followingconditions are satisfied (where indices are treated modulo n ):(1) A i A i +1 = 1;(2) A i A j = 0 if j = i, i ± A + . . . + A n = − K X .Note that a cyclic shift ( A k , A k +1 , . . . , A n , A , . . . , A k − ) of a toric system ( A , . . . , A n ) is also atoric system. Also, note that by our definition any toric system has maximal length. Example 3.2 (See [HI13, Section 3.1]) . Let Y be a smooth projective toric surface. Its torus-invariant prime divisors form a cycle, denote them T , . . . , T n in the cyclic order. Then ( T , . . . , T n )is a toric system on Y .The following is proved in [HP11, Lemma 3.3], see also [EL16, Propositions 2.8 and 2.15]. Proposition 3.3.
Let X be a surface with χ ( O X ) = 1 . Then a sequence ( A , . . . , A n ) in Pic( X ) isa toric system if and only if the corresponding collection (3.1) is numerically exceptional of maximallength for any D . In particular, if A is a toric system then A i + 2 = − A i · K X for any i . Definition 3.4.
Let X be a surface such that the sheaf O X is exceptional (that is, H ( O X ) = k , H ( O X ) = H ( O X ) = 0). A toric system ( A , . . . , A n ) on X is called exceptional (resp. strongexceptional ) if the corresponding collection ( O X ( D ) , . . . , O X ( D n )) is exceptional (resp. strong ex-ceptional). Note that exceptional toric systems are stable under cyclic shifts while strong exceptionaltoric systems are not in general.A toric system ( A , . . . , A n ) on X is called cyclic strong exceptional if the corresponding collection( O X ( D k +1 ) , . . . , O X ( D k + n ))is strong exceptional for any k ∈ Z . Equivalently: if all cyclic shifts( A k , A k +1 , . . . , A n , A , . . . , A k − )are strong exceptional. Definition 3.5.
We will use the following notation: let k, l ∈ [1 , . . . , n ] and k l + 1 (mod n ). Bya cyclic segment [ k, . . . , l ] ( [1 , . . . , n ] we mean the following set of indices:[ k, . . . , l ] := ( [ k, k + 1 , . . . , l − , l ] , if k l, [ k, k + 1 , . . . , n, , , . . . , l ] , if k > l. Note that [1 , . . . , n ] is not a cyclic segment of [1 , . . . , n ].For a toric system ( A , . . . , A n ) and k, l ∈ [1 , . . . , n ] denote A k...l := X i ∈ [ k,...,l ] A i . Proposition 3.6.
Let X be a surface with χ ( O X ) = 1 . For any toric system A on X , the divisor A k...l is numerically left-orthogonal with A k...l + 2 = X i ∈ [ k,...,l ] ( A i + 2) . Proof.
Follows from Propositions 2.4 and 3.3. (cid:3)
Corollary 3.7.
Let X be a surface with χ ( O X ) = 1 and A be a toric system on X . If one has A i > − for all i , then one has A k...l > − for any cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] . Proposition 3.8.
Let X be a surface such that the sheaf O X is exceptional. Then for any strongexceptional toric system ( A , . . . , A n ) on X we have A i > − for all i = 1 , . . . , n − and for anycyclic strong exceptional toric system ( A , . . . , A n ) on X we have A i > − for all i = 1 , . . . , n .Proof. The divisor A i under consideration is strong left-orthogonal. Hence by Proposition 2.3 A i = χ ( A i ) − h ( A i ) − > − (cid:3) The next proposition is a straightforward consequence of definitions.
Proposition 3.9.
Let X be a surface such that the sheaf O X is exceptional. (1) A toric system ( A , . . . , A n ) on X is exceptional if and only if the divisor A k...l is left-orthogonal for all k < l n − and if and only if the divisor A k...l is left-orthogonal forany cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] . (2) A toric system ( A , . . . , A n ) on X is strong exceptional if and only if the divisor A k...l is strongleft-orthogonal for all k < l n − . (3) A toric system ( A , . . . , A n ) on X is cyclic strong exceptional if and only if the divisor A k...l is strong left-orthogonal for any cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] . For weak del Pezzo surfaces we prove handy criteria for exceptionality of toric systems in terms ofeffectiveness of some divisors.
Theorem 3.10.
Let A = ( A , . . . , A n ) be a toric system on a weak del Pezzo surface X of degree d .Suppose A i > − for i n − . Then: (1) A is exceptional if and only if the following two conditions hold: (a) for any k l n − such that A k = . . . = A l = − , the divisor − A k...l is noteffective, (b) if A n − then for any l < k n such that A k = . . . = A n − = A = . . . = A l = − the divisor − A k...n...l is not effective (if l = 0 this reads as − A k...n ). (2) A is strong exceptional if and only if A is exceptional and the following holds: for any k l n − such that A k = . . . = A l = − , the divisors A k...l and − A k...l are not effective.Suppose moreover that A i > − for all i . Then (3) A is exceptional if and only if the following holds: for any cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] such that A i = − for all i ∈ [ k, . . . , l ] , the divisor − A k...l is not effective. (4) A is cyclic strong exceptional if and only if the following holds: for any cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] such that A i = − for all i ∈ [ k, . . . , l ] , the divisors A k...l and − A k...l are not effective.Proof. (1) By Proposition 3.9, A is exceptional if and only if for any 1 k l n − A k...l is left-orthogonal. Furthermore, if D is left-orthogonal then − D is not effective by definition.This proves the “only if” part. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 15
To prove the “if” part take any 1 k l n −
1, set D = A k...l and D ′ = − D − K X = A l +1 ...k − . Note that D + ( D ′ ) = d − D > − D = − D is left-orthogonal by assumption (a) and Proposition 2.9; if − D d − D isleft-orthogonal by Proposition 2.9. Finally, if D > d − D ′ ) −
2. We claim that − D ′ isnot effective. One can decompose D ′ as D ′ = A l +1 ...n...k − = A l +1 ...l − + A l ...n...k + A k +1 ...k − =: D − + D ′′ + D + , such that the following holds:(1) A l = . . . = A n − = A = . . . = A k = − l = l + 1 (i.e., D − = 0) then A l − > − k = k − D + = 0) then A k +1 > − A i > − i = 1 , . . . , n −
1) it follows that ( D ′′ ) = A n , A n ( D ′ ) − D − ) > − D − = 0), ( D + ) > − D + = 0). Note that D − , D + are effective by Lemma 2.12(2). Assume now that − D ′ is effective, then − D ′′ = − D ′ + D − + D + iseffective which contradicts to assumption (b). We conclude that − D ′ = K X + D is not effective andhence D is left-orthogonal by Proposition 2.9. Thus, in all cases D is left-orthogonal and therefore A is exceptional.(2) Suppose 1 k l n −
1, then by (1) D = A k...l is left-orthogonal. By Proposition 3.6,one has D > − D = − A k = . . . = A l = −
2. If D = −
2, then D is strongleft-orthogonal by assumptions and Proposition 2.9, if D > − D is strong left-orthogonal byProposition 2.9.(3) follows from (1), (4) is similar to (2). (cid:3) Corollary 3.11.
Let A = ( A , . . . , A n ) be a strong exceptional toric system with A n > − on a weakdel Pezzo surface. Then A is cyclic strong exceptional.Proof. By Proposition 2.9, we have A i > − i n − A i is strong left-orthogonal. Also A n > − > −
2. Use Theorem 3.10(4). Clearly, any cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ] such that A k = . . . = A l = − , . . . , n − ± A k...l are not effective byTheorem 3.10(2). (cid:3) Corollary 3.12.
Any toric system ( A , . . . , A n ) with A i > − for all i on a del Pezzo surface iscyclic strong exceptional.Proof. Use Theorem 3.10(4) and Corollary 2.15. (cid:3)
We finish this section with a very important theorem which is due to Hille and Perling [HP11,Theorem 3.5] for rational surfaces and to Charles Vial [Vi17, theorem 3.5] for the general case.
Theorem 3.13.
