Research Projects

This project focuses on studying the geometry of Richardson varieties. These varieties are formed by taking the intersection of a Schubert variety and an opposite Schubert variety inside a generalized flag variety G/B. They are important objects in algebraic geometry and representation theory because of their rich structure and connections to combinatorics. In a recent joint work with M. B. Can, we provide a type-free classification of toric Richardson varieties. This project will extend the study to spherical, horospherical, and weak Fano Richardson varieties, with a focus on their birational geometry and Mori-theoretic properties. The specific objectives are: 1. Describe the effective and nef cones of Richardson varieties in a generalized flag variety using the combinatorics of Bruhat intervals; study their face structures, and analyze contractions and extremal rays. 2. Describe necessary and sufficient conditions for a Richardson variety to be spherical under a suitable Levi subgroup of a parabolic subgroup of G using techniques from invariant theory and spherical geometry. 3. Give a complete classification of spherical and horospherical Richardson and Schubert varieties across all Lie types. 4. Study smooth and Gorenstein Richardson varieties and analyze the positivity properties of divisors (nef, ample, and basepoint-free).