Abstract Toroidal varieties (called toroidal embeddings in [20] ), roughly speaking, are algebraic varieties that are locally (formally) toric in structure, and toroidal morphisms are those morphisms of varieties which are locally determined by toric morphisms. A toroidal lifting of a morphism φ : X → Y of algebraic varieties, which can be obtained by performing sequences of blowups of nonsingular subvarieties above X and Y, is called a toroidalization of φ. The problem of existence of toroidalization, proposed first in [1] , has been proved only when Y is a curve, or when φ is dominant and X , Y are of dimension ⩽3. In general, specially when the dimension of Y is larger than two, the problem seems hard enough to be considered in rather restricted classes of morphisms such as strongly prepared or locally toroidal morphisms. The latter notion is originated from Cutkosky's proof of toroidalization, locally along a fixed valuation, in all dimensions in [8] . In this paper, we will prove the existence of toroidalization of locally toroidal morphisms from n–folds to 3–folds.

中文翻译：

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从 n 折到 3 折的局部环面态射的环面化

摘要 环形簇（在[20]中称为环形嵌入），粗略地说，是结构上局部（形式上）复曲面的代数簇，环形态射是那些由环面态射局部确定的簇的态射。代数变体的态射 φ : X → Y 的环形提升，可以通过执行 X 和 Y 之上的非奇异子变体的爆炸序列来获得，称为 φ 的环形化。在[1]中首先提出的环化存在问题仅在Y是曲线时，或者当φ占主导并且X，Y的维数为⩽3时才被证明。一般来说，特别是当 Y 的维数大于 2 时，这个问题似乎很难在相当受限制的态射类中考虑，例如强准备或局部环形态射。后一个概念源自 Cutkosky 的环形化证明，局部沿固定估值，在 [8] 中的所有维度上。在本文中，我们将证明从 n 折到 3 折的局部环面态射的环面化的存在性。