Scheduled Lectures

(Section 12) MWF 12:50—1:55
(x-hour) Tu 1:20—2:10
Kemeny Hall 120


Professor Sarah Frei
Office: 314 Kemeny Hall
Office Hours: Tues 1:20—2:10 (x-hour), Wed 2-3, Fri 10-11
Contact via email: sarah.frei(at) (Replace the (at) with an @).

Course Description

The goal of this course is to understand the classification of complex algebraic surfaces. The first third of the course will be an introduction to algebraic geometry before we focus on the study of surfaces. Along the way, we'll see: affine and projective varieties and morphisms between them; sheaves and sheaf cohomology; divisors and the Picard group; curves in surfaces, the intersection pairing, and how they govern geometry; Kodaira dimension and birational invariants of surfaces; and various families of surfaces (rational surfaces, ruled surfaces, K3 surfaces, abelian surfaces, Enriques surfaces, surfaces of general type).

Expected Background

Prior experience with algebra will be necessary. I will not assume any background in algebraic geometry, although without it, the course will be a bit fast-paced.


We will make use of the following texts:

(I may add to this list throughout the course.)

Please do not buy all of the textbooks! You should be able to find them in the library or around the department. If you have trouble accessing any of them, please let me know.


Your grade will be based on class participation and homework. As part of class participation, you will be expected to take careful notes for 3-4 lectures to be shared with the class.


I believe in the axioms laid out by SFSU Professor Federico Ardila, and I will use them to guide my instruction of this course.

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

4. Every student deserves to be treated with dignity and respect.

Using these as guiding principles, I expect students in Math 125 to collaborate with me in developing and maintaining an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. I expect all participants in this course (students and faculty alike) to treat each other with courtesy and respect.


There will be homework due approximately every other week. You will be given a list of exercises, and the number of problems you are expected to complete will be dependent on your standing (e.g. undergraduate, graduate with no AG background, graduate with AG background).

The assignments will be posted on Canvas, to be submitted in person or on Gradescope.

Covid Expectations

Please wear a mask or stay home if you have symptoms of respiratory illness, and take a covid test if possible. If you test positive for covid, please follow the guidelines given by Dartmouth. You will not be penalized for missing class due to illness, and I will work with you in making a plan get you caught up on whatever you miss.

The Honor Principle

Cooperation on homework is encouraged! However, you should write up the solutions on your own. Please acknowledge the people you worked with at the end of each assignment.

Plagiarism, collusion, or other violations of the Academic Honor Principle will be referred to the The Committee on Standards. If you have any questions as to whether some action would be acceptable under the Academic Honor Principle, please speak to me beforehand. For more information about standards of conduct, please see these resources.

Student Accessibility and Accommodations

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Getting Started with SAS webpage; ; 1-603-646-9900) and to request that an accommodation email be sent to their instructor in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with their instructor to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

Mental Health and Wellness

The academic environment is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including: the Counseling Center which allows you to book triage appointments online, the Student Wellness Center which offers wellness check-ins, and your undergraduate dean. The student-led Dartmouth Student Mental Health Union and their peer support program may be helpful if you would like to speak to a trained fellow student support listener. If you need immediate assistance, please contact the counselor on-call at (603) 646-9442 at any time. Please make me aware of anything that will hinder your success in this course.

Title IX

At Dartmouth, we value integrity, responsibility, and respect for the rights and interests of others, all central to our Principles of Community. We are dedicated to establishing and maintaining a safe and inclusive campus where all have equal access to the educational and employment opportunities Dartmouth offers. We strive to promote an environment of sexual respect, safety, and well-being. In its policies and standards, Dartmouth demonstrates unequivocally that sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are not tolerated in our community.

The Sexual Respect Website at Dartmouth provides a wealth of information on your rights with regard to sexual respect and resources that are available to all in our community.

Please note that, as a faculty member, I am obligated to share disclosures regarding conduct under Title IX with Dartmouth's Title IX Coordinator. Confidential resources are also available, and include licensed medical or counseling professionals (e.g., a licensed psychologist), staff members of organizations recognized as rape crisis centers under state law (such as WISE), and ordained clergy (see

Should you have any questions, please feel free to contact Dartmouth's Title IX Coordinator or the Deputy Title IX Coordinator for the Guarini School. Their contact information can be found on the sexual respect website at:

Religious Observances

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible, or before the end of the second week of the term (at the latest) to discuss appropriate adjustments. Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices.