On the automorphism group of certain algebraic varieties

iii Chapter 0. Introduction 1 Chapter 1. Stable curves 3 1 Preliminaries 5 1.1 Stable curves 5 1.2 Stable graphs 6 1.3 Program 7 2 Generating stable graphs 8 2.1 Description of the algorithm 8 2.2 The program generates all graphs 10 2.3 Description of the ranges 14 2.4 Performance 19 3 Computing automorphisms in low genera 20 3.1 Results for smooth curves 21 3.2 Automorphisms of stable curves 25 Chapter 2. Numerical Godeaux surfaces 31 1 Classification and construction 33 1.1 Preliminaries 33 1.2 Torsion group of Godeaux surfaces 33 1.3 Torsion of order five 36 1.4 Torsion or order four 37 1.5 Torsion of order three 38 2 Automorphism groups of known surfaces 38 2.1 Computing automorphisms 39 2.2 The results 46 2.3 Moduli stacks 50 2.4 Inertia stacks 55 3 General bounds on the automorphism groups 58 3.1 Preliminaries 59 3.2 Assuming no fixed part 61 3.3 Actions of cyclic groups on a curve 69