Rationality of Cubic Hypersurfaces in Low Dimension

The study of cubic hypersurfaces has been a central topic in algebraic geometry for over a century, dating back to Cayley and Salmon’s work on cubic surfaces and their 27 distinguished lines. In the 20th century, Clemens and Griffiths proved the irrationality of cubic threefolds via intermediate Jacobians, marking a significant breakthrough. More recently, cubic fourfolds have gained attention due to their connections with K3 surfaces, Hodge theory, and derived categories. A key question in this field is the rationality problem, which asks whether a variety is birationally equivalent to projective space. While cubic surfaces are always rational, cubic threefolds were shown by Clemens and Griffiths to be irrational using Prym varieties and quadric fibrations. The rationality of cubic fourfolds, however, remains open. Hassett’s work on special cubic fourfolds provides a framework linking their rationality to associated K3 surfaces and moduli space structures. This thesis explores cubic hypersurfaces with a focus on their geometric and cohomological properties, particularly in relation to rationality. Chapter 1 covers foundational results, including Serre duality, the Lefschetz hyperplane theorem, and Hodge theory. Chapter 2 introduces abelian and Prym varieties, which play a key role in understanding cubic hypersurfaces. Chapter 3 examines cubic threefolds, focusing on their irrationality and Torelli-type theorems, while Chapter 4 investigates cubic fourfolds, their connection to K3 surfaces, and conjectures regarding their rationality.