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106 8. TUBULAR EXCEPTIONAL CURVES2Theorem 8.2.2 is to realize these K-theoretical automorphisms by tubular mutationson the derived level. In the cases where the rank of the Grothendieck group isgreater or equal than four this can be accomplished without problem, so that thereare also at most two orbits on the derived level. But if the rank of the Grothendieckgroup equals three (that is, in each tubular family there is precisely one exceptionaltube, and this is of rank two) the index depends also on the arithmetic of thebase field, which leads to cases of index three (see Proposition 8.2.5 below; see alsoAppendix B for a list of the 17 tubular cases). Moreover, the analysis in [59] showsthe following propositions which stress the special role of the rank three case.Proposition 8.2.5. Let X be a tubular exceptional curve such that theGrothendieck group is of rank three. Then the index of X is at most three.Assume that there exists q ∈ ̂Q such that the numerical type of X〈q〉 is ε =1and such that there exists a unirational point in X〈q〉. Then the index of X is atmost two.0If the symbol of X equals @ 2 14A and if, for example, X and X〈2〉 contain unirationalpoints, then the index is one.Proof. Let q ∈ ̂Q such that the numerical type of X〈q〉 is ε = 1. Switchingto X〈q〉 we can assume that the numerical type of X itself is ε = 1. The Φ-image of the tubular mutations with respect to the exceptional tubes containingthe structure sheaf L and an exceptional simple object together with the Φ-imageof the translation T gives the subgroup 〈t, l, s 4 〉 of B 3 . This subgroup acts withthree orbits on the rational helix. The remaining assertions follow from the analysisin [57, 10.].□It is easy to see that each (2, 2)-bimodule over a finite field k is non-simple,hence there is a unirational point for the associated curve (compare 0.6.2). Thesame is true for the field k = R of real numbers. Thus we getCorollary 8.2.6. The index of a tubular algebra (or tubular exceptional curve)over any finite field k, or over the field R of real numbers, is at most two. □Proposition 8.2.7 ([59]). Let X be a tubular exceptional curve and r be therank of its Grothendieck group. There is a subgroup U of Aut(D b (X)) acting transitivelyon each Aut(D b (X))-orbit in the set of all slope categories such that additionallythe following holds:If r ≥ 4, or if the index is three, then U is generated by the translation T andtwo tubular mutations associated to exceptional tubes.□Proposition 8.2.8. Let X be a tubular exceptional curve such that theGrothendieck group is of rank three.(1) Let U be the subgroup of Aut(D b (X)) which is generated by the translationT and all tubular mutations associated to exceptional tubes (for all slopes). ThenU acts with three orbits on the set of all slope categories.(2) Moreover, for K 0 (X) there are three possible cases:0a) The symbol is@ 2 42@ 2 441A. Then the index of X is one, two or three (the precisevalue depending on the arithmetic of k).0 1«b) The symbol is2 . Then the index of X is two or three.A or „ 22
c) The symbol is „ 248.3. A TUBULAR CURVE OF INDEX THREE 1070«or @ 2 2 221A. Then the index of X is two or three. □Remark 8.2.9. (1) For the case c) in the preceding proposition we constructedexamples for indices two and three, respectively ([59]; see the following sections).We remark that in [59] we stated that in cases a) and b) examples of index threedo not exist. This was based on an argument which turned out to be wrong. Wenow do not see any reason why examples of index three in these cases should notexist.(2) Which index actually occurs depends on the question which of the 6 cosetsof the subgroup 〈t, l, s 4 〉 in B 3 have representatives which are realizable by automorphismsof the derived category. If (besides the identity) none of these representativesis realizable the index is three. If additionally only s 2 is realizable then the index istwo. If any other coset is realizable then the index is one (only possible in case a).Another question (related to Problem 5.2.2) is whether such realizations (assumingexistence) are always possible by tubular mutations (compare Proposition 6.3.4).8.3. A tubular curve of index threeIn this section we exhibit our example [59] of a tubular exceptional curve X ofindex three and list its further properties. Knowledge of the action of Aut(D b (X))on the rational helix allows to determine Aut(D b (X)) itself.Recall that for sequences p = (p 1 ,...,p t )andd = (d 1 ,...,d t ) of positiveintegers the abelian group Z [ ]d 1p 1,..., d tp tis also denote by L(p, d).Proposition 8.3.1. There is a tubular exceptional curve X over k = Q suchthat the following holds:(1) The index of X is three.(2) A projective coordinate algebra of X is given by the graded factorial algebraR = Q[X, Y, Z, U]/(X 2 + Y 2 + Z 2 ,U 2 − X 2 − 3Y 2 ),which is graded by L(p, d) with p =(1, 1, 1, 2) and d =(1, 1, 1, 2). (This propertyuniquely determines X.)(3) There is a tilting bundle whose endomorphism ring is the canonical algebraΛ given by the speciesKKFQFplus certain relations, where K = Q( √ −3, √ 2), F = ( )−1, −1Q be the skew field ofquaternions over Q on generators i and j with relationsi 2 = −1, j 2 = −1, ij = −ji,and moreover, M is the bimodule from 5.7.3, that is, M = K (K ⊕ K) F with thecanonical K-action and the F -action defined by(x, y) · i = √ 1 ( √ 2x + y, x − √ 2y), (x, y) · j =(y, −x)−3for all x, y ∈ K.(4) For the automorphism groups we have Aut(X) ≃ V 4 , the Klein four group,andAut(D b (X)) ≃ (Z 2 × V 4 ) ⋊ (F 2 × Z),M
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M EMOIRSof theAmerican Mathematical
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M EMOIRSof theAmerican Mathematical
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Dedicated to the memory of my belov
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viiiCONTENTS5.1. The automorphism g
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IntroductionCurves of genus zero. I
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INTRODUCTION 3(although not under t
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INTRODUCTION 5ideal P y associated
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INTRODUCTION 7It is important to st
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INTRODUCTION 9of the commutativity
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12 0. BACKGROUNDFor each λ ∈ k l
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14 0. BACKGROUND0.3. Canonical alge
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16 0. BACKGROUND0.3.8 (Exceptional
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18 0. BACKGROUNDWe say that M is a
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20 0. BACKGROUNDThen there is a nat
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22 0. BACKGROUND0.4.8. Let x ∈ X
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0.6. RATIONAL POINTS 25By [89] ther
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CHAPTER 1Graded factorialityIn this
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1.1. EFFICIENT AUTOMORPHISMS 31I M
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1.2. PRIME IDEALS AND UNIVERSAL EXT
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1.3. PRIME IDEALS AS ANNIHILATORS 3
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1.3. PRIME IDEALS AS ANNIHILATORS 3
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40 1. GRADED FACTORIALITYProof. Let
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42 1. GRADED FACTORIALITYOne can su
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44 1. GRADED FACTORIALITYThe non-si
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46 1. GRADED FACTORIALITYProof. The
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CHAPTER 2Global and local structure
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2.2. LOCALIZATION AT PRIME IDEALS 5
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2.2. LOCALIZATION AT PRIME IDEALS 5
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Page 73 and 74: CHAPTER 3Tubular shifts and prime e
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Page 88 and 89: 76 5. AUTOMORPHISM GROUPSan involut
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Page 109 and 110: CHAPTER 7Exceptional objectsIn this
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Page 131: APPENDIX BThe tubular symbols( ) (
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Page 139 and 140: Indexp-cycle, 89q-symbol, 104Algebr
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