## Math 643 Algebraic Geometry I, Fall 2018

Welcome to Math 643! This course serves as an introduction to Algebraic Geometry. Algebraic Geometry is a central subject in modern mathematics, with close connections with number theory, combinatorics, representation theory, differential and symplectic geometry. We will study basic properties of projective algebraic varieties such as dimension, degree and singularities. At the same time, we will develop a large body of examples that motivate the study of the subject. Depending on time, we will develop the classical theory of curves and surfaces. This course should be enough preparation for a course on the theory of schemes and cohomology.

** Lecturer: ** Xudong Zheng, xzheng@math.jhu.edu

** Office hours: ** TBD

** Venue: ** Gilman Hall 77, TTh 10:30-11:45

** Text book: ** The three recommended texts for this course are:

- Hartshorne, Algebraic Geometry, Spring 1977.

- Ulrich GĂ¶rtz and Torsten Wedhorn, Algebraic Geometry, Part I: Schemes. With Examples and Exercises, Springer 2010.

- Joe Harris, Algebraic Geometry: a first course, Springer 1992.

- Igor Shafarevich, Basic Algebraic Geometry I, Varieties in Projective Space, Springer-Verlag 1994.

** Prerequisites: ** A first year graduate course in algebra: familiarity with commutative rings and modules. We will develop the necessary commutative and homological algebra in the course. Familiarity with differential geometry or topology is helpful, but not required.

** Homework: ** There will be ten homework sets. The homework is due on Thursdays at the beginning of class. You may (in fact, you are encouaged to) work on problems together; however, the write-up must be your own and should reflect your own understanding of the problem.

** Grading: ** The grade will be entirely based on the homework.

** Additional references: ** The following is a list of references that are more advanced, but you might wish to consult them for more in depth treatments of the subject.

- D. Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer-Verlag.

- P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.

** Course materials: **
Homework 1: Hartshorne I. 1.1(a, b), 1.3, 1.4, 1.5, 1.8.
Homework 2, due Thursday September 21.
Homework 3, due Thursday September 28.
Homework 4, due Thursday October 12.
Homework 5, due Thursday October 26.
Homework 6: [GW] page 62, Exercises 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, due Monday November 6.
Homework 7: [GW] page 63, Exercise 2.13, page 89-90, Exercises 3.5, 3.6, 3.11, 3.12, 3.18, due Thursday November 16.
Homework 8, due Thursday November 30.
Homework 9, due Thursday December 7.