Let A = ( A , . . . , A n ) be a toric system on a smooth projective surface X such that χ ( O X ) = 1 ( X is not necessarily rational). Then there exists a smooth projective toric surface Y with torus-invariant divisors T , . . . , T n such that A i = T i for all i . We remark that a toric system A is uniquely defined by the sequence A in the following sense: if A = ( A , . . . , A n ) , B = ( B , . . . , B n ) are two toric systems on rational surfaces X and Y respectivelyand A i = B i for all i = 1 , . . . , n then there exists an isomorphism Pic( X ) → Pic( Y ) preserving theintersection form and mapping A i to B i (and thus K X to K Y ). To see this, one should consider Z n with the bilinear form q defined in the standard basis ( e , . . . , e n ) by q ( e i , e j ) := A i , i = j ;1 , if | i − j | = 1;0 , otherwise.There is a surjective map α : Z n → Pic( X ) sending e i to A i . Moreover, α is compatible with q andthe intersection form. Therefore, the triple (Pic( X ) , the intersection form , ( A i )) is identified with thetriple ( Z n / ker q, the form induced by q, (im e i )), and the latter depends only on the sequence A .3.3. Admissible sequences.
For a sequence ( a , . . . , a n ) of integers we define the m -th elementaryaugmentation as follows: • augm ( a , . . . , a n ) = ( − , a − , a , . . . , a n − , a n − • augm m ( a , . . . , a n ) = ( a , . . . , a m − , a m − − , − , a m − , a m +1 , . . . , a n ) for 2 m n ; • augm n +1 ( a , . . . , a n ) = ( a − , a , a , . . . , a n − , − Definition 3.14.
We call a sequence admissible if it can be obtained from the sequence (0 , k, , − k )or ( k, , − k,
0) where k ∈ Z by applying several elementary augmentation.It is not hard to see that admissible sequences are stable under cyclic shifts:sh( a , . . . , a n ) := ( a , . . . , a n , a )and symmetries: sym( a , . . . , a n ) := ( a n − , a n − , . . . , a , a n ) . We define symmetry like this and not in a more natural way because such symmetry preserves thecondition a i > − i = 1 , . . . , n −
1. This condition describes admissible sequences coming fromstrong exceptional toric systems, see Theorem 3.10 or Proposition 3.8. The analogous symmetryfor toric systems corresponds to the operation which sends an exceptional collection of line bundles( O X ( D ) , . . . , O X ( D n )) to ( O X ( − D n ) , . . . , O X ( − D )).Operations sh and sym act on the set of admissible sequences of fixed length n . They generate thegroup isomorphic to the n -th dihedral group: the group of all permutations of the index set [1 , . . . , n ]preserving or inverting the cyclic order. This group is isomorphic to Z /n Z ⋊ Z / Z .For a toric system A = ( A , . . . , A n ) on a surface X , we denote A := ( A , . . . , A n ) . The motivation for considering admissible sequences is the following discovery due to Hille andPerling.
Proposition 3.15.
Let X be a surface with χ ( O X ) = 1 , suppose X = P . Then for any toric system A on X the sequence A is admissible.Proof. First, assume that X is a toric surface with torus invariant divisors T , . . . , T n . Then thesequence T = ( T , . . . , T n ) is admissible. Indeed, for X a Hirzebruch surface one has n = 4and the statement is clear. Otherwise X has a torus-invariant ( − E by [HP11, Theo-rem 2.1]. Let E = T k , and consider the blow-down X ′ of E . The torus-invariant divisors on X ′ are T ′ , . . . , T ′ k − , T ′ k +1 , . . . , T ′ n , and ( T ′ k ± ) = T k ± + 1, ( T ′ i ) = T i otherwise. Therefore T =augm k (( T ′ ) ), and we proceed by induction. The general case follows from the toric case and Theo-rem 3.13. (cid:3) Definition 3.16.
We say that an admissible sequence ( a , . . . , a n ) is of the first kind if a i > − i n . We say that a toric system A is of the first kind if the sequence A is of the first kind. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 17
By Proposition 3.8, any cyclic strong exceptional toric system A is of the first kind. One couldsay also that toric systems of the first kind are numerically cyclic strong exceptional . Proposition 3.17.
All admissible sequences of the first kind are, up to cyclic shifts and symmetries,in Table 3. In particular, if a surface X with χ ( O X ) = 1 has a toric system ( A , . . . , A n ) of the firstkind, then n and deg( X ) > . Table 3.
Admissible sequences of the first kind type sequence P × P (0 , , , F (0 , − , , F (0 , − , , (0 , , − , − , − (0 , − , − , − , ( − , − , − , − , − , − ( − , − , − , − , − , (0 , , − , − , − , − (0 , − , − , − , − , ( − , − , − , − , − , − , − ( − , − , − , − , − , − , ( − , − , − , − , − , − , − , − ( − , − , − , − , − , − , − , − ( − , − , − , − , − , − , − , ( − , − , − , − , − , − , − , − , − Proof.
In the obvious notation we have(5 a ) = augm ( P × P ) , (5 b ) = augm ( F ) , (6 a ) = augm (5 a ) , (6 b ) = augm (5 b ) , (6 c ) = augm (5 a ) , (6 d ) = augm (5 b ) , (7 a ) = augm (6 a ) , (7 b ) = augm (6 b ) , (8 a ) = augm (7 a ) , (8 b ) = augm (7 a ) , (8 c ) = augm (7 b ) , (9) = augm (8 b ) . The reader is welcome to check that all other augmentations of the listed sequences are either not ofthe first kind or contained in the table up to some shifts or/and symmetries. (cid:3)
We remark that the admissible sequences are exactly the sequences of self-intersection numbers oftorus-invariant divisors on toric surfaces. Moreover, by [HP11, Prop. 2.5] the sequence is of the firstkind if and only if the corresponding toric surface has nef anti-canonical class. Therefore our Table 3coinsides with Table 1 in [HP11] which lists such surfaces.4.
Operations with toric systems
Transpositions and shifts.
Let ( O X ( D ) , . . . , O X ( D n )) be an exceptional collection of linebundles on a surface X . Recall that the line bundles O X ( D k ) , O X ( D k +1 ) , . . . , O X ( D l ) form a block if they are completely orthogonal to each other: Hom i ( O X ( D p ) , O X ( D q )) = 0 for any k p < q l and i . One can reorder bundles in a block and get essentially the same exceptionalcollection. Note that for mutually orthogonal O X ( D k ) , O X ( D k +1 ) the difference D k +1 − D k = A k isa strong left-orthogonal divisor with χ ( A k ) = 0. Hence A k = − a block of orthogonal line bundles O X ( D k ) , O X ( D k +1 ) , . . . , O X ( D l ) one has a chain A k , . . . , A l − ofstrong left-orthogonal ( − Definition 4.1.
Let A = ( A , . . . , A n ) be a toric system. Suppose A k = − k . Denotetran ( A ) = ( − A , A + A , A , . . . , A n − , A n + A ) if k = 1 , tran k ( A ) = ( A , . . . , A k − , A k − + A k , − A k , A k + A k +1 , A k +2 , . . . , A n ) if 1 < k < n, tran n ( A ) = ( A + A n , A , . . . , A n − , A n − + A n , − A n ) if k = n. It is easy to see that tran k ( A ) is also a toric system. If A corresponds to the numerically exceptionalcollection ( O X ( D ) , . . . , O X ( D n )) then tran k ( A ) corresponds to the numerically exceptional collection( O X ( D ) , . . . , O X ( D k − ) , O X ( D k +1 ) , O X ( D k ) , O X ( D k +2 ) , . . . , O X ( D n )) for 1 k n − , ( O X ( D n + K X ) , O X ( D ) , . . . , O X ( D n − ) , O X ( D − K X )) for k = n. The operation tran k is called k -th transposition . It is an involution and tran k , . . . , tran l define anaction of the symmetric group S l − k +2 on the set of toric systems A satisfying A k = . . . = A l = − permutation . Note also that(4.1) (tran k ( A )) = A . Lemma 4.2.
