A flag variety is a type of geometric object that arises in algebraic geometry, specifically in the study of projective spaces. It represents a collection of nested subspaces of varying dimensions. For example, a flag variety can consist of lines, planes, and higher-dimensional spaces that are contained within a larger space, allowing mathematicians to explore their relationships and properties. These varieties are often denoted by Grassmannians, which are spaces that parameterize all possible subspaces of a given dimension within a larger vector space. Flag varieties play a crucial role in various areas of mathematics, including representation theory and intersection theory, providing insights into the structure and behavior of algebraic objects.