Math 595AGI Fall 2025 (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 Description

Credit Hours: 3.00. This course is the first course in a 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 typed. For the first few homework assignments, neatly handwritten submissions will also be accepted. In addition, please read the Homework Guidelines on Brightspace.

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 problem sessions throughout the semester. Please see the course schedule below for more precise dates.
• Attendance/Participation is required for these 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 on the syllabus.

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 16 or Friday, October 17. Individual scheduling will be available a week prior to the exam.
• Final Exam: Wednesday, December 10 or Friday, December 12. Individual scheduling will be available a week prior to the exam.

Grades

Your total score will be determined as follows:

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

Attendance Policy

• Attendance during problem sessions is required. If you need to miss a 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.

Lecture Schedule and Homework Due Dates for Math 595AG:

Note: This is a tentative schedule and is subject to change.