Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

References

A classical textbook, predating schemes:

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• Hodge, W. V. D., and Pedoe, Daniel, Methods of Algebraic Geometry: Volume 1, Cambridge University Press, 1994, ISBN 0521469007
• Hodge, W. V. D., and Pedoe, Daniel, Methods of Algebraic Geometry: Volume 2, Cambridge University Press, 1994, ISBN 0521469015
• Hodge, W. V. D., and Pedoe, Daniel, Methods of Algebraic Geometry: Volume 3, Cambridge University Press, 1994, ISBN 0521467756

Modern textbooks that do not use the language of schemes:

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• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms (second edition), Springer, 1997, ISBN 0387946802
• Griffiths, Phillip, and Harris, Joe, Principles of Algebraic Geometry, Wiley-Interscience, 1994, ISBN 0471050598
• Harris, Joe, Algebraic Geometry: A First Course, Springer-Verlag, 1995, ISBN 0387977163
• Mumford, David, Algebraic Geometry I: Complex Projective Varieties, 2nd ed., Springer-Verlag, 1995, ISBN 3540586571
• Reid, Miles, Undergraduate Algebraic Geometry, Cambridge University Press, 1988, ISBN 0521356628
• Shafarevich, Igor, Basic Algebraic Geometry I: Varieties in Projective Space, Springer-Verlag, 2nd ed., 1995, ISBN 0387548122

Textbooks and references for schemes:

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• Eisenbud, David, and Harris, Joe, The Geometry of Schemes, Springer-Verlag, 1998, ISBN 0387986375
• Grothendieck, Alexander, Éléments de géométrie algébrique, Publications mathématiques de l'IHÉS, vols. 4, 8, 11, 17, 20, 24, 28, 32, 1960, 1961, 1963, 1964, 1965, 1966, 1967
• Grothendieck, Alexander, Éléments de géométrie algébrique, vol. 1, 2nd ed., Springer-Verlag, 1971, ISBN 3540051139
• Hartshorne, Robin, Algebraic Geometry, Springer-Verlag, 1997, ISBN 0387902449
• Mumford, David, The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians, 2nd ed., Springer-Verlag, 1999, ISBN 354063293X
• Shafarevich, Igor, Basic Algebraic Geometry II: Schemes and Complex Manifolds, Springer-Verlag, 2nd ed., 1995, ISBN 0387548122

On the internet:

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• Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System
• Algebraic geometry entry on PlanetMath
• Algebraic Equations and Systems of Algebraic Equations at EqWorld: The World of Mathematical Equations