Preface ........................................................ ix
Chapter 1 Introduction ......................................... 1
1.1 The Monster construction ................................... 1
1.2 Coordinates and the contact case ........................... 1
1.3 Symmetries. Equivalence of points of the Monster ........... 2
1.4 Prolonging symmetries ...................................... 2
1.5 The basic theorem .......................................... 2
1.6 The Monster and Goursat distributions ...................... 3
1.7 Our approach ............................................... 4
1.8 Proof of the basic theorem ................................. 5
1.9 Plan of the paper .......................................... 6
Acknowledgements ............................................... 11
Chapter 2 Prolongations of integral curves. Regular,
vertical, and critical curves and points ............ 13
2.1 From Monster curves to Legendrian curves .................. 13
2.2 Prolonging curves ......................................... 13
2.3 Projections and prolongations of local symmetries ......... 15
2.4 Proof of Theorem 2.2 ...................................... 15
2.5 Prom curves to points ..................................... 16
2.6 Non-singular points ....................................... 17
2.7 Critical curves ........................................... 17
2.8 Critical and regular directions and points ................ 20
2.9 Regular integral curves ................................... 20
2.10 Regularization theorem .................................... 22
2.11 An equivalent definition of a non-singular point .......... 23
2.12 Vertical and tangency directions and points ............... 24
Chapter 3 RVT classes. RVT codes of plane curves. RVT and
Puiseux ............................................. 27
3.1 Definition of RVT classes ................................. 27
3.2 Two more definitions of a non-singular point .............. 28
3.3 Types of RVT classes. Regular and entirely critical
prolongations ............................................. 28
3.4 Classification problem: reduction to regular RVT
classes ................................................... 29
3.5 RVT classes as subsets of  c 2 ........................... 29
3.6 Why tangency points? ...................................... 30
3.7 RVT code of plane curves .................................. 31
3.8 RVT code and Puiseux characteristic ....................... 33
Chapter 4 Monsterization and Legendrization. Reduction
theorems ............................................ 39
4.1 Definitions and basic properties .......................... 39
4.2 Explicit calculation of the legendrization of RVT
classes ................................................... 41
4.3 Prom points to Legendrian curves .......................... 42
4.4 Simplest classification results ........................... 43
4.5 On the implications and shortfalls of Theorems 4.14 and
4.15 ...................................................... 44
4.6 Prom points to Legendrian curve jets.
The jet-identification number ............................. 45
4.7 The parameterization number ............................... 47
4.8 Evaluating the jet-identification number .................. 50
4.9 Proof of Proposition 4.44 ................................. 52
4.10 Prom Theorem В to Theorem 4.40 ............................ 53
4.11 Proof that critical points do not have a
jet-identification number ................................. 55
4.12 Proof of Proposition 4.26 ................................. 55
4.13 Conclusions. Things to come ............................... 55
Chapter 5 Reduction algorithm. Examples of classification
results ............................................. 57
5.1 Algorithm for calculating the Legendrization and the
parameterization number ................................... 57
5.2 Reduction algorithm for the equivalence problem ........... 59
5.3 Reduction algorithm for the classification problem ........ 60
5.4 Classes of small codimension consisting of a finite
number of orbits .......................................... 61
5.5 Classification of tower-simple points ..................... 63
5.6 Classes of high codimension consisting of one or two
orbits .................................................... 67
5.7 Further examples of classification results; Moduli ........ 69
Chapter 6 Determination of simple points ...................... 71
6.1 Tower-simple and stage-simple points ...................... 71
6.2 Determination theorems .................................... 71
6.3 Explicit description of stage-simple RVT classes .......... 74
6.4 Local simplicity of RVT classes ........................... 79
6.5 Proof of Theorem 6.4 ...................................... 81
6.6 Proof of Theorem 6.6 ...................................... 82
6.7 Proof of Theorem 6.30 ..................................... 83
Chapter 7 Local coordinate systems on the Monster ............. 85
7.1 The KR coordinate system .................................. 85
7.2 Critical curves in the KR coordinates ..................... 88
7.3 RVT classes and KR coordinates ............................ 89
7.4 Monsterization in KR coordinates .......................... 90
Chapter 8 Prolongations and directional blow-up. Proof of
Theorems A and В .................................... 95
8.1 Directional blow-up and KR coordinates .................... 96
8.2 Directional blow-up and the maps  T,V, L ................. 99
8.3 Proof of Theorem A for Puiseux characteristics
[λ0; λ1] ................................................. 100
8.4 Further properties of the directional blow-up ............ 101
8.5 Proof of Theorem A for arbitrary Puiseux
characteristics .......................................... 104
8.6 Proof of Theorem В of section 4.8 ........................ 105
8.7 Proof of Propositions 8.10 and 8.11 ...................... 106
Chapter 9 Open questions ..................................... 109
9.1 Unfolding versus prolongation ............................ 109
9.2 Prolongation = blow-up? .................................. 109
9.3 Puiseux characteristic of Legendrian curves .............. 113
9.4 The infinite Monster ..................................... 114
9.5 Moduli and projective geometry ........................... 116
9.6 RVT and the growth vector ................................ 116
Appendix A Classification of integral Engel curves ........... 119
Appendix B Contact classification of Legendrian curves ....... 123
B.l Reduction theorems for curves ............................ 123
B.2 Reduction theorems for jets .............................. 124
B.3 Proof of Proposition 5.6, part (i) ....................... 126
B.4 Proof of Proposition 5.6, part (ii) ...................... 127
B.5 Proof of Proposition 5.5 ................................. 127
Appendix C Critical, singular and rigid curves ............... 131
C.l Critical ⇒ locally rigid ................................. 131
C.2 Singular ⇒ critical ...................................... 132
C.3 Another proof that vertical curves are rigid ............. 133
Bibliography .................................................. 135
Index ......................................................... 137
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