Rationality Properties of Some Hypersurfaces and Complete Intersections

Abstract

Solving equations has always been one of the core areas of mathematics. One important class of equations are so-called polynomial equations. These appear naturally in a broad collection of fields, such as number theory, particle physics and optimization. Studying the solutions to such equations is the domain of the mathematical subfield called algebraic geometry. When attempting to solve polynomial equations, one quickly notices that it is hard to give an explicit description of the solutions, a so-called parametrization. In fact, such a description is often outright impossible! But occasionally, one can find a clever trick that allows for a complete and explicit parametric description of the solutions. For any given polynomial equation, it is therefore natural to ask whether an explicit description is provably impossible, or if searching for a clever trick for describing the solution is warranted. This thesis answers such questions by computing so-called birational invariants. These are computed for many types of polynomial equations where they were not previously known. By doing so, new examples of polynomial equations are found where the solutions cannot be described in an explicit, parametric form.