Mathematics 6170

Algbraic Geometry

Course Description
Introduction to algebraic geometry, including affine and projective varieties, rational maps, morphisms, differentials and divisors. Additional topics might include Bezout's Theorem, the Riemann-Roch Theorem, elliptic curves, and sheaves and schemes.  Prereq., MATH 6140. Undergraduates must have approval of the instructor.
Schedule MWF 12 - 1 PM,  ECCR 116
(University of Colorado Academic Calendar)
Sebastian Casalaina-Martin
Mathematics 221
Office Hours
MWF 2 PM - 3 PM or by appointment.
We will work primarily from:

1.  Mumford, D., The red book of varieties and schemes.  Second, expanded edition.  Includes the Michigan lectures (1974) on curves and their Jacobians.  With contributions by Enrico Arbarello.  Lecture Notes in Mathematics, 1358. Springer-Verlag, Berlin, 1999. x+306 pp.

Some other introductory books on the subject one should know about:

1.  Shafarevich, I., Basic algebraic geometry. 1. Varieties in projective space. Second edition. Translated from the 1988 Russian edition and with notes by M. Reid. Springer-Verlag, Berlin, 1994. xx+303 pp.

2.  Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp.

3.  Griffiths, P. and Harris, J.  Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp.

4.  Harris, J.  Algebraic geometry. A first course. Graduate Texts in Mathematics, 133. Springer-Verlag, New York, 1992. xx+328 pp.

5.  Eisenbud, D. and Harris, J.  The geometry of schemes. Graduate Texts in Mathematics, 197. Springer-Verlag, New York, 2000. x+294 pp.

Some backgroud on commutative algebra will be assumed (although we will attempt to review the results we need as we go).  Some standard references are:

1.  Atiyah, M. F. and Macdonald, I. G. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.

2.  Matsumura, H. Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989. xiv+320 pp

3.  Eisenbud, D. Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp.

There will be homework assignments posted online.
This syllabus is a rough guide to the topics we will cover.
Please read the following class policies.