Let X be a surface such that the sheaf O X is exceptional. Suppose A = ( A , . . . , A n ) is a toric system on X and A k = − for some k , k n . (1) If A is exceptional then tran k ( A ) is exceptional if and only if the divisor A k is strong left-orthogonal. (2) If k = n then A is strong exceptional if and only if tran k ( A ) is strong exceptional. (3) A is cyclic strong exceptional if and only if tran k ( A ) is cyclic strong exceptional.Proof. It follows easily from Proposition 3.9. Note also that A k is strong left-orthogonal if and onlyif − A k is strong left-orthogonal. Indeed, both mean that h i ( A k ) = h i ( − A k ) = 0 for i = 0 , , χ ( A k ) = χ ( − A k ) = 0 for a ( − (cid:3) We introduce also the natural operation of cyclic shift:sh(( A , . . . , A n )) = ( A , A , . . . , A n , A ) . It corresponds to the following operation with collections of line bundles:( O X ( D ) , . . . , O X ( D n )) = ⇒ ( O X ( D ) , . . . , O X ( D n ) , O X ( D n +1 )) , where D n +1 = D − K X .It follows from Proposition 3.9 that cyclic shift preserves exceptional and cyclic strong exceptionaltoric systems but may not preserve strong exceptional toric systems.4.2. Augmentations.
Following Hille and Perling [HP11], we define augmentations. They providea wide class of explicitly constructed toric systems.
Definition 4.3.
Let A ′ = ( A ′ , . . . , A ′ n ) be a toric system on a surface X ′ , and let p : X → X ′ bethe blow up of a point with the exceptional divisor E ⊂ X . Denote A i = p ∗ A ′ i . Then one has thefollowing toric systems on X :augm p, ( A ′ ) =( E, A − E, A , . . . , A n − , A n − E );augm p,m ( A ′ ) =( A , . . . , A m − , A m − − E, E, A m − E, A m +1 , . . . , A n ) for 2 m n ;augm p,n +1 ( A ′ ) =( A − E, A , . . . , A n − , A n − E, E ) . N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 19
The toric systems augm p,m ( A ′ ) (1 m n + 1) are called elementary augmentations of the toricsystem A ′ . Lemma 4.4.
Augmentations of toric systems and sequences of integers are compatible: for any toricsystem A = ( A , . . . , A n ) on a surface and any k n one has (augm k ( A )) = augm k ( A ) . Moreover, if augm k ( A ) is of the first kind (see Definition 3.16) then A also is.Proof. Follows from the definitions. (cid:3)
Proposition 4.5 (See [EL16, Proposition 3.3]) . In the notation of Definition 4.3, let A be a toricsystem on a surface X such that A m = E for some m . Then A = augm p,m ( A ′ ) for some toric system A ′ on X ′ . Definition 4.6.
A toric system A on a rational surface X is called a standard augmentation if X is aHirzebruch surface or A is an elementary augmentation of some standard augmentation. Equivalently: A is a standard augmentation if there exists a chain of blow-ups X = X n p n −→ X n − → . . . X p −→ X where X is a Hirzebruch surface and A = augm p n ,k n (augm p n − ,k n − ( . . . augm p ,k ( A ′ ) . . . ))for some k , . . . , k n and a toric system A ′ on X . In this case we will say that A is a standardaugmentation along the chain p , . . . , p n . Remark 4.7.
To be more accurate, one should add that (the unique) toric system (
L, L, L ) on P isalso considered as a standard augmentation. To simplify the forthcoming definitions and statements,we will ignore this issue. Proposition 4.8.
Let X ′ be a surface such that the sheaf O X ′ is exceptional, let p : X → X ′ be ablow-up of a point. Assume A ′ is a toric system on X ′ and A = augm k ( A ′ ) . Then (1) A is exceptional if and only if A ′ is exceptional; (2) if A is strong exceptional then A ′ is strong exceptional; (3) if A is cyclic strong exceptional then A ′ is cyclic strong exceptional.Proof. (1) and (2) are in [EL16, Prop. 2.21]. For (3), note that cyclic shifts commute with elementaryaugmentation and use (2) and Definition 3.4. (cid:3) An exceptional collection of line bundles is called a standard augmentation if the associated toricsystem is a standard augmentation. One of main results in [HP11] is that any strong exceptionalcollection of line bundles on a toric surface comes from a standard augmentation. But this “comesfrom” does not literally means “is” (as this is false, see Example 4.16 below). Instead it means (see[HP11, Theorem 8.1]) that such collection has a normal form which is a standard augmentation.This normal form is obtained from the original collection by reordering bundles which are mutuallyorthogonal. In other words, the toric system associated to the normal form is obtained from theoriginal toric system by transpositions.Instead of using normal forms we prefer to introduce the following two definitions that we findmore convenient.
Definition 4.9.
A toric system A on a rational surface X is called an augmentation in the weaksense if A can be obtained from a toric system on a Hirzebruch surface by several transpositions,cyclic shifts and elementary augmentations (applied in any order). Definition 4.10.
An exceptional (resp. strong exceptional, cyclic strong exceptional) toric system A = ( A , . . . , A n ) on a rational surface X is called an exceptional (resp. strong exceptional, cyclicstrong exceptional) augmentation if X is a Hirzebruch surface or A can be obtained by a transposition,cyclic shift or an elementary augmentation from some toric system which is an exceptional (resp.strong exceptional, cyclic strong exceptional) augmentation. Remark 4.11.
If a toric system is a standard augmentation, it is an augmentation in the weaksense.
Remark 4.12.
In the both above definitions, a sequence A (0) , A (1) , . . . , A ( m ) = A of toric systems is required to exist, where A (0) is a toric system on a Hirzebruch surface and any A ( k ) isobtained from A ( k − by one of the operations: elementary augmentation, cyclic shift or transposition.Suppose A is exceptional, then A ( k ) -s are not automatically exceptional. The difference betweenDefinitions 4.9 and 4.10 is that in Definition 4.10 all intermediate toric systems A ( k ) are required tobe exceptional while in Definition 4.9 they are not.Next Proposition follows from results of [EL16]. Proposition 4.13.
Let ( O X ( D ) , . . . , O X ( D n )) be an exceptional collection of line bundles of maxi-mal length on a rational surface X . Suppose that the corresponding toric system A is an exceptionalaugmentation. Then this collection is full.Proof. According to Definition 4.10, there exists a sequence A (0) , A (1) , . . . , A ( m ) = A of exceptional toric systems such that A (0) is a system on a Hirzebruch surface and any A ( k ) isobtained from A ( k − by a cyclic shift, a transposition or an elementary augmentation. Thus we haveto check the following:(1) any exceptional toric system on a Hirzebruch surface is full;(2) any cyclic shift of a full exceptional toric system is full;(3) any transposition of a full exceptional toric system is full;(4) any elementary augmentation of a full exceptional toric system is full.Propositions 2.16, 2.19 in [EL16] imply (1). Statement (2) holds because a cyclic shift of a fullexceptional collection is the mutation of the first object to the last position. For (3), note thattran k ( A , . . . , A n ) does not change the set of line bundles in the corresponding exceptional collectionsfor 1 k n −
1. For k = n use (2) and the equality tran n = sh ◦ tran ◦ sh − . Finally, (4) is [EL16,Prop. 2.21(2)]. (cid:3) Therefore Conjecture 1.2 implies that any strong exceptional collection of line bundles of maximallength is full.
Lemma 4.14.
Let X be a rational surface and A be a cyclic strong exceptional toric system on X .Then A is an augmentation in the weak sense if and only if A is a cyclic strong exceptional augmen-tation.Proof. “If” is trivial. For “only if”, suppose A = t k ◦ . . . ◦ t ( A ′ ) where A ′ is a toric system on aHirzebruch surface and any t i is either tran , sh or augm. Recall that tran and sh preserve cyclicstrong exceptional toric systems (Lemma 4.2(3) and a remark after it). By Proposition 4.8, ifaugm k ( B ) is cyclic strong exceptional then so is B . Therefore all toric systems t l ◦ . . . ◦ t ( A ′ ) and A ′ where 1 l k are cyclic strong exceptional and A is a cyclic strong exceptional augmentationby definition. (cid:3) N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 21
Hille and Perling in their original paper [HP11, Theorem 8.1] proved that any strong exceptionaltoric system on a toric surface is a strong exceptional augmentation. In [EL16, Theorem 1.4] it isproved that any toric system on a del Pezzo surface is a standard augmentation. Hochenegger andIlten in [HI13] prove that any exceptional toric system on a toric surface of Picard rank Remark 4.15.
