Algebraic curves

Description

Algebraic curves are among the most fundamental and beautiful objects in mathematics. They have connections with many mathematical subjects, including (but not limited to) algebra, geometry, combinatorics, and number theory. Classic examples include conics, elliptic curves, and plane quartics, each of which exhibits rich geometric and arithmetic properties. Algebraic curves have deep theoretical significance and broad applications. They play a central role in areas such as cryptography, coding theory, and mathematical physics. Elliptic curves, for instance, are both at the heart of modern cryptographic protocols and were crucial in the proof of Fermat’s Last Theorem, so this topic is a very versatile one.

While modern algebraic geometry often approaches curves through abstract frameworks (like schemes and cohomology), a more concrete, hands-on approach remains both accessible and rewarding. Many fundamental results can be understood through classical methods, including intersection theory, divisor theory, and the Riemann-Roch theorem, which offer powerful ways to analyze the geometry of curves. This perspective has been advocated by mathematicians such as William Fulton and Phillip Griffiths, whose works provide an entry point into the field without requiring heavy machinery.

In this project, students will explore algebraic curves using both classical and modern perspectives. They will begin with foundational texts and then investigate a particular class of curves or an application of their choice. Depending on their interests, they may study real, complex, or arithmetic aspects of curves, or delve into computational techniques for working with explicit examples. Through this exploration, they will develop their own understanding of this fascinating subject and its connections to broader areas of mathematics.

Essential modules

It is essential that students have taken Algebra II, TopologyII and Geometry III.
Helpful modules are Galois Theory III and Codes & Cryptography III, but they are not essential for the success of the project.

Resources

- “Algebraic geometry and arithmetic curves” by Q. Liu, especially Chapter 7. [Library link]
- “Algebraic curves : an introduction to algebraic geometry” by W. Fulton [Library link]
- “Geometry of Algebraic Curves” by E. Arbarello et Al [Library link]