"Toric varieties are arguably the simplest and most accessible varieties. They often appear in applications, both within mathematics and across the sciences. A toric variety is an irreducible variety that is parametrized by a vector of monomials. The relations among these monomials are binomials, i.e. polynomials with only two terms. Thus, an irreducible variety is toric if and only if its prime ideal is generated by binomials. Monomials and binomials correspond to points in an integer lattice, and we think of these as the vertices of a lattice polytope. Toric varieties appear prominently in optimization and statistics, thanks to the purely combinatorial description given above. This description also makes them a perfect “model organism” for algebraic geometers. They use toric varieties to test conjectures, teach geometric concepts, and compute invariants. For instance, the dimension and degree of a toric variety are the dimension and volume of the associated lattice polytope." [Michalek, Sturmfels 2021]

"The study of toric varieties is a wonderful part of algebraic geometry. There are elegant theorems and deep connections with polytopes, polyhedra, combinatorics, commutative algebra, symplectic geometry, and topology. Toric varieties also have unexpected applications in areas as diverse as physics, coding theory, algebraic statistics, and geometric modeling. Moreover, as noted by Fulton [105], “toric varieties have provided a remarkably fertile testing ground for general theories.” At the same time, the concreteness of toric varieties provides an excellent context for someone encountering the powerful techniques of modern algebraic geometry for the first time." [Cox, Little, Schenck 2010]