On Splitting Coverings of Algebraic Varieties

這篇文章的主要目的是介紹一些三次覆蓋的性質。首先，我們會介紹一些關於三次覆蓋的篡本＇隨質，並且利用三次覆蓋的技巧去構造出一個 Pg = q → 2 ，磯＝ 5 的代數曲面。最後，我們會把類似的技巧誰廣到四次覆蓋去構造一個 P 〞 ＝ q → 2 ，磯＝ 6 的代數曲面。The main purpose of this article is to introduce some properties of triple coverings. In this article, we review some properties of a triple covering, and ten we consider to use a triple covering to construct a complex surface of general type with pg = q = 2 and K2X = 5. Finally, we try to use similar techniques to develop a quadruple cover over a complex abelian surface.Table of Contents Table of Contents iii Introduction iv 1 Triple covers in algebraic geometry 1 1.1 local analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The ramification and branch locus . . . . . . . . . . . . . . . . . . . . 3 2 An triple cover example 7 2.1 defining equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 local analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 An quadruple cover example 11 3.1 defining equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 ramification and branch locus . . . . . . . . . . . . . . . . . . . . . . 14 3.3 geometric invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Bibliography 1