Math 591-- Algebraic Geometry

Fall 2010,  MWF 10:15--11:05 am,  2229 SAS Hall

Instructor: Seth Sullivant, office: 3114 SAS Hall, email: smsulli2@ncsu.edu

Office Hours: W 1:30-2:30, Th 2:30-3:30

Text: B. Hassett, Introduction to Algebraic Geometry, Cambridge University Press

Alternate/Additional Reading: Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms; Eisenbud and Harris, The Geometry of Schemes; Hartshorne, Algebraic Geometry

Prerequisites: MA 521 or equivalent.

Course Description: Algebraic geometry is the study of solutions to systems of polynomial equations. The algebra/geometry dictionary establishes a bijection between such solution sets and certain ideals in a polynomial ring. Hence, techniques from commutative algebra can be used to analyze these geometric objects. The course will provide an introduction to algebraic geometry. We will emphasize not only the theory, but also practical aspects of how to compute with polynomial ideals using Groebner bases. We will follow the book closely and discuss the following topics: Groebner bases, affine varieties, morphisms and rational maps, elimination theory, the Nullstellensatz, primary decomposition, projective varieties, Grassmannians, and Hilbert Functions.

Schedule: Is available here.

Homework: Homework will be assigned regularly at the beginning of the semester and is due in class on Fridays. Students must write up their own solutions. Please indicate on your homework any sources that you used in preparing solutions (e.g. if another student helped with a solution, or you found the solution in a book). Students are encouraged to prepare homework solutions in LateX. A file explaining how to prepare your homework can be found here. Homework assignments can be found here.

Final Project: Students will be expected to prepare a final project. This will consist of critically reading a paper on a topic related to the material from this course or working on a research problem in algebraic geometry, preparing a paper describing your project, and giving a twenty minute presentation to the class on your project during the final weeks of class. Project topics will be chosen halfway through the course, based on some suggested topics on the course website and in discussion with the instructor. Suggested topics can be found here.

Exams: There will be no exams. However, students will be expected to attend the final exam period as this will be used for student final presentations (Friday, 12/10 8–11 AM).

Grades: Grades will be based on Homework (50%) and a Final Project and Presentation (50%).

Attendance: Students are expected to arrive on time, to contribute to group work and class discussions, and to stay until the class ends. Attendance at all meetings of the class is expected. Occasional absences will be approved if they meet University policies.

Adverse Weather: Announcements regarding scheduled delays or the closing of the University due to adverse weather conditions will be broadcast on local radio and television stations and posted on the University homepage.

H1N1 Policy: If you are ill with symptoms of H1N1 influenza (i.e. fever over 100, sore throat, cough, stuffy or runny nose, fatigue, headache, body aches, vomiting and diarrhea) please do not come to class. Instead, immediately contact your medical provider or Student Health Services (515-7107) for advice or to arrange an appointment. If you are diagnosed with H1N1, please inform me immediately. You will be required to be isolated away from class for up to 7 days or 24 hours after symptoms subside, whichever is longest.

Cell Phones: Pagers, cellular phones and other types of telecommunication equipment are prohibited from use during class. Make sure that any pagers, phones or other equipment are turned off during the class period. If you have a special need to have your pager or phone on during class, please let me know.

Academic Integrity Statement: Students are required to follow the NCSU policy . “Academic dishonesty is the giving, taking, or presenting of information or material by a student that unethically or fraudulently aids oneself or another on any work which is to be considered in the determination of a grade or the completion of academic requirements or the enhancement of that student's record or academic career.’’ (NCSU Code of Student Conduct). The Student Affairs website has more information.

Students with Disabilities: Reasonable accommodations will be made for students with verifiable disabilities. In order to take advantage of available accommodations, students must register with Disabilities Services for Students.

Class Evaluations: Online class evaluations will be available for students to complete during the last two weeks of class. Students will receive an email message directing them to a website where they can login using their Unity ID and complete evaluations. All evaluations are confidential; instructors will never know how any one student responded to any question, and students will never know the ratings for any particular instructors.

LateX Resources: