Introduction to Algebraic Geometry (Syllabus)

algebraic geometry
Cox--Little--O'Shea
Hyperelliptic curve
Châtelet surface
Description
We study systems of polynomial equations in several variables. As an introduction to this branch of mathematics, we will examine the computational foundations, the dictionary between ideals in a polynomial ring and affine varieties, and projective geometry.
Instructor
Gregory G. Smith (512 Jeffery Hall, ggsmith@mast.queensu.ca)
Lectures
(slot 012)
Monday at 12:30–13:20 in 306 MacLaughlin
Wednesday at 11:30–12:20 in 306 MacLaughlin
Thursday at 13:30–14:20 in 306 MacLaughlin
Office Hour
Thursday at 16:30–17:20 in 201 Jeffery Hall (aka Math Help Center)
 
Assessment
The course grades will be computed as follows:
  • 40% Homework
  • 40% Project
  • 20% Exam
Homework
Problem sets are posted in PDF on the lectures webpage. Your browser can be trained to open these files with the free program Acrobat Reader (or other PDF viewer). Solutions to each problem set will be submitted via the Crowdmark system. Instructions for using this software are available. The solution to each problem must be uploaded separately. Solutions are due on Fridays before 17:00; late homework will receive no credit. Your best five of six solution sets will determine your homework grade.
Exam
There will be a three-hour exam on Sunday, 16 April 2023 at 14:00–17:00 in Gym 5.
 
Learning Outcomes
Upon successful completion of the course, students will be able to do the following:
  • Execute accurate and efficient calculations with ideals in a multivariate polynomial ring involving Gröbner bases, membership, intersections, and quotients.
  • Explain and use elimination theory to solve systems of polynomial equations.
  • Define and illustrate the correspondence between ideals and varieties by translating between algebraic and geometric statements.
  • Describe and demonstrate a basic understanding of projective geometry.
  • Create and present rigorous solutions to problems and coherent proofs of theorems.
To develop these abilities, students are expected to
  • independently read the course notes or at least one textbook,
  • regularly participate in the lectures and office hours,
  • persistently complete all homework assignments and all steps in the project, and
  • continually discuss mathematics with other students.
Writing
We write to communicate. Please bear this in mind as you complete the homework, the project, and the exam. Work must be neat and legible to receive consideration. You must explain your work in order to obtain full credit; an assertion is not an answer.
Academic Integrity
Students are responsible for familiarizing themselves with all of the regulations concerning academic integrity and for ensuring that their assignments and their behaviour conform to the principles of academic integrity. Students are welcome to discuss problems, but should write up the solutions individually. Students must explicitly acknowledge any assistance including books, software, technology, websites, students, friends, professors, references, etc.
Accommodations
The instructor is committed to achieving full accessibility for people with disabilities. Part of this commitment includes arranging academic accommodations for students with disabilities to ensure they have an equitable opportunity to participate in all of their academic activities. If you think that you may need academic accommodations, then you are strongly encouraged to contact the instructor and the Queenʼs Student Accessibility Services (QSAS) as soon as possible.
Licensing
Materials generated by the instructor may not be used for commercial advantage or monetary compensation. Some material is clearly copyrighted and may not be reproduced or retransmitted in any form without express written consent. Other material, licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, may be remix, adapt, or build upon it, as long as appropriate credit is given and the new creation is distributed under the identical terms.
Technology
Students are encouraged to use any available technology on the homework and project, but these aids will not be allowed during the exam.
Primary Reference
David A. Cox, John B. Little, and Donal O’Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Fourth Edition, Springer, 2015.