Hochenegger and Ilten provide examples of a strong exceptional toric system A =( A , . . . , A ) on a weak del Pezzo toric surface of Picard rank 5 such that A is not an elementaryaugmentation, see [HI13, Example 5.6] and [Ho13, Section 4]. We remark that the transpositiontran ( A ) = ( − A , A + A , A , . . . , A , A + A ) is also a strong exceptional toric system and is anelementary (and moreover standard) augmentation. Therefore the cited examples do not contradictto Conjecture 1.2.Here we give another example demonstrating that the use of transpositions is necessary. Moreover,one cannot get a “normal form” only by reordering of line bundles in the collection. Therefore theuse of cyclic shifts (or of transpositions affecting the last term) is necessary. Example 4.16.
Let X = X , A , be the weak del Pezzo surface of degree 4 of type 2 A with 8 lines.Explicitly, let P , P , P , P ∈ P be points and H ⊂ P be a line such that P , P , P ∈ H , P / ∈ H .Let X ′ be the blow-up of P , P , P , P with exceptional divisors E , E , E , E . Let P ∈ E be ageneral point and X be the blow-up of P . Consider the following toric system on X (recall that E k ,k ,...,k m denotes E k + E k + . . . + E k m ): A = ( A , . . . , A ) = ( L − E , E , L − E , L − E , E − E , L − E , E − E , − L + E ) . We have A = ( − , − , − , , − , − , − , − , hence A is of the first kind, type (8c) from Table 3.There are 8 irreducible ( − X , see [CT88, Prop. 6.1(3)]: E = A , E = A , E = A , E = A , L − E = A , L − E = A ,L − E = A , L − E = A . It follows that no A i is a ( − A is not an elementary augmentation. Further, byLemma 5.2(1) there are no irreducible ( − A k...l where 1 k l
7. Thismeans (it would be clear in Section 5, see Lemma 5.1) that there exist no elementary augmentation B that can be sent to A by transpositions tran , . . . , tran . On the other hand, let B = tran tran ( A ),then B = E is an irreducible curve and B is an elementary augmentation.One can check (using Theorem 3.10) that the toric system A is cyclic strong exceptional.5. Toric systems of the first kind are augmentations
Below we prove that any toric system A = ( A , . . . , A n ) of the first kind (Definition 3.16) ona rational surface X is an augmentation in the weak sense (Definition 4.9). Moreover, if A isexceptional/strong exceptional/cyclic strong exceptional then A is an exceptional/strong excep-tional/cyclic strong exceptional augmentation in the sense of Definition 4.10.The proof is by induction in n . To do one step, it suffices to find a toric system B obtained from A by several transpositions, such that B is an elementary augmentation. For any toric system A on asurface X , denote by I ( X, A ) ⊂ I ( X ) the subset of ( − A by several transpositions: I ( X, A ) = { D ∈ I ( X ) | ∃ k , . . . , k r , i ∈ [1 , . . . , n ] D = B i , where B = tran k r . . . tran k ( A ) } . Lemma 5.1.
In the above notation one has (5.1) I ( X, A ) = { A k...l | ∃ m, k m l, a k = . . . = a m − = a m +1 = . . . = a l = − , a m = − } , where a i = A i .Proof. First, we prove ⊃ inclusion. Suppose a k = . . . = a m − = a m +1 = . . . = a l = − , a m = − A (for I ( X, A ) it holds because trancommutes with sh: tran i ◦ sh = sh ◦ tran i +1 ). Hence, applying cyclic shifts to A , we may and willassume that 1 k l n −
1. Let(5.2) ( O X ( D ) , . . . , O X ( D n ))be the corresponding numerically exceptional collection. Then the line bundles in any of the seg-ments O X ( D k ) , . . . , O X ( D m ) and O X ( D m +1 ) , . . . , O X ( D l +1 ) are numerically completely orthogonal.Permuting line bundles in (5.2) one can obtain the numerically exceptional collection( O X ( D ) , . . . , O X ( D k − ) , O X ( D k +1 ) , O X ( D k +2 ) , . . . , O X ( D m ) , O X ( D k ) , O X ( D l +1 ) , O X ( D m +1 ) , . . . , O X ( D l ) , O X ( D l +2 ) , . . . , O X ( D n )) . Let B be the corresponding toric system, explicitly one has B = tran m +1 . . . tran l − tran l (tran m − . . . tran k +1 tran k ( A ))(recall that tran i ( A ) = A and thus one has (tran k r . . . tran k ( A )) i = − i, k , . . . , k r ∈ [ k, . . . , l ] \ { m } ). Then B m = D l +1 − D k = A k...l . Since A k...l is a ( − A k...l ∈ I ( X, A ).Now we prove ⊂ inclusion. As above, applying cyclic shifts to A we may assume that a n = − A = ( − , − , . . . , − A ′ m is a ( − A ′ = tran k r . . . tran k ( A ) . Let ( O X ( D ) , . . . , O X ( D n )) and ( O X ( D ′ ) , . . . , O X ( D ′ n )) be the corresponding collections of line bun-dles. The sets { D , . . . , D n } and { D ′ , . . . , D ′ n } are the same because a n = − n is not used in(5.3). Assume D ′ m = D k and D ′ m +1 = D l +1 . Hence A ′ m = D ′ m +1 − D ′ m = D l +1 − D k = A k...l . Further,the line bundles D k , D k +1 , . . . , D m are in one block, and D m +1 , D m +2 , . . . , D l +1 are in another block.It follows that a k = a k +1 = . . . = a m − = a m +1 = . . . = a l = −
2. Also A m = ( A ′ m ) = − (cid:3) The following lemma is very encouraging.
Lemma 5.2.
Let A be a toric system of the first kind on a rational surface X . (1) Suppose C = A k...l and C ′ = A k ′ ...l ′ are ( − -classes and C = C ′ . Then k = k ′ and l = l ′ . (2) Assume deg( X ) , then I ( X, A ) = I ( X ) .Proof. (1) Denote a i := A i . By Lemma 5.1, there exist numbers p ∈ [ k, . . . , l ], p ′ ∈ [ k ′ , . . . , l ′ ] suchthat(5.4) a p = a p ′ = − , a i = − i = k, . . . , p − , p + 1 , . . . , l and i = k ′ , . . . , p ′ − , p ′ + 1 , . . . , l ′ . We consider several cases distinguished by the relation between [ k, . . . , l ] and [ k ′ , . . . , l ′ ]. Essentially,there are three cases (note that [ k, . . . , l ] ∪ [ k ′ , . . . , l ′ ] = [1 , . . . , n ] by (5.4) and explicit description ofadmissible sequences of the first kind in Proposition 3.17).First, suppose that [ k, . . . , l ] ⊂ [ k ′ , . . . , l ′ ]. Then C ′ = A k ′ ...k − + A k...l + A l +1 ...l ′ =: D − + C + D + and C · C ′ = C + D − · C + D + · C. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 23
Note that D − · C = 1 unless k = k ′ and D − = 0, similarly for D + · C . Since C = C · C ′ = −
1, itfollows that k = k ′ and l = l ′ , we are done.Now assume that k < k ′ l < l ′ . Then C = A k...k ′ − + A k ′ ...l =: D − + D and C ′ = A k ′ ...l + A l +1 ...l ′ =: D + D + , where D − , D + are nonzero. We have − C · C ′ = D + D · ( D − + D + ) + D − · D + = D + 2 + 0 = D + 2 . Since D = − − k, . . . , l ] ∩ [ k ′ , . . . , l ′ ] = ∅ . Then − C · C ′ = A k...l · A k ′ ...l ′ = 0 or 1, acontradiction.(2) Note that the sequence A is admissible and of the first kind, see Definition 3.16. For anyadmissible sequence A of the first kind (up to a symmetry and a shift, see Table 3) we find thecardinality of I ( X, A ) using Lemma 5.1. In Table 4 we present all divisors A k...l satisfying numericalconditions in the r.h.s. of (5.1). By part (1), they are pairwise different. Also, their number equalsto the cardinality of I ( X ), see Table 1. It follows that I ( X ) = I ( X, A ). Table 4. I ( X, A ) for toric systems of the first kindtype A I ( X, A ) | I ( X ) |
5a (0 , , − , − , − A , A , A
35b (0 , − , − , − , A , A , A
36a ( − , − , − , − , − , − A , A , A , A , A , A
66b ( − , − , − , − , − , A , A , A , A , A , A
66c (0 , , − , − , − , − A , A , A , A , A , A
66d (0 , − , − , − , − , A , A , A , A , A , A
67a ( − , − , − , − , − , − , − A , A , A , A , A , A , A , A , A , A − , − , − , − , − , − , A , A , A , A , A , A , A , A , A , A − , − , − , − , − , − , − , − A k , A k − , k , A k, k +1 , A k − , k, k +1 k ∈ { , , , }
8b ( − , − , − , − , − , − , − , − A , A , A , A , A k...l for k ∈ { , } , l ∈ { , , } , 16 A k...l for k ∈ { , , } , l ∈ { , }
8c ( − , − , − , − , − , − , − , A k...l for k ∈ { , } , l ∈ { , , , } , A k...l for k ∈ { , , , } , l ∈ { , } − , − , − , − , − , − , − , − , − A k...l for k ∈ { , , } , j ∈ { , , } , 27 A k...l for k ∈ { , , } , l ∈ { , , } , A k...l for k ∈ { , , } , l ∈ { , , } (cid:3) Theorem 5.3.
