The Inequality 8p_g < μ for Hypersurface Two‐dimensional Isolated Double Points

Abstract

The Milnor number p and the geometric genus pB of normal 2-dimensional double points are studied by using Zariski's canonical resolution. By using formulas due to E. HORIKAWA and H. LAUFER, we represent p -8pg in terms of the number of blowing-ups along IP' and the number I of "even" components in the resolution process. A key point of our arguments is the fact that if I is small then the resolution process is restricted very much. For rational double points and double points with p , = 1, each classes are characterized by numerical invariants appearing in this resolution process. For the case p , = 1, we can make our inequality sharper and can prove 12. pB -3 < p. This is an another proof of Xu-Yau's inequality for the singularity with p , = 1 in our situation.(l985), 297-354 9-329

Singularity invariants related to Milnor fibers: survey

“…This was shown to be true in the following cases: quasi-homogeneous [131], weakly elliptic, f = g(x, y) + z N [8,97], double point [123], triple point [7], absolutely isolated [88].…”

Durfee's conjecture on the signature of smoothings of surface singularities

“…For hypersurfaces we have the following 'positive' results: 8p g < µ for (X, 0) of multiplicity 2, Tomari [30], 6p g ≤ µ − 2 for (X, 0) of multiplicity 3, Ashikaga [3], 6p g ≤ µ − ν + 1 for quasi-homogeneous singularities, Xu-Yau [33], 6p g ≤ µ for suspension singularities {g(x, y) + z k = 0}, Némethi [24,25], 6p g ≤ µ for absolutely isolated singularities, Melle-Hernández [21].…”

Dedekind sums and the signature of f (x, y) + zN

“…Actually, for any germ g = f (x, y) + z 2 , Tomari [24] proved the inequality σ ≤ −µ/2. Notice that our coefficient (a + 1)/3a of µ in (4.1) generalizes Tomari's coefficient 1/2 (case a = 2).…”