Hypersurface Singularities In General Two-dimensional Circulation Durfee Conjecture And Related Issues

In 1978, Durfee conjectured that:Let (V, p) be a normal 2-dimensional hypersurface singularities, then we have v(V,p)>6pg(V,p).In recent thirty years, this "geogtaphical" problem concerning (ps,/x) which was posed by Durfee have been the important research domains for algebraic geometry mathematicians at all times. The problem describes the relations between the Milnor number ^ and the geometric genus pg of the singularities. In 1994, Tan proved the relations between n and the Betti number 62 (-Ep) for the surface singularities of cyclic type. In this paper, we band them together to study the relations of these three aspects and gain the inequalities we want.In chapter 1,2, we make a systematic introduction for the cyclic coverings, the singularities of cyclic type and the background of the Durfee problem, we also set out all important results about these.Chapter 3 discusses the cyclic hypersurface triple points. By the canonical resolution of the triple covering and induction, we get the best inequality between the surface formal Milnor number /a(S) and pg , then we prove the Durfee conjecture in this case.In chapter 4, we continue to discuss the cyclic hypersurface quartic points. Using the results on Ring Theory, we combine induction with local analyses of singularities, then we get the best inequality of integral coefficient between n(S] and pg . So we solve the Durfee conjecture in this case.The results on the surface singularities of n-fold cyclic coverings will appear in chapter 5. We use plenty of tricks on Number Theory, induction and local analyse, then for general hypersurface singularities of cyclic type, we get the Milnor inequality. Moreover, we also have some interesting results on Number Theory.