Any toric system A of the first kind on a smooth rational projective surface X is anaugmentation in the weak sense. Moreover, if X = P then A is an augmentation in the weak sensealong any given chain of blow-downs to a Hirzebruch surface.Proof. If X is a Hirzebruch surface or P there is nothing to prove. Assume now there is a blow-up X → X ′ with the exceptional curve E . By Lemma 5.2, I ( X, A ) = I ( X ). The class of E belongs to I ( X, A ), hence by Lemma 5.1 E = A k...l for some k, l . By definition of I ( X, A ), there exists a toricsystem of the form B = tran k r . . . tran k ( A ) , such that B m = E for some m . By Proposition 4.5, B = augm m ( C ) for some toric system C on X ′ .We have B = A by (4.1), hence B is of the first kind. Now C is also of the first kind by Lemma 4.4.By induction in rank Pic( X ) we may assume that C is an augmentation in the weak sense, hence B and A are also such. (cid:3) Corollary 5.4.
Any cyclic strong exceptional toric system A on a rational surface X is a cyclicstrong exceptional augmentation along any given chain of blow-downs of X to a Hirzebruch surface.Proof. Any cyclic strong exceptional toric system is of the first kind by Proposition 3.8. By The-orem 5.3, A is an augmentation in the weak sense along any chain of blow-downs to a Hirzebruchsurface. From Lemma 4.14 it follows that A is a cyclic strong exceptional augmentation along thatchain. (cid:3) Corollary 5.5.
Any cyclic strong exceptional collection of line bundles of maximal length on arational surface is full.Proof.
It follows from Corollary 5.4 and Proposition 4.13. (cid:3)
Corollary 5.6.
Let X ′ be a rational surface and X be a blow-up of a point on X ′ . Suppose that X possesses a cyclic strong exceptional toric system. Then X ′ also possesses such system.Proof. Let A be a cyclic strong exceptional toric system on X . By Corollary 5.4, A is a cyclic strongexceptional augmentation along the blow-down X → X ′ . By Definition 4.10, it means that thereexists a cyclic strong exceptional toric system B on X (obtained from A by transpositions and cyclicshifts) such that B = augm i ( B ′ ) for some cyclic strong exceptional toric system B ′ on X ′ . (cid:3) Corollary 5.7.
A rational surface X possessing a cyclic strong exceptional toric system cannot beblown down to F d for d > . Moreover, X ∼ = F , F or X has P as a minimal model.Proof. First, we note that Hirzebruch surfaces F d do not have cyclic strong exceptional toric systemsfor d >
2. It follows from [HP11, Proposition 5.2]. Indeed, let
F, S ∈ Pic( F d ) denote the standardbasis such that F is a fiber and S is a d -curve, d >
0. Then by [HP11, Proposition 5.2] all exceptionaltoric systems on F d have the form (up to cyclic shifts)( F, S + sF, F, S − ( d + s ) F ) , moreover, such system is cyclic strong exceptional if and only if s, − ( d + s ) > −
1. The existence ofa cyclic strong exceptional toric system implies that − d = s − ( d + s ) > −
2, that is, d X can be blown down to P or to some F d .Assume the latter case. It follows from the above remark and Corollary 5.6 that 0 d
2. If X isnot F d then X can be blown down to X ′ which is a blow up of F , F or F at one point. Note thatsuch X ′ can be always blown down to F or F . The latter case is impossible by Corollary 5.6 so X ′ can be blown down to F and further to P . (cid:3) By Theorem 5.3, any exceptional toric system of the first kind can be obtained by elementaryaugmentations and transpositions from some toric system on a Hirzebruch surface. But Theorem 5.3does not track exceptionality of toric systems. The following result guarantees that the intermediatetoric systems can be chosen to be also exceptional (and thus we do not leave the world of exceptionalcollections of line bundles).
Theorem 5.8. (1)
Any exceptional toric system A of the first kind on a rational surface X isan exceptional augmentation. (2) Any strong exceptional toric system A of the first kind on a weak del Pezzo surface X is astrong exceptional augmentation. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 25
Proof. (1) In fact, all intermediate toric systems from the proof of Theorem 5.3 are also exceptional.To make the arguments more clear, we give an independent proof.We prove that any exceptional toric system A of the first kind on a rational surface X is anexceptional augmentation by induction in rank Pic( X ). Cases X = P or X is a Hirzebruch surfaceare trivial. Assume X is the blow-up of X ′ and E is the ( − X ) > E = A k...l for some cyclic segment [ k, . . . , l ]. For anyfixed rank Pic( X ) we argue by induction in the length of that cyclic segment. For l = k we havethat A k = E is a ( − A = augm k ( C ) by Proposition 4.5 where the toric system C on X ′ isexceptional by Proposition 4.8 and we may use induction in rank Pic( X ). For l = k by Lemma 5.1we have(5.5) A m = − m , k m l, A i = − i = k, k + 1 , . . . , m − , m + 1 , . . . , l. We may assume without loss of generality that m = l . The ( − A l is not effective. Indeed,otherwise E = A k...l = A k...l − + A l and thus is reducible, because A k...l − is a ( − − A k...l is a class of anirreducible curve.By Lemma 2.14(2) we have A l ∈ R lo \ R eff = R slo . It follows (see Lemma 4.2(1)) that A ′ = tran l ( A )is also an exceptional toric system of the first kind on X and ( A ′ ) k...l − = A k...l = E . By the inductionhypothesis, A ′ is an exceptional augmentation.(2) The proof is analogous to (1). We prove that any strong exceptional toric system A =( A , . . . , A n ) of the first kind on a weak del Pezzo surface X is a strong exceptional augmentation.We argue by induction in rank Pic( X ). For the induction step, assume X → X ′ is a blow-up withan exceptional curve E . As in (1), we have E = A k...l for some cyclic segment [ k, . . . , l ] ⊂ [1 , . . . , n ].For any fixed rank Pic( X ) we use induction in the length of [ k, . . . , l ]. As in (1), for k = l we havethat A = augm k ( C ) where C is a strong exceptional toric system on X ′ by Propositions 4.5 and 4.8.One can use induction in rank Pic( X ).For k = l we can assume as in (1) that (5.5) holds, m = l and A l is a strong left-orthogonal ( − l ( A ) may be not strong exceptional.If l = n then tran l ( A ) is strong exceptional by Lemma 4.2(2) and there are no problems. Nowsuppose l = n . We claim that the system sh( A ) = ( A , . . . , A n , A ) is strong exceptional (recallthat sh( A ) is automatically exceptional). Assume the contrary. Then by Proposition 3.9(1),(2) thereexists a divisor D = A p...q with 2 p q n which is left-orthogonal but not strong left-orthogonal(note that Proposition 3.9 is applied to sh( A ), not A ). If q < n then A p...q is strong left-orthogonalsince A is strong exceptional. Hence, q = n = l . If p m then D is a left-orthogonal r -class with r = D = − n X i = p ( A i + 2) > − A m = − D is strong left-orthogonal by Proposition 2.9. Hence, m + 1 p .It means that D is a ( − D is effective. Both D and A k...p − (as a ( − A k...l = A k...p − + D is reducible, a contradiction. Thus the toric system sh( A ) =( A , . . . , A n , A ) is strong exceptional. Now the toric system A ′′ = tran n − sh( A ) = ( A , . . . , A n − , A n − ,n , − A n , A ,n )is also strong exceptional (by Lemma 4.2(2)) and ( A ′′ ) k − ,...,n − = A k...n = E is irreducible. Weproceed by induction in l − k . (cid:3) Surfaces with cyclic strong exceptional toric systems
In this section we classify surfaces with cyclic strong exceptional collections of line bundles havingmaximal length. We prove that such collections can exist only on weak del Pezzo surfaces anddetermine which types of weak del Pezzo surfaces possess such collections and which do not.First we demonstrate that a surface with a cyclic strong exceptional collection of line bundleshaving maximal length must be rational. It is proven by Morgan Brown and Ian Shipman in [BS17,Theorem 4.4] that a surface with a full strong exceptional collection of line bundles is rational. Itseems that fullness is not really needed in the proof in [BS17], nevertheless, we prefer to give aseparate simple proof for the case of cyclic strong exceptional collections.
