viii CONTENTS

5.1. The automorphism group of a homogeneous curve 76

5.2. The structure of Aut(H) 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.1. Tables for the domestic and tubular cases 113

Appendix B. The tubular symbols 119

Bibliography 121

Index 127