Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

Complex Surface Singularities with Integral Homology Sphere Links

The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. We study a large class of such complete intersections, those of “splice type,” and boldly conjecture that all Gorenstein singularities with homology sphere links are equisi...

An asymptotic existence theorem for plane curves with prescribed singularities

Let d,m1, . . . ,mr be (r + 1) positive integers. Denote by V (d;m1, . . . ,mr) the variety of irreducible (complex) plane curves of degree d having exactly r ordinary singularities of multiplicities m1, . . . ,mr. In most cases, it is still an open problem to know whether this variety is empty or not. In this paper, we will concentrate on the case where the r singularities can be taken in a ge...

On the Existence of Plane Curves with Prescribed Singularities

Plane Curves of Minimal Degree with Prescribed Singularities

We prove that there exists a positive α such that for any integer d ≥ 3 and any topological types S1, . . . , Sn of plane curve singularities, satisfying μ(S1) + · · ·+ μ(Sn) ≤ αd, there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S1, . . . , Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove tha...

Line Bundles Associated with Normal Surface Singularities

In [15] L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (X, 0) (whose link M is a rational homology sphere) with the Seiberg-Witten invariant of M associated with the “canonical” spin structure of M . (The interested reader is invited to see the articles [15, 16, 17, 13, 19, 20] for the verification of the c...