This master’s project studies the geometry and arithmetic of surfaces defined by two quadratic equations in 4 variables. The project combines algebra, geometry, and number theory to better understand how arithmetic constraints interact with geometric structure. Expected prerequisites include MATH321 and MATH411, or equivalent courses covering modern algebra and Galois theory. Experience coding (or willingness to learn) would be beneficial.
The goal of the project is to understand and classify when these surfaces can have solutions defined over quadratic field extensions (so-called quadratic points), and under what circumstances such solutions fail to exist. To investigate this, the project examines how symmetries arising from field extensions act on the lines contained in the surface (there are 16 of them). Using tools from group theory, all possible symmetry actions can be classified. For each case, the goal is either to construct explicit examples with no quadratic points or prove that such examples cannot occur.
Supervisors
Primary Supervisor: Brendan Creutz
Key qualifications and skills
Prerequisities: MATH321 (or knowledge of groups, rings and fields)
Desired: MATH411 (Galois theory) and some experience coding (e.g., Python or Sage)
Does the project come with funding
No - Student must be self-funded
Final date for receiving applications
Ongoing
How to apply
Direct inquiries to brendan.creutz@canterbury.ac.nz
Keywords
Number Theory, Algebraic Geometry, Group Theory
