Del i serien Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge

Algebraic Varieties

Häftad, Engelska, 2012

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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.

Produktinformation

Utgivningsdatum2012-07-21
Mått155 x 235 x 12 mm
Vikt324 g
FormatHäftad
SpråkEngelska
SerieErgebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Antal sidor196
Upplaga1956
FörlagSpringer-Verlag Berlin and Heidelberg GmbH & Co. KG
ISBN9783642527630

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Algebra inom Naturvetenskap och teknik

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 relatedquestions.- 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.