Related papers: Normal complex surface singularities with rational…
Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with…
In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing.…
A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional…
We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the…
Recent work ([18], [1]) has produced a complete list of weighted homogeneous surface singularities admitting smoothings whose Milnor fibre has only trivial rational homology (a "rational homology disk"). Though these special singularities…
It is known (Stipsicz-Szab\'o-Wahl) that there are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a rational homology disk ($\mathbb{Q}$HD) smoothing, i.e.,…
Using the picture deformation technique of De Jong-Van Straten we show that no singularity whose resolution graph has 3 or 4 large nodes, i.e., nodes satisfying d(v)+e(v)\leq -2, has a QHD smoothing. This is achieved by providing a general…
In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with…
In this paper, a $\mathbb{Q}$HD singularity is a weighted homogeneous normal surface singularity admitting a rational homology disk ($\mathbb{Q}$HD) smoothing. These singularities are rational but often not log canonical. We classify all…
By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…
The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). We then discuss the case when the normal surface singularity is an N-fold…
A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely…
We show that isolated surface singularities which are non-normal may have Milnor fibers which are non-diffeomorphic to those of their normalizations. Therefore, non-normal isolated singularities enrich the collection of Stein fillings of…
The rational blowdown operation in 4-manifold topology replaces a neighborhood of a configuration of spheres by a rational homology ball. Such configurations typically arise from resolutions of surface singularities that admit rational…
Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential…
We complete the classification of the smooth, closed, oriented 4-manifolds having Euler characteristic less than four and a horizontal handlebody decomposition of genus one. We use the classification result to find a large family of…
For an isolated hypersurface singularity $f=0$, the Milnor number $\mu$ is greater than or equal to the Tjurina number $\tau$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito…
We show that a complex normal surface singularity admitting a contracting automorphism is necessarily quasihomogeneous. We also describe the geometry of a compact complex surface arising as the orbit space of such a contracting…
While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity,…
We prove that the boundaries of the Milnor fibers of smoothings of non-isolated reduced complex surface singularities are graph manifolds. Moreover, we give a method, inspired by previous work of N\'emethi and Szilard, to compute associated…