Cubic fourfolds, K3 surfaces, and rationality questions

TL;DR

This survey explores the geometry of complex cubic fourfolds, focusing on their rationality, associated K3 surfaces, and classical constructions, providing insights into longstanding rationality questions in algebraic geometry.

Contribution

It synthesizes classical and modern approaches to understanding cubic fourfolds and their links to K3 surfaces, highlighting new perspectives on rationality problems.

Findings

Identification of special cubic fourfolds linked to K3 surfaces

Connections between rationality and Hodge structures in cubic fourfolds

Overview of classical and recent rationality constructions

Abstract

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and their geometric interpretations, and connections with rationality and unirationality constructions.