Tetrahedron equation and the algebraic geometry

The tetrahedron equation arises as a generalization of the famous Yang–Baxter equation to the $2+1$-dimensional quantum field theory and the $3$-dimensionaI statistical mechanics. Very little is still known about its solutions. Here a systematical method is described that does produce nontrivial solutions to the tetrahedron equation with spin-like variables on the links. The essence of the method is the use of the so-called tetrahedral Zamolodchikov algebras. Bibliography: 12 titles.