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 obtained from the graph of a rational double point or rational triple point by making (some) vertex weights more negative. For rational singularities we show one direction in general, and the other direction (simpleness) within the special classes of rational quadruple points and of sandwiched singularities.
There are strange dual pairs of bimodal singularities that are not assigned an invertible projectivisation in EbelingPloog. We study families of $K3$ surfaces associated to such pairs.
We consider del Pezzo surfaces with du Val singularities. We'll prove that a del Pezzo surface $X$ with du Val singularities has a $-K_X$-polar cylinder if and only if there exist tiger such that the support of this tiger does not contain anti-canonical divisor. Also we classify all del Pezzo surfaces $X$ such that $X$ has not any cylinders.
We prove the constancy of the Łojasiewicz exponent in non-degenerate $μ$-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.
We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.
On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle
通过计算余切空间$T^{1}$和$T^{2}$的一般维数公式中的修正项，证明了一类有理曲面奇点的余切空间的维数公式.我们得到了有理曲面奇点$X$的对偶图不包含特定类型的构形时，修正项为零，这推广了Theo de Jong关于有理行列式曲面奇点的修正项$c(X)$为零的一个结果.特别地，我们的结果意味着$c(X)$是零的Riemenschneiders拟行列式有理曲面奇性，这也推广了结果的qoutient奇性。
We prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.
We provide combinatorial/topological formula for the multiplicity of a complex analytic normal surface singularity whenever the analytic structure on the fixed topological type is generic.
Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Department of Mathematics, Hiyoshi Campus, Keio University（庆应义塾大学日吉校区数学系）；Tokyo Institute of Technology（东京工业大学）；Departement of Mathematics, Galatasaray University（加拉塔萨雷大学数学系）
We explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a cubic surface with singularities that are not ordinary double points lies on t…
We study symplectic resolutions of the Hilbert scheme of two points on a surface with one ADE-singularity. We also characterize such singularities by central fibers of their symplectic resolutions. As an application, we show that these singularities are isomorphic to the Slodowy slices which are transversal to the `sub-subregular' orbits in the nilpotent cones of ADE-types.
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 Némethi and Szilard, to compute associated plumbing graphs.
We study lattice polarizations of five exceptional pairs of families of K3 surfaces obtained via compactifications of strange dual pairs of bimodal singularities. We show that the polarizations induced by embedding these paired families in paired toric varieties obtained from polar dual reflexive polytopes cannot be mirror lattices and identify mirror sublattices.
The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to define singularities of splice type and splice-quotient singularities. The Casson Invariant Conjecture and Milnor Fiber Conjectures are discussed, as well as some open questions about hypersurface and complete intersection singularities with integral homology sphere link.
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm enumerating topological types of morsifications of real function singularities is promoted.
We give a new criterion for when a resolution of a surface of general type with canonical singularities has big cotangent bundle and a new lower bound for the values of $d$ for which there is a surface with big cotangent bundle that is deformation equivalent to a smooth hypersurface in $\mathbb{P}^3$ of degree $d$.
