Contents Introduction 1 Chapter 1. The canonical system 7 Chapter 2. Some known results 10 Chapter 3. Surfaces with K2 = 7, pg' 4, such that the canonical system doesn't have a fixed part 14 3.1. Surfaces of type (LI) 14 3.2. Surfaces of type (1.2) 18 3.3. Surfaces of type (1.3) 26 3.4. Surfaces of type (III) 27 Chapter 4. \K\ has a (non trivial) fixed part 43 4.1. Surfaces of type (F) 43 4.2. Surfaces of type (V) 48 Chapter 5. The moduli space 51 5.1. The "fine" classification of surfaces with Kg = 7 and p9 = 4 (revisited) 51 5.2. Counting the number of moduli 54 5.3. The irreducible components of the moduli space 64 5.4. The fundamental group 73 Bibliography 78
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