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Title
Abstract
Introduction 0 Algebraic Geometry
An Introduction to Morphisms
Morphisms and Dimension A. Examples and Discussion
Reduction to the Affine Case
Bézout's Theorem 0 Introduction
The Link with the Affine Case
Conclusion
Sheaf Cohomology 0 Introduction
Introduction: The Euler-Poincaré Characteristic
Normalisation: The Algebraic Method
Affine Blow-Ups A. Introduction
Changing Resolutions by Liaison: The Acm Case
Discussion of the Non-Acm Case
Introduction
Preliminary Results
References

Algebraic Geometry – An Introduction . Daniel Perrin

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References (44)

  1. The incidence variety and applications Let d be a positive integer and let R d = H 0 (P 3 , O P 3 (d)) be the vector space of homogeneous polynomials of degree d in X, Y, Z, T . We identify the space of degree d surfaces in P 3 with the projective space P(R d ).
  2. What is the dimension of P(R d )? We set V d = {(D, F ) ∈ G × P(R d ) | D ⊂ F } and denote by π (resp. p) the projection of V d onto P(R d ) (resp. onto G).
  3. Let H be a plane of equation αX + βY + γZ + δT = 0 and let D be a line which is not contained in H. Determine both the homogeneous coordinates of the point of intersection of D and H as a function of α, β, γ, δ and the Plücker coordinates of D. What happens when D ⊂ H?
  4. Prove that V d is a closed subvariety of G × P(R d ), called the incidence variety (use 1), for example, and let the plane H vary). Prove that the projections π and p are closed maps (i.e., they send closed sets to closed sets).
  5. Determine the fibres p -1 (D) for any D ∈ G (use Riemann-Roch). Deduce that p is surjective and V d is irreducible. Calculate the dimension of V d .
  6. Assume d 4. Prove that π is not dominant. Deduce that there is a non-empty open set in P(R d ) consisting of surfaces which do not contain any line. (We say that the "general" surface of degree 4 does not contain a line.)
  7. Assume d = 3. Prove there are only a finite number of lines in the surface XY Z -T 3 = 0. (You may either use a direct argument or use 1) and 2) above.) Deduce that π is surjective and that a general cubic surface contains a finite number of lines. Study the surface XY Z + T (X 2 + Y 2 + Z 2 ) -T 3 = 0. (Start by proving that the lines which do not meet the line X = T = 0 are defined by equations of the form Y = aX + bT , Z = cX + dT .)
  8. What happens when d = 1?
  9. We take d = 2. Determine the Plücker coordinates of the lines contained in the quadric of equation XY -ZT = 0. Prove they form a closed one-dimensional subset of G. Deduce that π is surjective. Are the fibres of π all of the same dimension? References
  10. E. Ballico, G. Bolondi & J. Migliore -The Lazarsfeld-Rao problem for liaison classes of two-codimensional subschemes of P n , Amer. J. Math. 113 (1991), p. 117-128.
  11. N. Bourbaki -Elements of Mathematics. Commutative Algebra. Chapters 1-7, Springer, Berlin, 1998.
  12. A. Grothendieck & J. Dieudonné -Éléments de géométrie algébrique, vol. 4, 8, 11, 17, 20, 24, 28, 32, Publications Mathématiques de l'IH ÉS.
  13. C. Godbillon -Topologie algébrique, Hermann, Paris, 1971.
  14. R. Godement -Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958.
  15. A. Gramain -Topology of surfaces, BCS Associates, Moscow, ID, 1984, (French edition: 1971).
  16. L. Gruson & C. Peskine -Genre des courbes de l'espace projectif (II), Ann. Sci. École Norm. Sup. (4) 15 (1982), p. 401-418.
  17. R. Hartshorne -Algebraic Geometry, Springer, Berlin, 1977.
  18. S. Lang -Algebra, Addison-Wesley, Reading, MA, 1965.
  19. M. Martin-Deschamps & D. Perrin -Sur la classification des courbes gauches, Astérisque, vol. 184-185, Société Mathématique de France, Paris, 1990.
  20. H. Matsumura -Commutative Algebra, Benjamin, 1970.
  21. D. Mumford -The Red Book of Varieties and Schemes, Lect. Notes in Math., vol. 1358, Springer, 1988.
  22. D. Perrin -Cours d'algèbre, Ellipses, Paris, 1996.
  23. C. Peskine -An algebraic introduction to complex projective geome- try. 1, Cambridge Studies in Advanced Mathematics, vol. 47, Cambridge University Press, Cambridge, 1996.