Lemma 6.1.
Let X be a smooth projective surface such that the sheaf O X is exceptional. Assume X admits a cyclic strong exceptional toric system. Then X is a rational surface.Proof. Assume that ( A , . . . , A n ) is a cyclic strong exceptional toric system on X . By Proposi-tion 3.15, the sequence A is admissible. By Proposition 3.8, we have A i > − i . There-fore, A is of the first kind, see Table 3. Then we can group A + . . . + A n into two groups D := A + . . . + A m , D := A m +1 + . . . + A n for some m such that D , D are both non-zerostrong left-orthogonal divisors and D > − , D > −
1. Thus h ( D i ) = D i + 2 > i = 1 , D , D are both effective, hence − K X = A + . . . + A n is effective and − K X is effective andnonzero. It follows immediately that h (2 K X ) = 0. Recall that h ( O X ) = 0 by the assumptions. ByCastelnuovo’s rationality criterion, we conclude that X is a rational surface. (cid:3) Proposition 6.2.
Let X be a smooth projective surface such that the sheaf O X is exceptional. Assume X admits a cyclic strong exceptional toric system. Then X is a weak del Pezzo surface of degree deg( X ) > .Proof. By Lemma 6.1, X must be a rational surface. By Corollary 5.7, X is either F , F , or ablow-up of P . Since F , F are weak del Pezzo surfaces, we have to consider X being a blow-up of P . By [Do12, Prop. 8.1.23] or [CT88, Prop. 0.4], it suffices to show that this blow-up is in almostgeneral position . It means that there is a sequence of s X = X s → X s − → . . . → X → X = P where X m → X m − is a blow-up of one point P m ∈ X m − which does not lie on a smooth rational( − X m − .Note that in our case s X ) > X possesses a cyclic strong exceptionaltoric system, see Proposition 3.17. Denote by L, E , . . . , E m the standard basis in Pic( X m ). Thenfor 0 m − X m ) are of the form E i − E j and ± ( L − E i − E j − E k ) (seeAppendix A or [Ma74, a table in 25.5.2]). Suppose that for some 0 m P m belongsto a ( − R ⊂ X m − which is of the form E i − E j or L − E i − E j − E k , where i < j < k < m .In the first case, consider the divisor D = L − E j − E m = ( L − E i ) + ( E i − E j − E m ) . This divisor is a ( − D ∈ I ( X, A ) by Lemma 5.2 (where A is a cyclic strong exceptionaltoric system on X ). It follows from Proposition 3.9 and Lemma 5.1 that D is strong left-orthogonal.On the other hand, the class E i − E j − E m is effective and we have (using Proposition 2.3) h ( D ) > h ( L − E i ) = 2 > D + 2 = χ ( D ) , thus D cannot be strong left-orthogonal.In the second case the class L − E i − E j − E k − E m ∈ Pic( X m ) is represented by a ( − C ⊂ X m (because the class L − E i − E j − E k ∈ Pic( X m − ) is represented by a ( − N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 27 the point P m ). Let f l : X l → P denote the blow-down. Then f m ( C ) is a line on P and L = f ∗ m L = f ∗ m f m ( C ) = C + m X l =1 c l E l , where c l > f l ( P l ) ∈ f m ( C ), otherwise c l = 0. On the other hand, L = C + E i + E j + E k + E m . It follows that for any l m, l / ∈ { i, j, k, m } the point f l ( P l ) does not belongto f m ( C ), while for l ∈ { i, j, k, m } we have f l ( P l ) ∈ f m ( C ). Hence the blow-ups of P i , P j , P k , P m commute with the blow-ups of other P l -s. Changing the order of blow-ups we can assume that( i, j, k, m ) = (1 , , ,
4) and other blow-ups are performed after these four. By Corollary 5.6, X alsohas a cyclic strong exceptional toric system, denote it by ( A , . . . , A ). We claim that there exists acyclic segment [ p, . . . , q ] ⊂ [1 , . . . ,
7] such that A p...q = 2 L − E − E − E − E . It would follow fromProposition 3.9 that the divisor D = 2 L − E − E − E − E is strong left-orthogonal. Since thedivisor D − L = L − E − E − E − E is effective, we have h ( X , D ) > h ( X , L ) = 3 > D + 2 = χ ( X , D ) , and D cannot be strong left-orthogonal.Now we prove the claim. Note that D is a 0-class in Pic( X ). Actually we will check that all0-classes in Pic( X ) are of the form A k...l . By writing D = aL + P i =1 b i E i and solving equations D = 0 , D · K X = −
2, one can check that there are totally five 0-classes: 2 L − E − E − E − E and L − E i , i = 1 , , ,
4. By Proposition 3.17, the sequence ( A , . . . , A ) can be either( − , − , − , − , − , − , −
1) or ( − , − , − , − , − , − , B , . . . , B ) = ( A , A , A , A , A ) , ( C , . . . , C ) = ( A , A , A , A , A )respectively. Indeed, B i and C i are 0-classes by Proposition 3.6. In the first case the matrix ( A i · B j ) i ∈{ , , } , j has pairwise different columns, therefore B , . . . , B are also different. In the secondcase we use the matrix ( A i · C j ) i ∈{ , , } , j . The matrices can be easily computed using the definitionsof a toric system and of A k...l :( A i · B j ) i ∈{ , , } , j = , ( A i · C j ) i ∈{ , , } , j = . (cid:3) Now we classify surfaces possessing a cyclic strong exceptional toric system. We refer to Appen-dix A for the sets of ( − >
3. We provideexplicit examples of cyclic strong exceptional toric systems on any surface where such a toric systemexists. Recall notation (2.1): E i ...i k = E i + . . . + E i k , L i ...i k = L − E i ...i k . Proposition 6.3.
Toric systems from Table 5 are cyclic strong exceptional. Therefore weak del Pezzosurfaces from Table 5 admit cyclic strong exceptional toric systems. Weak del Pezzo surfaces X fromTable 6 do not admit cyclic strong exceptional toric systems.Proof. First note that the sequences from Table 5 are toric systems. Moreover, it is easy to see thatthey are obtained from each other by elementary augmentations; thus any sequence in Table 5 is astandard augmentation. Next one needs to check that they are cyclic strong exceptional. Any toricsystem A in Table 5 is of the first kind. According to Theorem 3.10, we have to check that all divisors Table 5.
