Smoothness Of The Normalization Of A Finite Abelian Covering With Normal Crossing

The covering theories of an algebraic surface are the important topics in algebraic geometry. And the theory of Abelian coverings is an unclear realm. We investigated the smoothness of the normalization of a finite Abelian covering with normal crossing of an algebraic surface, after Yun Gao. In her Ph.D thesis, Gao gave the four kinds of elementary transformations, and a result on the smoothness of an Abelian covering. Using these results, we can separate variables of the definition equations of an Abelian covering under some conditions we imposed on, and this procedure is exactly the normalization of the covering. The conditions we imposed on are just the conditions for smoothness.We firstly determined the smoothness of two dimensional varieties. Because the elementary transformations we used don’t ask for dimensions, we can get the corresponding results in any situations by an analogy.