Math 595AG Fall 2024 (Algebraic Geometry I): Syllabus Summary
Instructor: Takumi Murayama

This is a summary of the course syllabus and does not contain every component of the official course syllabus on Brightspace. Please look at Brightspace for a complete syllabus.

Course Information

• Math 59500AG: Algebraic Geometry I
• Location and Times
• Section 010: Math 215, MWF 4:30 PM – 5:20 PM
• Office Hours (Tentative)
• Monday 1:50PM – 3:20PM in Math 646
• Friday 1:50 PM – 3:20PM in Math 646
• Extra Zoom Office Hour: Tuesday 10:00AM – 11:00AM (link on Brightspace)
• Course Webpage: Brightspace – Please check regularly for announcements.

Instructor Contact Information

• Name of Instructor: Takumi Murayama
• Office: Math 646
• Purdue Email:
• Please put "AG Math 595 Fall 2024" in the subject line. This will help me find your email more easily.
• Emails are read between 10AM and 6PM and will generally be responded to within 24 hours. Emails sent after 6PM in any given day may not be read until the following day.

Course Description

Credit Hours: 3.00. This course is the first course in a (planned) two semester introductory sequence in algebraic geometry. Algebraic geometry is the geometric study of solutions to systems of polynomial equations. Algebraic geometry has interactions with many other fields of mathematics, including commutative algebra, algebraic topology, number theory, several complex variables, and complex geometry.

This first course will mainly focus on the theory of algebraic varieties over algebraically closed fields but I plan to transition to the theory of schemes by the end of the semester. Planned topics include the following:

• Affine varieties.
• Projective varieties.
• Morphisms.
• Rational maps.
• Nonsingular varieties and the Jacobian criterion.
• Nonsingular curves.
• Intersections in projective space, Hilbert polynomials, and Bézout's theorem.
• Sheaves.
• Locally ringed spaces and schemes.

Textbook: Course notes will be provided. The notes will largely draw from Algebraic geometry by Robin Hartshorne, available via the Purdue Library here: https://doi.org/10.1007/978-1-4757-3849-0.

While the core material comes from the textbook and I will try to incorporate the conventions in the textbook, there will inevitably be differences stemming from the instructor's mathematical perspective. My suggestion would be to consider the lecture notes as the main text and the textbook as a very good supplementary reference that can provide additional examples and explanations.

Optional Textbooks: All texts listed below have free access options for Purdue students.

For algebraic varieties:

• Algebraic geometry: A first course by Joe Harris, available via the Purdue library here: https://doi.org/10.1007/978-1-4757-2189-8.
• Basic algebraic geometry 1 (third edition) by Igor R. Shafarevich, available via the Purdue library here: https://doi.org/10.1007/978-3-642-37956-7.

• Éléments de géométrie algébrique by Alexander Grothendieck and Jean A. Dieudonné, available here: http://www.numdam.org/item/PMIHES_1960__4__5_0/ (and then keep clicking the "followed-by" links).
• Eléments de géométrie algébrique I (second edition) by Alexander Grothendieck and Jean A. Dieudonné, available for short term loan here: https://n2t.net/ark:/13960/t42s6kw4b.

Learning Outcomes: We will cover Hartshorne Chapter I. Time permitting, we will also cover the first two sections of Hartshorne Chapter II. Topics to be covered are listed in the course description.

Note: Algebraic geometry is best learned by solving as many exercises as possible. I expect homework solutions to be thorough and either typed or handwritten neatly. In addition, please read the Homework Guidelines on Brightspace.

Important Dates

Please see the Purdue University Fall 2024 Add/Drop Calendar for important administrative dates.

Learning Resources

Lecture notes will be provided on BrightSpace.

Copyright Policy

Effective learning environments provide opportunities for students to reflect, explore new ideas, post opinions openly, and have the freedom to change those opinions over time. Stu- dents and instructors are the authors of the works they create in the learning environment. As authors, they own the copyright in their works subject only to the university’s right to use those works for educational purposes. Students may not copy, reproduce, or post to any other outlet (e.g., YouTube, Facebook, or other open media sources or websites) any work in which they are not the sole or joint author or have not obtained the permission of the author(s).

Course Policies

Please make sure to keep up with the conventions and definitions from the lecture notes. While the core material comes from the textbook and I will try to incorporate the conventions in the textbook, there will inevitably be differences stemming from the instructor’s mathematical perspective. My suggestion would be to consider the lecture notes as the main text and the textbook as a very good supplementary reference that can provide additional examples and explanations.

The homework will be graded according to the conventions, notations, and definitions from the lecture notes.

Course Participation Information and Policies

• There will be a problem session on most Fridays. Please see the course schedule (on Brightspace) for more precise dates.
• Attendance/Participation is required for these Friday problem sessions and will make up the Participation portion (30%) of your final grade. If you cannot make a problem session, please email me.
• You will work in small groups of 3–5 students on a worksheet. Some of the trickier homework problems will be on the worksheet, so this is a great opportunity to make progress on the homework with your peers.

Homework Information and Policies

• There will typically be one homework assignment due each week.
• The due date for homework will typically be Wednesday by 11:59PM (West Lafayette time). However, due dates may vary depending on the proximity to breaks. It is your responsibility to check the due dates and complete the assignments by the due dates.
• Homework assignments must be submitted on Gradescope.
• Each student will be allowed 2 late homework submissions, no questions asked. The student must give notice of late submission to the instructor by the homework due date. Additional extensions may be granted on a case-by-case basis.
  Please note that there may be grading delays for late homeworks.
• Collaboration is strongly encouraged for homework assignments. However, the work you turn in must be your own. Please list the names of any collaborators as well as any outside resources that you have consulted on the first page of your homework. Please make sure to read the section on Academic Honesty.

Exam Information and Policies

There will be one 30–40 minutes oral midterm exam and one comprehensive 30–40 minutes oral final exam. More details on the oral exams will be provided closer to the exam dates.

Exam Dates

• Midterm Exam: Thursday, October 10 or Friday, October 11. Individual scheduling will be available a week prior to the exam.
• Final Exam: Wednesday, December 4 or Friday, December 6. Individual scheduling will be available a week prior to the exam.

Grades

Your total score will be determined as follows:

• 30% Participation during Friday Problem Sessions
• 30% Homework
• 20% Oral Midterm Exam
• 20% Oral Final Exam

Accessibility/Accommodations

Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247, as soon as possible.

If the Disability Resource Center (DRC) has determined reasonable accommodations that you would like to utilize in this class, you must send your Course Accommodation Letter to the instructor. Instructions on sharing your Course Accommodation Letter can be found by visiting: https://www.purdue.edu/drc/students/course-accommodation-letter.php. Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.

Attendance Policy

• Attendance during Friday problem sessions is required. If you need to miss a Friday problem session, please email me.
• Attendance during lectures is not required. However, it is the student’s responsibility to keep up with the course in the event that the student misses any lectures. I will not re-lecture any material.

Illness Policy

Students should not come to class if they feel very ill. If you are sick and cannot make a Friday problem session, please email me.

Lecture Schedule and Homework Due Dates for Math 595AG: