# MATHS 734 : Algebraic Geometry

## Science

### Course Prescription

Algebraic geometry is a branch of mathematics studying zeros of polynomials. The fundamental objects in algebraic geometry are algebraic varieties i.e., solution sets of systems of polynomial equations.

### Course Overview

Algebraic geometry is the study of polynomials and their solutions, and has connections to many branches of mathematics including number theory, complex analysis and topology. For example, algebraic curves over the complex numbers correspond to Riemann surfaces. Computational algebraic geometry has many applications in diverse fields, including cryptography, coding theory, combinatorics, integer programming, signal processing, robotics and various areas in computer science.

The basic objects of study in algebraic geometry are varieties (sets of solutions to systems of polynomial equations) and rational maps between them. The course will give a general overview of the subject, with particular focus on curves. We will introduce projective space, conics, Hilbert’s nullstellensatz, the Hilbert basis theorem, Groebner bases, the Zariski topology, irreducibility and dimension. Applications in geometry may include elliptic curves, Bezout’s theorem, Riemann-Roch theory and resolution of singularities. The majority of the course will be developing tools in commutative algebra (both theoretical and computational), but there will also be geometrical examples and motivations.

The course is suitable for students who are pursuing advanced study in a wide range of areas in pure mathematics, especially number theory, algebra, geometry or topology.

### Course Requirements

Prerequisite: MATHS 332 and at least one of MATHS 320, 328 Restriction: MATHS 334

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Apply concepts, theorems and calculations in commutative algebra and algebraic geometry to solve conceptual and theoretical questions. (Capability 1, 2, 3 and 5)
2. Demonstrate and apply correct use of advanced algebraic techniques. (Capability 1, 2, 3 and 5)
3. Be able to analyse and solve equations both algebraically and geometrically. (Capability 1, 2, 3 and 5)
4. Write logical and rigorous mathematical arguments, using appropriate language and sufficient detail and structure. (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Tests 60% Individual Test
Presentation 10% Individual Coursework
1 2 3 4
Assignments
Tests
Presentation

### Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

For further information on the Tuākana Maths programme, please visit
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. For this course, you can expect a total of 33 hours of lectures, 77 hours of reading and thinking about the content and 40 hours of work on assignments and other assessments.

### Delivery Mode

#### Campus Experience

• Attendance is expected at scheduled activities, including tutorials, to complete components of the course.
• The activities for the course are scheduled as a standard weekly timetable.

### Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

Recommended Texts:
• David Cox, John Little and Donal O’Shea "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra"
• Miles Reid "Undergraduate Algebraic Geometry"
• William Fulton "Algebraic Curves: An Introduction to Algebraic Geometry"

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Course will be made more geometric, less commutative algebra.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Class Representatives

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page .

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter .

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 31/10/2022 09:30 a.m.