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Ⅰ．Invariantive Theory 1

1．Ordinary singularities．Projective characters 1

2．Adjoint primals in S4 2

3．Linear systems of surfaces 3

4．Characters of a linear system 4

5．Jacobian systems．The canonical system 5

6．The arithmetic genus．The irregularities 9

7．The virtual arithmetic genus 12

8．The adjoint systems 13

9．The relation P3 ＝ Pa 14

10．Fundamental property of adjoint systems．Absolute invariance of Pa 16

11．Examples 17

12．Threefolds of linear genus unity 19

Ⅱ．Systems of Equivalence 21

1．Introduction 21

2．Covariant point sets 23

3．Invariant series and systems 24

4．Further developments 28

5．Extension to varieties of higher dimension 29

Ⅲ．Systems of Surfaces 31

1．The RIEMANN-ROCH theorem 31

2．Applications and extensions 32

3．Varieties which contain only complete intersections 34

4．Theory of the base 35

5．Applications of the PICARD variety 37

Ⅳ．Criteria of Rationality 39

1．Introduction 39

2．The standard forms of NOETHER and ENRIQUES 41

3．Varieties which are unirational in K(Sh) 43

4．Varieties containing congruences of rational curves 46

5．ENRIQUES＇S theorem and its applications 48

6．Congruences of index greater than unity 50

7．The general problem of unisecants 51

8．Varieties containing systems of elliptic or hyperelliptic curves 55

9．Threefolds of given curve section 58

10．Threefolds containing systems of rational surfaces 61

11．FANL＇S theorem and allied results 65

12．Threefolds containing systems of surfaces with P(1)≦1 71

13．Linear systems of surfaces of maximum dimension 74

Ⅴ．The Adjunction Problem 76

1．Introduction 76

2．Some varieties on which adjunction terminates 77

3．Threefolds which possess anticanonical systems 79

4．The FANO threefolds 83

5．Classification of the FANO threefolds 85

6．Conditions for unirationality or birationality 88

7．Threefolds of the first species 90

8．Threefolds whose prime sections are ENRIQUES surfaces 93

9．Questions of irrationality 94

10．Some unsolved problems 99

Ⅵ．Continuous Transformation Groups 101

1．Groups of auto-collineations 101

2．Automorphisms in general 104

3．Abelian and quasi-Abelian Varieties 106

4．Pseudo-Abelian varieties 109

5．Elliptic and hyperelliptic threefolds 115

6．Threefolds which possess finite continuous groups of automorphisms 116

7．Extensions to varieties of higher dimension 119

8．Other types of automorphism 121

Appendix 124

Bibliography 137

Index 141