Cyclic strong exceptional toric systemsdegree types toric system9 P L, L, L F H , H , H , H F L , E , L , L F F, S − F, F, S − F (where F = 0 , S = 2 , F S = 1)7 any L , E , L , E , L L , E , L , E , L , E ∅ ); ( A ); (2 A ); ( A ); ( A + A ) L , E , E − E , L , E , L , E ∅ ); ( A ); (2 A , A , A ); L , E , E − E , L , E − E , E , L , E (3 A ); ( A + A ); ( A , A );(2 A + A ); ( A + A ); (2 A + A )3 ( ∅ ); ( A ); (2 A ); ( A ); (3 A ); E − E , L , E , E − E , L , E , E − E , L , E ( A + A ); (4 A ); (2 A + A );(2 A ); ( A +2 A ); (3 A ) Table 6.
Surfaces with no cyclic strong exceptional toric systemsdeg( X ) type of X type of X ′ P ∈ X ′ A A A , A general4 A A general4 D A general on L D A general on E A A , A + A A , E A A general3 D D general3 2 A + A A , E ∩ Q A + A A general on Q A A general on E D D general3 A + A A E ∩ Q E D general on E of the form A k...l where [ k, . . . , l ] ⊂ [1 , . . . , n ] is a cyclic segment such that A k = A k +1 = . . . = A l = − X ) > X ) = 5 , , X ) = 5 : L , E − E ;deg( X ) = 4 : L , E − E , E − E , L ;deg( X ) = 3 : E − E , L , L , E − E , L , L , E − E , L , L . We refer to Tables 9, 10, 11 for the sets R irr ( X ) of ( − , , ± R eff . It may be useful to note that R eff ( X ,A + A ) is maximal N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 29 among all surfaces of degree 5 from Table 5, therefore we need to check the above claim only for X ,A + A . Likewise, for degree 4 the maximal R eff is owned by the surfaces X , A + A and X , A + A .For degree 3 the surfaces with maximal R eff in Table 5 are X , A and X , A .Now we prove the negative part of the Proposition. First, we check that the surfaces X ,A and X ,A have no cyclic strong exceptional toric system. Assume the contrary, A is such system. Thenby Propositions 3.8 and 3.17, A is ( − , − , − , − , − , − , −
1) or ( − , − , − , − , − , − ,
0) (upto a shift and a symmetry). In both cases we see that there exist two ( − D = A i and D = A i +2 which are strong left-orthogonal and such that D · D = 0. On the other hand, we have R slo ( X ,A ) = ±{ L , L , L , L } and R slo ( X ,A ) = ∅ due to Lemma 2.14(1) and Table 9. It iseasy to see that such D and D cannot exist.We claim that all other surfaces X from Table 6 are blow-ups of X ,A or X ,A . It would followthen from Corollary 5.6 that they admit no cyclic strong exceptional toric systems. In the right twocolumns of Table 6 we present a surface X ′ and a point P ∈ X ′ such that the blow-up of X ′ at P has the same type as X (here Q denotes the ( − L − E ∈ Pic( X )). We refer to [CT88],Propositions 6.1 and 8.4, for the verifying.Indeed, let p : X → X ′ be the blow-up of a point P ∈ X ′ with the exceptional divisor E . Thenone has R irr ( X ) = p ∗ ( R irr ( X ′ )) G { p ∗ ( C ) − E | C ∈ I irr ( X ′ ) , C ∋ P } . The r.h.s. of the above formula can be read from the diagrams in [CT88, Propositions 6.1 and 8.4]. (cid:3) An application to dimension of D b (coh X )The paper [BF12] by Matthew Ballard and David Favero initiated the study of the relation be-tween the dimension of triangulated categories in the sense of Rafael Rouquier [Ro08] and full cyclicstrong exceptional collections. Recall the definition of dimension. For an object G of a triangulatedcategory T define full subcategories h G i i , i > F ∈ h G i if F is a directsummand in some finite direct sum of shifts of G . For i > F ∈ h G i i if there exists anexact triangle F → F ′ → F i − → F [1] with F ∈ h G i , F i − ∈ h G i i − , and F is a direct summandin F ′ . Denote h G i = ∪ i h G i i . Object G is called a classical generator if T = h G i . Generator G issaid to have generation time n if n is the minimal number such that T = h G i n . Rouquier defineddimension dim T of a triangulated category T as the minimal possible generation time for all classicalgenerators in T . Conjecture 7.1 (Orlov) . For a smooth projective variety X one has dim D b (coh( X )) = dim X. The bound D b (coh( X )) > dim X was proven by Rouquier in [Ro08], while the equality is knownonly for some special varieties. A useful tool for proving the conjecture was found by Ballard andFavero in [BF12]: Theorem 7.2 (See [BF12, Theorem 3.4]) . Let X be a smooth proper variety over a perfect field k ,let T ∈ D b (coh( X )) be a classical generator such that Hom i ( T, T ) = 0 for i = 0 . Denote i = max { i | Hom i ( T, T ⊗ O X ( − K X )) = 0 } . Then the generation time of T is dim X + i . Corollary 7.3.
Let X be a smooth proper variety over a perfect field k , let ( E , . . . , E n ) be a fullstrong exceptional collection on X . Suppose the generation time of the generator T = ⊕ i E i is equalto dim X . Then the collection ( E , . . . , E n ) is cyclic strong exceptional. We are able to prove the converse statement for collections of line bundles on surfaces.
Proposition 7.4.
Let X be a smooth projective surface with a full cyclic strong exceptional collection ( O X ( D ) , . . . , O X ( D n )) of line bundles. Then the generator T = ⊕ ni =1 O X ( D i ) of the category D b (coh( X )) has generationtime two. Therefore, dimension of D b (coh( X )) is equal to .Proof. By Proposition 6.2, X is a weak del Pezzo surface and d := deg( X ) >
3. By Theorem 7.2, wehave to check that H i ( X, O X ( D − K X )) = 0 for any i > D of the form D k − D l . Thereare three possible cases: D = D k − D l , k < l, ,D k − D l , k > l. For D = D k − D l where k < l we have H i ( X, O X ( D k − D l − K X )) = Hom i ( O X ( D l ) , O X ( D k − K X )) = 0for any i > O X ( D l ) , O X ( D l +1 ) , . . . , O X ( D n ) , O X ( D − K X ) , O X ( D − K X ) , . . . , O X ( D k − K X ) , . . . , O X ( D l − − K X ))is strong exceptional by assumption.For D = D k − D l where k > l we borrow the arguments from Lemma 3.9 of [BF12]. By [Do12,Theorem 8.3.2], the linear system | − K X | has no base points. By Bertini’s Theorem, the generaldivisor Z ∈ | − K X | is a smooth irreducible curve. By the adjunction formula, Z is a curve of genus 1.Consider the standard exact sequence(7.1) 0 → O X ( D ) → O X ( D − K X ) → O Z ( D − K X ) → . One has H i ( X, O X ( D )) = 0 for i > O X ( D ) , . . . , O X ( D n )).Also, Z · ( D − K X ) = K X ( K X − D ) = d − D · K X = d + χ ( D ) > d > D is a strongleft-orthogonal divisor and hence χ ( D ) >
0. It follows that H i ( X, O Z ( D − K X )) = H i ( Z, L )where L is a line bundle on Z of degree >
3. Since Z has genus 1, we get H i ( Z, L ) = 0 for i > H i ( X, O X ( D − K X )) = 0for i > D = 0. Alternative, one can use Kawamata-Viehweg vanishing: H i ( X, O X ( − K X )) ∼ = H − i ( X, O X (2 K X )) ∗ = 0 for i > − K X is nef and big, see [KM98,Theorem 2.64]. (cid:3) Corollary 7.5.
