Introduction .................................................... 1
Chapter 0. Background ......................................... 11
0.1 Notation .................................................. 11
0.2 One-parameter families, generic modules and tameness ...... 11
0.3 Canonical algebras and exceptional curves ................. 14
0.4 Tubular shifts ............................................ 19
0.5 Tame bimodules and homogeneous exceptional curves ......... 22
0.6 Rational points ........................................... 24
Part 1. The homogeneous case .................................. 27
Chapter 1. Graded factoriality ................................ 29
1.1 Efficient automorphisms ................................... 30
1.2 Prime ideals and universal extensions ..................... 33
1.3 Prime ideals as annihilators .............................. 35
1.4 Noetherianness ............................................ 38
1.5 Prime ideals of height one are principal .................. 39
1.6 Unique factorization ...................................... 41
1.7 Examples of graded factorial domains ...................... 43
The non-simple bimodule case .............................. 44
The quaternion case ....................................... 46
The square roots case ..................................... 46
Chapter 2. Global and local structure of the sheaf category ... 49
2.1 Serre's theorem ........................................... 49
2.2 Localization at prime ideals .............................. 51
2.3 Noncommutativity and the multiplicities ................... 56
2.4 Localizing with respect to the powers of a prime
element ................................................... 58
2.5 Zariski topology and sheafification ....................... 59
Chapter 3. Tubular shifts and prime elements .................. 61
3.1 Central prime elements .................................... 61
3.2 Non-central prime elements and ghosts ..................... 62
Chapter 4. Commutativity and multiplicity freeness ............. 69
4.1 Finiteness over the centre ................................ 69
4.2 Commutativity of the coordinate algebra ................... 70
4.3 Commutativity of the function field ....................... 71
Chapter 5. Automorphism groups ................................ 75
5.1 The automorphism group of a homogeneous curve ............. 76
5.2 The structure of Aut( ) .................................. 77
5.3 The twisted polynomial case ............................... 78
5.4 On the Auslander-Reiten translation as functor ............ 80
5.5 The quaternion case ....................................... 82
5.6 The homogeneous curves over the real numbers .............. 82
5.7 Homogeneous curves with finite automorphism group ......... 85
Part 2. The weighted case ..................................... 87
Chapter 6. Insertion of weights ............................... 89
6.1 p-cycles .................................................. 89
6.2 Insertion of weights into central primes .................. 91
6.3 Automorphism groups for weighted curves ................... 95
Chapter 7. Exceptional objects ................................ 97
7.1 Transitivity of the braid group action .................... 97
7.2 Exceptional objects and graded factoriality ............... 99
Chapter 8. Tubular exceptional curves ........................ 101
8.1 Slope categories and the rational helix .................. 102
8.2 The index of a tubular exceptional curve ................. 105
8.3 A tubular curve of index three ........................... 107
8.4 A related tubular curve of index two ..................... 109
8.5 Line bundles which are not exceptional ................... 110
Appendix A. Automorphism groups over the real numbers ........ 113
A.l Tables for the domestic and tubular cases ................ 113
Appendix B. The tubular symbols .............................. 119
Bibliography .................................................. 121
Index ......................................................... 127
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