Rigidity of Pham-Brieskorn Threefolds

Algebraic geometry is generally concerned with understanding the properties of geometric objects defined by polynomial equations. These geometric objects, called “varieties”, are points, curves, surfaces and higher dimensional objects. To each variety there is an associated abstract object called a “commutative ring”. It is interesting to try to determine whether a variety contains a certain type of structure that we call a “special cylinder”. A given variety contains a special cylinder if and only if the associated commutative ring is “non-rigid”. The goal of our research is to understand whether a particular family of varieties called Pham-Brieskorn varieties contains these special cylinders, or equivalently, to understand whether the commutative rings associated to them (Pham-Brieskorn rings) are rigid.

Understanding which Pham-Brieskorn rings are rigid has been an open question in the field of algebraic geometry for over 20 years. In the 3-dimensional situation, only three open cases remain. Our immediate goal and expectation is to solve these three remaining 3-dimensional cases, and over the longer term is to bring new insights and techniques towards characterizing rigid rings.

Faculty Supervisor:

Daniel Daigle

Student:

Partner:

Université Bourgogne Europe

Discipline:

Mathematics

Sector:

Other

University:

University of Ottawa

Program: