《Algebraic Threefolds With Special Regard Toproblems of Rationality》

Ⅰ．Invariantive Theory1

1．Ordinary singularities．Projective characters1

2．Adjoint primals in S42

3．Linear systems of surfaces3

4．Characters of a linear system4

5．Jacobian systems．The canonical system5

6．The arithmetic genus．The irregularities9

7．The virtual arithmetic genus12

8．The adjoint systems13

9．The relation P3 ＝ Pa14

10．Fundamental property of adjoint systems．Absolute invariance of Pa16

11．Examples17

12．Threefolds of linear genus unity19

Ⅱ．Systems of Equivalence21

1．Introduction21

2．Covariant point sets23

3．Invariant series and systems24

4．Further developments28

5．Extension to varieties of higher dimension29

Ⅲ．Systems of Surfaces31

1．The RIEMANN-ROCH theorem31

2．Applications and extensions32

3．Varieties which contain only complete intersections34

4．Theory of the base35

5．Applications of the PICARD variety37

Ⅳ．Criteria of Rationality39

1．Introduction39

2．The standard forms of NOETHER and ENRIQUES41

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

4．Varieties containing congruences of rational curves46

5．ENRIQUES＇S theorem and its applications48

6．Congruences of index greater than unity50

7．The general problem of unisecants51

8．Varieties containing systems of elliptic or hyperelliptic curves55

9．Threefolds of given curve section58

10．Threefolds containing systems of rational surfaces61

11．FANL＇S theorem and allied results65

12．Threefolds containing systems of surfaces with P(1)≦171

13．Linear systems of surfaces of maximum dimension74

Ⅴ．The Adjunction Problem76

1．Introduction76

2．Some varieties on which adjunction terminates77

3．Threefolds which possess anticanonical systems79

4．The FANO threefolds83

5．Classification of the FANO threefolds85

6．Conditions for unirationality or birationality88

7．Threefolds of the first species90

8．Threefolds whose prime sections are ENRIQUES surfaces93

9．Questions of irrationality94

10．Some unsolved problems99

Ⅵ．Continuous Transformation Groups101

1．Groups of auto-collineations101

2．Automorphisms in general104

3．Abelian and quasi-Abelian Varieties106

4．Pseudo-Abelian varieties109

5．Elliptic and hyperelliptic threefolds115

6．Threefolds which possess finite continuous groups of automorphisms116

7．Extensions to varieties of higher dimension119

8．Other types of automorphism121

Appendix124

Bibliography137

Index141