Math 767 — Spring 2002 Algebraic Geometry

This is an introductory course in Algebraic Geometry. We plan to cover the following standard topics: algebraic sets and affine varieties, Zariski topology, Hilbert's Basis Theorem and Nullstellensatz, projective and quasi-projective varieties, products of algebraic varieties, separated and complete varieties, regular and rational morphisms, dimension theory, singular and smooth points, coherent sheaves and localization, Serre's lemma, algebraic curves, nonsingular model of curves, Bezout's theorem.

References

1. R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, 1977.
2. I. R. Shafarevich, Basic Algebraic Geometry I: Varieties in Projective Space, Springer-Verlag, Berlin, 1994.
3. G. Kempf, Algebraic Varieties, LMS Lecture Note Series 172, Cambridge Univ. Press, 1993.
4. V. I. Danilov, Algebraic Varieties and Schemes, Encyclopaedia Math. Sci. 23, Springer, Berlin, 1994, pp. 167-297.
5. J. S. Milne, Algebraic Geometry, Lecture Notes, (see http://www.jmilne.org/math/).
   

Algebraic geometry is a mixture of the ideas of two Mediterranean cultures. It is a superposition of the Arab science of the lightning calculation of solutions of equations over the Greek art of position and shape. This tapestry was originally woven on European soil and is still being refined under the influence of international fashion. Algebraic geometry studies the delicate balance between the geometrically plausible and the algebraically possible. Whenever one side of this mathematical teeter-totter outweighs the other, one immediately loses interest and runs off in search of a more exciting amusement... (G. Kempf, Algebraic Varieties)