Algebraic Geometry 1

In this lecture course we will introduce the basic concepts of algebraic geometry, namely, the theory of schemes. The so-called affine schemes are essentially just commutative rings, considered through their spectrum as geometric objects. More general schemes are built out of affine schemes through a gluing process. There are also very concrete constructions of non-affine schemes via classical projective geometry. We will present the foundations of this theory, motiviated and illustrated through concrete examples and applications to classical problems.

 

Participants should have a good understanding of the material in linear algebra 1 and 2 as well as algebra 1. We will also be using the material covered in algebra 2, that is, commutative algebra. For those who have not taken algebra 2 but still would like to take this course, we are offering a pre-course in commutative algebra in the weeks before the beginning of lectures, during October 6.-16. For further information regarding this pre-course, please see the preparatory course: commutative algebra.

 

Texts

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

Eisenbud, David; Harris, Joe, Schemes. The language of modern algebraic geometry. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. xii+157 pp.

Mumford, David, The red book of varieties and schemes. Lecture Notes in Mathematics, 1358. Springer-Verlag, Berlin, 1988. vi+309 pp

Görtz, Ulrich; Wedhorn, Torsten, Algebraic geometry I. Schemes with examples and exercises. Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010.

Shafarevich, Igor R., Basic algebraic geometry. 1. Varieties in projective space. Third edition. Translated from the 2007 third Russian edition. Springer, Heidelberg, 2013.

 

Supplementary Texts in commutative algebra:

Atiyah, M. F.; Macdonald, I. G., Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969

Zariski, Oscar; Samuel, Pierre, Commutative algebra. Vol. 1. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958 edition. Graduate Texts in Mathematics, No. 28. Springer-Verlag, New York-Heidelberg-Berlin, 1975.

Matsumura, Hideyuki, Commutative algebra. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.

Eisenbud, David, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995.

Bosch, Siegfried, Algebraic geometry and commutative algebra. (online)