  24. C. Peskine & L. Szpiro -Liaison des variétés algébriques, Invent. Math. 26 (1974), p. 271-302.
  25. A.P. Rao -Liaison among curves in P 3 , Invent. Math. 50 (1979), p. 205- 217.
  26. W. Rudin -Real and complex analysis, McGraw-Hill, New York, NY, 1974. References
  27. P. Samuel -Algebraic theory of numbers, Houghton Mifflin, Boston, 1970 (French edition: 1967).
  28. I.R. Shafarevich -Basic Algebraic Geometry, Springer, 1977.
  29. A. Grothendieck -Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957), p. 119-221.
  30. Universitext Aguilar, M.; Gitler, S.; Prieto, C.: Algebraic Topology from a Homotopical Viewpoint Aksoy, A.; Khamsi, M. A.: Methods in Fixed Point Theory Alevras, D.; Padberg M. W.: Linear Opti- mization and Extensions Andersson, M.: Topics in Complex Analysis Aoki, M.: State Space Modeling of Time Series Arnold, V. I.: Lectures on Partial Differen- tial Equations
  31. Arnold, V. I.; Cooke, R.: Ordinary Differen- tial Equations
  32. Audin, M.: Geometry Aupetit, B.: A Primer on Spectral Theory Bachem, A.; Kern, W.: Linear Programming Duality Bachmann, G.; Narici, L.; Beckenstein, E.: Fourier and Wavelet Analysis Badescu, L.: Algebraic Surfaces
  33. Balakrishnan, R.; Ranganathan, K.: A Text- book of Graph Theory Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differ- ential Equations
  34. Bapat, R.B.: Linear Algebra and Linear Models Benedetti, R.; Petronio, C.: Lectures on Hy- perbolic Geometry
  35. Benth, F. E.: Option Theory with Stochastic Analysis Berberian, S. K.: Fundamentals of Real Analysis Berger, M.: Geometry I, and II
  36. Bhattacharya, R., Waymire, E.C.: A Basic Course in Probability Theory Bliedtner, J.; Hansen, W.: Potential Theory Blowey, J. F.; Coleman, J. P.; Craig, A. W. (Eds.): Theory and Numerics of Differential Equations Blowey, J. F.; Craig, A.; Shardlow, T. (Eds.): Frontiers in Numerical Analysis, Durham 2002, and Durham 2004
  37. Blyth, T. S.: Lattices and Ordered Algebraic Structures Börger, E.; Grädel, E.; Gurevich, Y.: The Classical Decision Problem Böttcher, A; Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices Boltyanski, V.; Martini, H.; Soltan, P. S.: Excursions into Combinatorial Geometry Boltyanskii, V. G.; Efremovich, V. A.: Intu- itive Combinatorial Topology Bonnans, J. F.; Gilbert, J. C.; Lemarchal, C.; Sagastizbal, C. A.: Numerical Optimization Booss, B.; Bleecker, D. D.: Topology and Analysis Borkar, V. S.: Probability Theory Bridges/Vita: Techniques of Constructive Analysis Brunt B. van: The Calculus of Variations Bühlmann, H.; Gisler, A.: A Course in Cred- ibility Theory and its Applications Carleson, L.; Gamelin, T. W.: Complex Dynamics Cecil, T. E.: Lie Sphere Geometry: With Applications of Submanifolds Chae, S. B.: Lebesgue Integration Chandrasekharan, K.: Classical Fourier Transform Charlap, L. S.: Bieberbach Groups and Flat Manifolds Chern, S.: Complex Manifolds without Potential Theory Chorin, A. J.; Marsden, J. E.: Mathematical Introduction to Fluid Mechanics Cohn, H.: A Classical Invitation to Alge- braic Numbers and Class Fields Curtis, M. L.: Abstract Linear Algebra Curtis, M. L.: Matrix Groups
  38. Cyganowski, S.; Kloeden, P.; Ombach, J.: From Elementary Probability to Stochastic Differential Equations with MAPLE Da Prato, G.: An Introduction to Infinite Dimensional Analysis Dalen, D. van: Logic and Structure Das, A.: The Special Theory of Relativity: A Mathematical Exposition Debarre, O.: Higher-Dimensional Algebraic Geometry Deitmar, A.: A First Course in Harmonic Analysis. 2 nd edition Demazure, M.: Bifurcations and Cata- strophes
  39. Devlin, K. J.: Fundamentals of Contempo- rary Set Theory
  40. DiBenedetto, E.: Degenerate Parabolic Equations Diener, F.; Diener, M.(Eds.): Nonstandard Analysis in Practice Dimca, A.: Sheaves in Topology Dimca, A.: Singularities and Topology of Hypersurfaces DoCarmo, M. P.: Differential Forms and Applications Duistermaat, J. J.; Kolk, J. A. C.: Lie Groups Dumortier.: Qualitative Theory of Planar Differential Systems Dundas, B. I.; Levine, M.; Østvaer, P. A.; Röndip, O.; Voevodsky, V.: Motivic Homo- topy Theory
