Geometry Beyond the Torelli Map: Families, Sygyzies, and Special Subvarieties

This project is situated within mathematics, more specifically in algebraic geometry, a central area that explores geometric structures using tools from algebra. The research focuses on understanding the geometric and arithmetic properties of a fundamental object known as the Torelli locus by studying its interactions with the larger space in which it sits. These spaces play a key role in modern geometry, number theory, and mathematical physics. More precisely, the Torelli locus consists of all higher-dimensional shapes, known as abelian varieties, that can be associated with smooth surfaces (1-dimensional complex objects called algebraic curves). Studying the Torelli locus allows to understand exactly how these surfaces relate to their associated abelian variety and how they fit inside the larger parameter space for abelian varieties.
Our goal is to describe the Torelli locus: what it looks like, how it is positioned within the larger space, and how it interacts with other notable structures. By examining these relations, we uncover deep connections between geometry, symmetries, and arithmetic, showing how the properties of an algebraic curve are reflected in the abelian variety it produces.
Through this approach, we hope to shed light on long-standing questions in algebraic geometry, differential geometry, and number theory, providing a clearer picture of the rich, intricate world of geometric structures. Beyond the boundaries of pure mathematics, the tools and ideas developed here contribute to the theoretical foundation of areas such as string theory and mirror symmetry in physics, where our techniques are essential.