Dimension of D b (coh( X )) is equal to for any weak del Pezzo surface X fromTable 5. A toric system which is not an augmentation
Here we give an example of a strong exceptional toric system A on a weak del Pezzo surface X ofdegree 2 such that A is not an augmentation in the weak sense. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 31
By [De14, Section 3.6], a weak del Pezzo surface X of degree 2 of type A + 2 A has the followingconfiguration of ( − ◦ ) and ( − • ).(8.1) • ⑦⑦⑦⑦⑦⑦⑦ ◦ • ❅❅❅❅❅❅❅ • ❅❅❅❅❅❅❅ ◦ • ◦ •• ◦ • ⑦⑦⑦⑦⑦⑦⑦ Two vertices here are connected by a line if and only if the corresponding curves intersect (then suchintersection is unique and transversal). There are four ( − X , and they do not intersect.Let p : X → X ′ be the blow-down of these curves. One can check that p maps ( − X to the following configuration of curves on X ′ (where ∗ denotes a 0-curve), and there are no othernegative curves on X ′ :(8.2) ◦ ⑧⑧⑧⑧⑧⑧⑧ ◦ ❄❄❄❄❄❄❄ ◦ ❄❄❄❄❄❄❄ ∗ ◦◦ ◦ ⑧⑧⑧⑧⑧⑧⑧ It means that X ′ is a blow-up of P at three generic points, and we may assume that the curves on(8.2) belong to the following classes (here we use notation (2.1) from Section 2.1): L ④④④④④④④④ E ❈❈❈❈❈❈❈❈ E ❈❈❈❈❈❈❈❈ L L L E ④④④④④④④④ Now p is the blow-up of the four points P = E ∩ L , P = E ∩ L , P = E ∩ C, P = L ∩ C on X ′ , where C is some 0-curve representing the class L . The curves on (8.1) belong to the followingclasses: L ttttttttt E E − E ❏❏❏❏❏❏❏❏❏ E − E ❏❏❏❏❏❏❏❏❏ E L E L L E E − E ttttttttt Thus, in the notation of Definition 2.6 we have(8.3) I irr ( X ) = { E , E , E , E } ,R irr ( X ) = { L ; L , E − E , L ; E − E , L , E − E } and using Proposition 2.8 we get that(8.4) R eff ( X ) = { L ; L , E − E , L , L , L , L − E ; E − E , L , E − E , L , L , L } . Consider the sequence A = ( L , L , − L , L − E , E , L , − L , L − E , E − E , − L + E ) . One can check that A is a toric system and A = ( − , − , − , − , − , − , − , − , − , − . Denote C i = 3 L − E − E i and Q ij = 2 L − E + E ij , these are ( − I ( X, A ) == { A ...j | j } ∪ { A i...j | i , j } ∪ { A i...j | i , j } == { L , Q , L , C } ∪ { E , Q , L , Q , L , C , Q , C , E , Q , L , Q }∪∪ { Q , L , Q , Q , L , Q } . It follows (see (8.3)) that there are no ( − I ( X, A ) and therefore A is not an augmentationin the weak sense, see Definition 4.9 and the definition of I ( X, A ) in Section 5. A forteriori, A isneither a standard augmentation, nor exceptional or strong exceptional augmentation.On the other hand, A is strong exceptional. To check this, we use the next Lemma 8.1.
Let X be a weak del Pezzo surface of degree . Let A be a toric system on X with A = ( − , − , − , − , − , − , − , − , − , − . Then A is strong exceptional if and only if all ( − -classes A k...l with k l are neithereffective nor anti-effective and the ( − -classes A , and A are not anti-effective.Proof. Direct consequence of Theorem 3.10. (cid:3)
By the above lemma, we have to check that the ( − A = L , A = − L , A = 2 L − E , A = E − E , A = L , A = 2 L − E ,A = L , A = − L , A = E − E , A = E − E are neither effective nor anti-effective and that the ( − A = − (2 L − E ) , A , = − (2 L − E )are not anti-effective. The first claim is straightforward by (8.4), for the second the argument isbased on the following trivial fact. Lemma 8.2.
Suppose D is an effective divisor on a surface and C is an irreducible reduced curve.If D · C < then D − C is also an effective divisor. Suppose now that D = − A = 2 L − E >
0. Since D · L = − L ∈ R irr ( X ) it followsfrom Lemma 8.2 that D − L = L + E >
0. Since ( L + E ) · E = − E ∈ I irr ( X ),it follows from Lemma 8.2 that ( L + E ) − E = L >
0, which contradicts to (8.4). Similarly(with 6 and 7 interchanged) one checks that − A , = 2 L − E is not effective. Therefore, A is astrong exceptional toric system by Lemma 8.1. Remark 8.3.
Note that the surface X was obtained from P in two steps of blow-ups. It agreeswith Conjecture 1.1. N CYCLIC STRONG EXCEPTIONAL COLLECTIONS OF LINE BUNDLES ON SURFACES 33
Appendix A. Classification of weak del Pezzo surfaces
In this Appendix we present classification of weak del Pezzo surfaces of degrees 7 to 3. Ourreferences here were [CT88] and [De14]. Recall that two weak del Pezzo surfaces X and X ′ are of thesame type if there exists an isometry Pic( X ) → Pic( X ′ ) preserving canonical class and the sets ofnegative curves. The set of ( − X is a set of simple roots of some root subsystem in R ( X ),see Proposition 2.8. It turns out (see [De14, Remark 2.1.4]) that for surfaces X of degree > d = deg( X ), the incidence graph Γ of ( − m of ( − d, Γ , m ) or just by (Γ , m ) when the degree is clear fromthe context. We omit m if d and Γ determine the type uniquely. Any weak del Pezzo surface (exceptfor Hirzebruch surfaces F and F ) is a blow-up of P (maybe at infinitesimal points), we denote thestandard basis in Pic( X ) by L, E , . . . , E n . In the following tables, for any type of degrees 7 to 3, wepresent the set R irr ( X ) ⊂ R ( X ) of ( − X ) corresponding to irreducible ( − X is a certain weak del Pezzo surface of that type.By [Ma74, a table in 25.5.2], R ( X ) = {± ( E − E ) } for X of degree 7, R ( X ) = { E i − E j , ± L ijk } for X of degree 6 , ,
4, and R ( X ) = { E i − E j , ± L ijk , ± Z } for X of degree 3, where L ijk = L − E i − E j − E k and Z := 2 L − E − E − E − E − E − E . Table 7.
Weak del Pezzo surfaces of degree 7type R irr ∅ ∅ A E − E Table 8.
Weak del Pezzo surfaces of degree 6type R irr ∅ ∅ A , E − E A , L A E − E , L A E − E , E − E A + A L , E − E , E − E Table 9.
Weak del Pezzo surfaces of degree 5type R irr ∅ ∅ A E − E A E − E , E − E A E − E , E − E A + A L , E − E , E − E A E − E , E − E , E − E A E − E , E − E , E − E , L Table 10.
Weak del Pezzo surfaces of degree 4type R irr ∅ ∅ A E − E A , E − E , E − E A , L , E − E A E − E , E − E A L , E − E , E − E A + A E − E , E − E , E − E A , E − E , E − E , E − E A , L , E − E , E − E A E − E , E − E , L , L A + A E − E , L , E − E , E − E A + A E − E , L , E − E , E − E A E − E , E − E , E − E , E − E A + A E − E , L , L , E − E , E − E D E − E , E − E , E − E , L D E − E , E − E , E − E , E − E , L Table 11.
Weak del Pezzo surfaces of degree 3type R irr ∅ ∅ A Z A E − E , E − E A E − E , E − E A E − E , E − E , E − E A + A E − E , E − E , E − E A E − E , E − E , E − E A E − E , E − E , E − E , Z A + A E − E , L , E − E , E − E A + A E − E , E − E , E − E , E − E A E − E , E − E , E − E , E − E A E − E , E − E , E − E , E − E D E − E , E − E , E − E , L A + A E − E , Z, E − E , E − E , E − E A + 2 A L , E − E , E − E , E − E , E − E A + A Z, E − E , E − E , E − E , E − E A E − E , E − E , E − E , E − E , E − E D E − E , E − E , E − E , E − E , L A E − E , E − E , E − E , E − E , L , L A + A Z, E − E , E − E , E − E , E − E , E − E E L , E − E , E − E , E − E , E − E , E − E References [BF12] M. Ballard, D. Favero, “Hochschild dimensions of tilting objects”,
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Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, RUSSIA
E-mail address : [email protected] Department of Mathematics, Indiana University, 831 E. Third St., Bloomington, IN 47405, USA
E-mail address : [email protected] Department of Mathematics, Indiana University, 831 E. Third St., Bloomington, IN 47405, USA
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