Simon Fraser University
Engaging the World

Course Outlines

Fall 2024 - MATH 818 G100

Algebra and Geometry (4)

Class Number: 5504

Delivery Method: In Person

Overview

  • Course Times + Location:

    Sep 4 – Dec 3, 2024: Wed, Fri, 2:30–4:20 p.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

An introduction to algebraic geometry with supporting commutative algebra. Possible topics include Hilbert basis theorem, Hilbert's Nullstellensatz, Groebner bases, ideal decomposition, local rings, dimension, tangent and cotangent spaces, differentials, varieties, morphisms, rational maps, non-singularity, intersections in projective space, cohomology theory, curves, surfaces, homological algebra.

COURSE DETAILS:

Algebraic geometry is the study of polynomials in more than one variable. This very old field of study has many connections to other areas of mathematics, and the modern perspective has a commutative algebra framework. To paraphrase Sophie Germain: "Algebra is nothing but written geometry; geometry is nothing but drawn algebra".
 
This is an introduction to algebraic geometry at the graduate level with an emphasis on the geometry of algebraic curves. Algebraic geometry goes hand in hand with commutative algebra, which we will study along the way. There is no prerequisite course, but it is assumed that students are, minimally, familiar with rings, ideals, quotient rings and the isomorphism theorems as well as prime and maximal ideals. The first assignment, called "Homework 0", will be due at the end of the first week of classes, and is posted on the instructor's website in advance, along with recommended readings, which may also be helpful for students wishing to refresh their memory on algebra background.

The following topics will be covered:
  • Hilbert's basis theorem
  • Nullstellensatz and the Zariski topology
  • Basic notions of affine and projective varieties
  • Morphisms and rational maps between varieties
  • Differentials and singularities
  • Weil divisors and valuation rings
  • The Riemann--Roch theorem for curves

COURSE-LEVEL EDUCATIONAL GOALS:

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Grading

  • Assignments 60%
  • Exam 40%

Materials

REQUIRED READING:

The course reading will primarily draw on course notes of Gathmann, particularly "Plane Algebraic Curves", "Commutative Algebra" and "Algebraic Geometry".
These notes are all available online at the author's website: https://agag-gathmann.math.rptu.de/en/notes.php
Course readings may also come from other sources, but any such readings will be made available electronically to all students.

REQUIRED READING NOTES:

Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

SFU’s Academic Integrity website http://www.sfu.ca/students/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating. Check out the site for more information and videos that help explain the issues in plain English.

Each student is responsible for his or her conduct as it affects the university community. Academic dishonesty, in whatever form, is ultimately destructive of the values of the university. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the university. http://www.sfu.ca/policies/gazette/student/s10-01.html

RELIGIOUS ACCOMMODATION

Students with a faith background who may need accommodations during the term are encouraged to assess their needs as soon as possible and review the Multifaith religious accommodations website. The page outlines ways they begin working toward an accommodation and ensure solutions can be reached in a timely fashion.