
Math 137: Algebraic Geometry, Spring 2014
MWF 1112 Room 411
Instructor: Melody Chan, office 233 Science Center
email: mtchan at math.harvard.edu
office hours: Tuesdays 12pm, Wednesdays 45pm
course website: www.math.harvard.edu/~mtchan/2014Spring_137.html
Course Assistant Michael Proulx
Problem session: Thursdays 89pm 310 Science Center
email: michaelproulx at college . harvard . edu
Required textbook
Cox, Little, and O'Shea, Ideals, Varieties and Algorithms, Springer, 2007.
This book is available as a free download to anyone with access to SpringerLink, which should include anyone with a Harvard PIN. Try this link.
Supplementary textbook
Harris, Algebraic Geometry: A First Course.
This book will be on reserve at the Cabot Science Library.
Topics
This is a one semester undergraduate course on algebraic geometry. Our main source will be the textbook of Cox, Little, and O'Shea, supplemented with additional material from Harris' book and other resources as time permits. In addition to learning the basics of affine and projective varieties, we will emphasize algorithms and computation. To get a general sense for what topics will be covered, please consult the CLO textbook.
The prerequisites are Math 122 and 123 (undergraduate algebra).
If you would like to take this class and don't know whether your background is sufficient, I'd be happy to discuss it with you.
Grades
Weekly homework (65%), a 15 minute inclass presentation (10%), and a final exam (25%). The homework is the most important part of the course work.
Homework
Due Fridays in class.
Please take note: no late homework will be accepted, unless you explicitly ask for (and receive) an extension from me.
Homework 1, due February 7
Homework 2, due February 14 (revised 1(b))
Homework 3, due February 21
Homework 4, due February 28
Homework 5, due March 7
Homework 6, due March 14
Homework 7, due March 28
Homework 8, due April 4
Homework 9, due April 11
Homework 10, due April 18
Homework 11, due April 25
Homework 12, due May 2
Solutions by Michael to selected problems are posted at iSites.
Please read them and discuss with Michael if you have any questions.
Collaboration
I urge you to collaborate on all homework. Please write up your solutions separately and indicate with whom you collaborated.
You certainly may not copy solutions from the Internet or otherwise represent the work of others as your own. Nor may you post homework questions or solutions. As a rule of thumb, you are welcome to consult any online resources at the level of generality of Wikipedia or Mathworld. If there is any room for doubt, it is incumbent upon you to check with me.
Accommodations for students with disabilities
If you need accommodations for a disability, please talk to me as soon as possible and within the first two weeks of the term.
Schedule
Week 1: Affine varieties, ideals, monomial orders, division algorithm, Groebner bases, Hilbert basis theorem.
Week 2: Buchberger's algorithm, elimination and extension, polynomial/rational implicitization, resultants.
Week 3: Proof of extension theorem using resultants, Hilbert Nullstellensatz, operations on ideals
Week 4: Ideal quotients and saturation, irreducible varieties and decomposition, primary decomposition
Week 5: Quotient rings, regular maps and coordinate rings, zerodimensional varieties, rational functions
Week 6: Rational maps and birational equivalence. Projective space, projective varieties, homogeneous ideals, projective Nullstellensatz
Week 7: Computing projective closures, projective elimination theorem. Student Talk: Bitangents of plane quartic curves.
Week 8: Regular maps of quasiprojective varieties. Student talks: Affine schemes; cubic curves and the Cayley Bacharach theorem; Grassmannians.
Week 9: More on regular maps of quasiprojective varieties. Complex curves and the degreegenus formula (guest lectures by Yaim Cooper)
Week 10: Products of quasiprojective varieties and graphs of regular maps. Rational maps of quasiprojective varieties. Quadric hypersurfaces.
Week 11: Intersection multiplicity and Bezout's theorem using resultants. Student talks: Elliptic curves, the group law, and cryptography; Newton polygons.
Week 12: Affine and projective Hilbert functions; Hilbert polynomial; dimension and degree.
Week 13: Dimension, tangent spaces, and nonsingularity.
