Toric Varieties

Toric varieties are specialized geometric objects in algebraic geometry built from combinatorial data called fans, which are collections of cones. They serve as a bridge between algebraic equations and geometric shapes, often representing spaces with symmetries similar to toruses (doughnut-shaped surfaces). These varieties allow mathematicians to study complex geometric structures using combinatorial tools, making them valuable in areas like mirror symmetry and optimization. Because they are defined by relatively simple data, toric varieties provide a concrete framework for understanding more intricate geometric concepts.