MA40188 Algebraic Curves

2015/16 Semester 1

Unit Description

This unit provides an introduction to algebraic geometry for students who have some background of abstract algebra. You should have taken Algebra 2B (MA20217 or equivalent) before taking this unit. Some familiarity of the theory of rings and fields will be assumed. See recommended texts for lecture notes of Algebra 2B in the last two years. You should also be comfortable with reading and writing rigorous mathematical proofs.

Lectures and Office Hours

The class meets three times a week. In general, Monday and Friday morning lectures are devoted to introducing new material, while the exercise classes on Friday afternoons will mainly focus on problem solving. See course calendar for more details on the contents of each lecture.

Your lecturer for this unit is Dr. Ziyu Zhang. You can contact me by email at zz505 at Bath for any questions or concerns. I hold two office hour sessions each week. You are welcome to come and ask questions at any time during the sessions. If you have questions to ask but cannot make to the office hours, please feel free to contact me.

There will be extra office hours after the Christmas vacation. Please see exam information.

Recommended Texts

• Reid, Miles. Undergraduate algebraic geometry. London Mathematical Society Student Text, 12. Cambridge University Press, Cambridge, 1988. ISBN: 0-521-35662-8. The online access of this book is provided by the university library.

• Fulton, William. Algebraic curves -- An introduction to algebraic geometry. Advanced Book Classics. Addison-Wesley Publishing Company, Redwood City, CA, 1989. ISBN: 0-201-51010-3. The author has made this book freely available to anyone through his webpage.

Course Calendar

More than any other discipline, mathematics requires that a student understands one concept before moving on to the next. Therefore it is critical that you attend this class on a regular basis in order to stay on track with the material we cover. Brief lecture notes will be available here usually after each lecture for your convenience. Although these notes roughly reflect what is being discussed in lectures, reading these notes cannot replace attending lectures yourself.

Here is a complete set of lecture notes, exercise sheets and solutions in one file. It comes in two different versions:

• Here is a long version: 115 pages. The notes for individual lectures were extracted from this version. It has a table of contents. You can use the clickable cross-reference links to jump back and forth. This version is more suitable for reading on a screen.
• Here is a short version: 70 pages. The table of contents and large chunks of blank spaces are removed. The font and page margins are smaller. Cross-reference links are also clickable. This version is more economic for printing purpose.

The following is a course calendar, including lecture notes for each individual lecture and handouts distributed in lectures.

Exercise Sheets

There will be 10 exercise sheets in total. Every exercise sheet contains 4 problems, most of which have multiple questions. These problems are usually given in increasing difficulty, with the first one involving routine questions and the last one comprehensive and sometimes slightly challenging. Explicit examples are heavily emphasized. Exercise sheets are designed to reinforce concepts covered in lecture as well as to encourage students to explore implications of the results discussed in class. Very few students will be able to go through the entire course without struggling on some problems, so do not be discouraged if you do not immediately know how to solve a problem. In confronting difficult questions you should consider how the problem at hand connects to topics, definitions and/or theorems discussed in class.

Although it is not mandatory to submit your work on a regular basis, it is strongly recommended that your try to work as much as possible on the exercises. If you have worked on a problem for a while and remain stuck, you are encouraged to discuss it with one another, or ask me for hints.  However if you have taken notes while discussing exercises with classmates, it will be a good idea to put these notes away when writing your own solution. Be warned: watching someone else solve a problem will not make exercises a good preparation for your exam. Writing down your own solution will help you think through the material and get a better understanding.

Exercise sheets are usually handed out on Monday and discussed in the exercise class on Friday of the following week. You are warmly welcome to submit your solutions for correction at the end of the exercise class, or anytime earlier. Please write your solutions neatly, with enough justification in each step. Don't forget to write your full name on your work. If it contains multiple pages, it will be helpful if you staple them before submission. At least two problems will be marked every week, but you should treat all four problems equally important. Submitted work will be handed back to you in the week after the deadline. Brief solutions will be posted here shortly after the exercise class.

Exam Information

Here is a mock exam paper and a few old exam papers for your preparation of the exam. Notice that the syllabus for this unit has minor changes every year, so please read the extra remarks for each exam paper to find out questions you don't need to worry about and notations that were used differently.

Here is an exam information sheet, which should give you a rough idea what types of questions you can expect in the exam.

For the convenience of your exam preparation, extra office hours will be provided after the Christmas vacation, until the day of the exam. See the following chart for details. The office hour on Sunday 17th January will take place in the lobby area outside the department office.