  41. Edwards, R. E.: A Formal Background to Higher Mathematics Ia, and Ib Edwards, R. E.: A Formal Background to Higher Mathematics IIa, and IIb Emery, M.: Stochastic Calculus in Mani- folds Emmanouil, I.: Idempotent Matrices over Complex Group Algebras Endler, O.: Valuation Theory Engel, K.-J.; Nagel, R.: A Short Course on Operator Semigroups Erez, B.: Galois Modules in Arithmetic Everest, G.; Ward, T.: Heights of Polynomi- als and Entropy in Algebraic Dynamics Farenick, D. R.: Algebras of Linear Trans- formations Foulds, L. R.: Graph Theory Applications Franke, J.; Hrdle, W.; Hafner, C. M.: Statis- tics of Financial Markets: An Introduction Frauenthal, J. C.: Mathematical Modeling in Epidemiology Freitag, E.; Busam, R.: Complex Analysis Friedman, R.: Algebraic Surfaces and Holo- morphic Vector Bundles Fuks, D. B.; Rokhlin, V. A.: Beginner's Course in Topology Fuhrmann, P. A.: A Polynomial Approach to Linear Algebra Gallot, S.; Hulin, D.; Lafontaine, J.: Rie- mannian Geometry Gardiner, C. F.: A First Course in Group Meyer-Nieberg, P.: Banach Lattices Mikosch, T.: Non-Life Insurance Mathe- matics Mines, R.; Richman, F.; Ruitenburg, W.: A Course in Constructive Algebra Moise, E. E.: Introductory Problem Courses in Analysis and Topology Montesinos-Amilibia, J. M.: Classical Tes- sellations and Three Manifolds Morris, P.: Introduction to Game Theory Mortveit, H., Reidys, C.: An Introduction to Sequential Dynamical Systems Nicolaescu, L.: An Invitation to Morse Theory Nikulin, V. V.; Shafarevich, I. R.: Geome- tries and Groups Oden, J. J.; Reddy, J. N.: Variational Meth- ods in Theoretical Mechanics Øksendal, B.: Stochastic Differential Equa- tions Øksendal, B.; Sulem, A.: Applied Stochastic Control of Jump Diffusions. 2nd edition
  42. Orlik, P.; Welker, V.: Algebraic Combina- torics Perrin, D.: Algebraic Geometry: An intro- duction
  43. Poizat, B.: A Course in Model Theory Polster, B.: A Geometrical Picture Book Porter, J. R.; Woods, R. G.: Extensions and Absolutes of Hausdorff Spaces Procesi, C.: Lie Groups Radjavi, H.; Rosenthal, P.: Simultaneous Triangularization
  44. Ramsay, A.; Richtmeyer, R. D.: Introduc- tion to Hyperbolic Geometry Rautenberg, W.: A concise Introduction to Mathematical Logic Rees, E. G.: Notes on Geometry Reisel, R. B.: Elementary Theory of Metric Spaces Rey, W. J. J.: Introduction to Robust and Quasi-Robust Statistical Methods Ribenboim, P.: Classical Theory of Alge- braic Numbers Rickart, C. E.: Natural Function Algebras Rotman, J. J.: Galois Theory Rubel, L. A.: Entire and Meromorphic Functions Ruiz-Tolosa, J. R.; Castillo E.: From Vectors to Tensors Runde, V.: A Taste of Topology Rybakowski, K. P.: The Homotopy Index and Partial Differential Equations Sabbah, C.: Isomonodromic Deformations and Frobenius Manifolds: An introduction Sagan, H.: Space-Filling Curves Samelson, H.: Notes on Lie Algebras Sauvigny, F.: Partial Differential Equa- tions I Sauvigny, F.: Partial Differential Equa- tions II Schiff, J. L.: Normal Families Schirotzek, W.: Nonsmooth Analysis Sengupta, J. K.: Optimal Decisions under Uncertainty Séroul, R.: Programming for Mathemati- cians Seydel, R.: Tools for Computational Finance Shafarevich, I. R.: Discourses on Algebra Shapiro, J. H.: Composition Operators and Classical Function Theory Simonnet, M.: Measures and Probabilities Smith, K. E.; Kahanpää, L.; Kekäläinen, P.; Traves, W.: An Invitation to Algebraic Geometry Smith, K. T.: Power Series from a Computa- tional Point of View Smorynski: Self-Reference and Modal Logic Smoryński, C.: Logical Number Theory I. An Introduction Srivastava: A Course on Mathematical Logic Stichtenoth, H.: Algebraic Function Fields and Codes Stillwell, J.: Geometry of Surfaces Stroock, D. W.: An Introduction to the The- ory of Large Deviations Sunder, V. S.: An Invitation to von Neu- mann Algebras Tamme, G.: Introduction to Étale Cohomology Tondeur, P.: Foliations on Riemannian Manifolds Toth, G.: Finite Mbius Groups, Minimal Immersions of Spheres, and Moduli Tu, L. W.: An Introduction to Manifolds Verhulst, F.: Nonlinear Differential Equa- tions and Dynamical Systems Weintraub, S. H.: Galois Theory Wong, M. W.: Weyl Transforms Xambó-Descamps, S.: Block Error-Cor- recting Codes Zaanen, A.C.: Continuity, Integration and Fourier Theory Zhang, F.: Matrix Theory Zong, C.: Sphere Packings Zong, C.: Strange Phenomena in Convex and Discrete Geometry Zorich, V. A.: Mathematical Analysis I Zorich, V. A.: Mathematical Analysis II

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