Algebraic Varieties, 1956 Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge Series, Vol. 12

Algebraic geometry has always been an ec1ectic science, with its roots in algebra, function-theory and topology. Apart from early resear­ ches, now about a century old, this beautiful branch of mathematics has for many years been investigated chiefly by the Italian school which, by its pioneer work, based on algebro-geometric methods, has succeeded in building up an imposing body of knowledge. Quite apart from its intrinsic interest, this possesses high heuristic value since it represents an essential step towards the modern achievements. A certain lack of rigour in the c1assical methods, especially with regard to the foundations, is largely justified by the creative impulse revealed in the first stages of our subject; the same phenomenon can be observed, to a greater or less extent, in the historical development of any other science, mathematical or non-mathematical. In any case, within the c1assical domain itself, the foundations were later explored and consolidated, principally by SEVERI, on lines which have frequently inspired further investigations in the abstract field. About twenty-five years ago B. L. VAN DER WAERDEN and, later, O. ZARISKI and A. WEIL, together with their schools, established the methods of modern abstract algebraic geometry which, rejecting the c1assical restriction to the complex groundfield, gave up geometrical intuition and undertook arithmetisation under the growing influence of abstract algebra.

I. A survey of the foundations.- 1. Algebraic varieties.- 2. Absolute and relative varieties.- 3. The local rings.- 4. Algebraic product.- 5. Normal varieties.- 6. Birational transformations.- 7. Simple points.- 8. The intersection multiplicity.- 9. The calculus of cycles.- II. The resolution of singularities.- 1. The local uniformisation theorem.- 2. Monoidal transformations.- 3. Zariski’s proof for threefolds.- III. Linear systems.- 1. Divisors.- 2. The definition of linear system.- 3. Linear equivalence.- 4. Complete systems.- 5. The multiples of a linear system.- 6. Ample linear systems.- 7. Bertini’s theorems.- IV. The geometric genus.- 1. The adjoint forms.- 2. The canonical system.- 3. The canonical system as a birational invariant.- 4. The arithmetic definition of the canonical system.- 5. Relations between canonical and adjoint systems.- V. The arithmetic genus.- 1. The definition.- 2. The modular property of the arithmetic genus.- 3. The definition of the virtual arithmetic genus of a cycle.- 4. The birational invariance of the arithmetic virtual genus.- 5. The absolute invariance of pa(V) (r ? 3).- 6. The virtual characters of a cycle.- 7. Virtual and effective dimensions.- 8. A second definition of the arithmetic genus.- 9. The virtual characters of K.- VI. Algebraic and rational equivalence.- 1. The associated variety.- 2. Specialisation of a cycle and algebraic systems.- 3. Algebraic correspondences.- 4. The degeneration principle of Enriques-Zariski.- 5. Fundamental and exceptional varieties.- 6. A property of Chow varieties.- 7. Algebraic equivalence.- 8. Rational equivalence.- 9. The intersection-product for equivalence classes.- 10. A theorem of Severi and its consequences.- VII. The Abelian varieties from the algebraic viewpoint, and related questions.- 1. Jacobi variety.- 2. The base for the group of algebraic equivalence for divisors.- 3. The first Picard variety.- 4. The total maximal algebraic families.- 5. A property of the arithmetic genus.- 6. Non-special total families.- 7. The first Picard variety according to Matsusaka.- 8. The second Picard variety and the superficial irregularity.- VIII. Theory and applications of the canonical systems.- 1. Introduction.- 2. A new definition of the canonical divisors.- 3. Todd’s canonical systems.- 4. Introduction to Segre’s theory.- 5. The covariant sequence.- 6. The algebra of covariant sequences.- 7. The canonical sequence.- 8. Some applications.- 9. The behaviour of the canonical systems under birational transformations.- 10. Irregular intersection problems.- 11. Miscellaneous results.- IX. The algebraic varieties as complex-analytic manifolds.- 1. The complex manifolds and Chow’s theorem.- 2. Hermìte’s and Kähler’s metrics.- 3. The currents.- 4. The fundamental existence theorems.- 5. The complex operators.- 6. The Hodge-Eckmann theory.- 7. Hodge’s birational invariants.- 8. Miscellaneous results.- 9. Chern’s classes as canonical classes.- X. The applications of stack theory to algebraic geometry.- 1. Complex line bundles.- 2. The stack concept.- 3. Cohomology groups over a stack.- 4. A theorem of Dolbeault.- 5. Positive complex line bundles.- 6. The Picard variety in stack theory.- 7. The theorem of Riemann-Roch for adjoint systems.- 8. The arithmetic genera.- 9. The Riemann-Roch theorem.- 10. Miscellaneous results.- XI. The superficial irregularity and continuous systems.- 1. The deficiency of a linear system.- 2. The Poincaré families.- 3. The superficial irregularity.- 4. Characteristic systems of complete continuous systems.- 5. Miscellaneous results.- 1. Treatises, Monographs and Reports.- 2. List of papers.