Professor: Dr. Hal Schenck
324 Illini Hall
Office hours: following class, or by appointment.
T-R 11-1220 in AH 143
Course Description. There is a beautiful interplay between algebra and geometry, which is essentially an extension of the fact that the parabola (a geometric object) is the set of points in the plane satisfying $y-x^2=0$ (an algebraic equation). In this class we'll quickly review some basic commutative algebra. Then we'll discuss graded objects and varieties in projective space; concentrating on building up a stable of good examples. We'll use these examples to illustrate fundamental geometric constructions--tangent spaces, smoothness, dimension and degree, and then discussing morphisms between varieties, divisors, line bundles and sheaf cohomology. A main objective of the class is to bring all the abstract concepts to life with lots of examples. Prerequisite: a graduate class in algebra, and mathematical maturity. Good preparatory reading is the book of Cox-Little-O'Shea ``Ideals, varieties, and algorithms".
Text. Cox-Little-O'Schenck, Toric Varieties, publicly available at http://www.cs.amherst.edu/~dac/toric.html. Although this book is devoted to toric varieties, each chapter begins with an introductory section, covering the general setting. We will cover the introductory sections of Chapters 1-9, including affine and projective varieties, normal varieties, divisors, line bundles, sheaves and morphisms, the cotangent and canonical sheaves, and sheaf cohomology.
Grading. Your grade will be determined by homework scores (70%),
and final (30%). Problems will assigned in class and collected every two weeks. This is a comp class, and the final will be equivalent to the comprehensive exam.
Academic Integrity I encourage group work on the homework problems. This does not include copying each others solutions.
Copying Course Materials: All printed hand-outs and web-materials are protected by US Copyright Laws. No multiple copies can be made without written permission by the instructor.
In addition to the online notes (CLS), the following material may be useful:
Schenck (S) "Computational Algebraic Geometry", Cambridge 2003.
Harris (H) "Algebraic Geometry: a first course", Springer 1992.
Shafarevich "Basic Algebraic geometry I", Springer 1994.
Cox-Little-O'Shea (CLO) "Ideals, Varieties, Algorithms", Springer 1992.
Schedule by week
1. Affine varieties (CLS chap 1, S chap 1 H chap 1)
2. Regular and rational maps (CLS chap 1, S chap 1 H chap 2,5,7)
3. Projective varieties (CLS chap 2, S chap 2 H chap 1)
4. Grobner bases, resultants, projection (S chap 4 H chap 3, CLO chap 2,3)
5. Normal varieties, gluing construction, blowups (CLS chap 3, H chap 7)
6. The Grassmannian (Handout, H chap 6)
7. Dimension and degree (S chap 2,3, H chap 11,13,18)
8. Combinatorial alg. geom: SR ring and torics (S chap 5, handout)
9. Divisors (CLS chap 4)
10. Line bundles (CLS chap 6)
11. More morphisms (CLS chap 7)
12. Tangent and Cotangent bundles (CLS chap 8)
13. Cech cohomology (CLS chap 9)
Updated 1/29/17 (hks).