Math 763 -- Algebraic Geometry I

Fall 2019

Homework assignments

• Homework 1, due Thursday, September 19th.
• Homework 2, due Thursday, September 26th.
• Homework 3, due Tuesday, October 8th.
• Homework 4, due Thursday, October 17th.
• Homework 5 will be assigned next week (either Tuesday, October 22nd or Thursday, October 24th) and due around Halloween.

Course description

This is a first course in algebraic geometry. While there are no formal prerequisites beyond a knowledge of the material covered in the first-year algebra and geometry sequence, familiarity with some basic commutative algebra will be helpful. The rough outline of the course is as follows (subject to change):

• Affine and projective varieties.
• Morphisms and rational maps.
• Local properties: smoothness and dimension. Tangent space.
• Divisors.
• Low-dimensional varieties: curves and surfaces. Blow-ups.
• The Riemann-Roch Theorem.

Here is a more detailed lecture-by-lecture list of topics that I covered in the past, of course, this is all subject to change.

Handouts

• Correspondence between sets and ideals
• Proof of the Nullstellensatz

References

• Shafarevich, Basic Algebraic Geometry.
• Algebraic Geometry (online notes) by Milne.
• Hartshorne, Algebraic Geometry, Chapter I (this is more advanced, so does not quite match the content).
• Here is a discussion on MathOverflow with more books on algebraic geometry, but most of them are going to be too advanced.
• Here are notes from the last time I taught this course. These were taken in class, so

there are probably typos.

Information for students

• Instructor: Dima Arinkin
• Office Hours: Tuesday 3-4pm, Wednesday 2-2:45pm, and by appointment in VV 603
• Lectures: TuTh 11am-12:15pm, VV B129
• Grade: There will be weekly homework assignments, but no exams in this course.