چكيده فارسي :
Toroidal varieties are algebraic varieties that are locally (for- mally) toric in structure, and toroidal morphisms are those morphisms of varieties which are locally determined by toric morphisms. The problem of toroidalization, proposed rst in [1], is to construct a toroidal lifting of a dominant morphism φ : X ! Y of algebraic varieties by blowing up non- singular subvarieties in the target and domain. This problem is evidently very di cult, and it has been solved only when Y is a curve, or when φ is dominant and X; Y are of dimension ⩽ 3 { see [7]. This article provides a comprehensive survey of the toroidalization problem. In addition, we discuss some recent results in toroidalization of locally toroidal morphisms [2], which is among patching type problems.
