Durfee's Conjecture On Singularities Of Cyclic Galois Covers Of Degree 5

In 1978 Durfee [5] conjectured that:Let (V,p)be a normal 2-dimensional hypersurface singularities, then we have:μ≥6p_g(V,p).Durfee'problem is always the important research domains for algebaric geometry mathematicians till now. The problem describes the relations between the Milnor's numberμand the geometric genus p_g(V,p) of the singularities. In 1994, Tan [12] proved the relations betweenμand the Betti's number b₂ (E_p) for the surface singularities of cyclic type. In 2002 Doctor Lan [8] proved Durfee's conjectured in her graduation thesis for cyclic type of degree 3 and 4.In this paper we will prove Durfee's conjecture on singularities of cyclic Galois covers of degree 5, that is,We will talk about some backgrounds in Sec. 1, and introduce cyclic Galois coverings and their singularities Sec. 2. Some properties about Dedekind's sums will be discussed in Sec. 3.In the Sec. 4, we will obtain the proof of Durfee's conjectured on singularities of cyclic Galois covers of degree 5.