Related papers: On taut singularities in arbitrary characteristics
We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose components intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called…
We prove that the group $\mathrm{SAut}_{\mathrm{k}}(\mathbb{A}^2)$ is simple as an algebraic group of infinite dimension, over any infinite field $\mathrm{k}$, by proving that any closed normal subgroup is either trivial or the whole group.…
We study the geometric and combinatorial effect of smoothing an intersection point in a collection of arcs or curves on a surface. We prove that all taut arcs with fixed endpoints and all taut 1-manifolds with at least two non-disjoint…
For a singular Riemannian foliation whose leaves are properly embedded, we show in the first part of this article the existence of global tubular neighbourhoods, and we develop a global description of the foliation as stratification by…
We extend fundamental inequalities related to the canonical map of surfaces of general type to positive characteristic. Next, we classify surfaces on the Noether lines, i.e., even and odd Horikawa surfaces, in positive characteristic. We…
We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular…
In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers and the Tjurina numbers for isolated hypersurface…
We classify normal stable surfaces with $K_X^2 = 1$, $p_g = 2$ and $q=0$ with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli…
We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular…
We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations…
It is a conjecture of Koll\'ar that a variety $X$ with rational singularities in some open subvariety $U$ has a rationalification; that is, a proper, birational morphism $f: Y \rightarrow X$ such that $Y$ has rational singularities, and…
It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of…
We prove semi-rationalification and semi-log-canonicalization for Gorenstein demi-normal surfaces. That is, given a Gorenstein demi-normal surface X with semi-rational (respectively, semi-log canonical) singularities in an open set U with…
We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…
We prove a conjecture of Teissier asserting that if $f$ has an isolated singularity at $P$ and $H$ is a smooth hypersurface through $P$, then $\widetilde{\alpha}_P(f)\geq \widetilde{\alpha}_P(f\vert_H)+\frac{1}{\theta_P(f)+1}$, where…
A description of Cohen-Macaulay modules over cusp surface singularities and over unimodule hypersurface singularities of type T is given. It is proved that among minimally elliptic singularities and their quotients only simple elliptic and…
K\"ahler's paper "\"Uber die Verzweigung einer algebraischen Funktion zweier Ver\"anderlichen in der Umgebung einer singul\"aren Stelle" offered a more perceptual view of the link of a complex plane curve singularity than that provided…
We show that if a cubic hypersurface with positive dual defect over the complex number field is not a cone, then either the hypersurface coincides with the secant variety of the singular locus, or the hypersurface contains a linear…
This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the…
Bonnet has characterized his surfaces by a geometric condition. What is done here is a characterization of the same surfaces by two analytic conditions: (i) the mean curvature $H$ of a surface in $\mathbb{R}^3$ should admit a reduction to…