Related papers: Deformations of weighted homogeneous surface singu…

We prove that small deformations of canonical singularities are canonical.

In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary…

Given a complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge…

We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces Jan Steven's list [Manuscripta Math. 1993] of the numbers…

In two parts, we present a bigness criterion for the cotangent bundle of resolutions of orbifold surfaces of general type. As a corollary, we obtain the \textit{canonical model singularities} (CMS) criterion that can be applied to determine…

The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local…

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…

Hypersurface simple K3 singularities defined by nondegenerate quasi-homogeneous polynomials are classified into ninety five classes in term of the weight of the polynomial by T. Yonemura. We consider versal deformations of them. It has been…

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.

Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan-Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic…

In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite…

Let $\mathcal{F}$ be written as $ f^{*}(\mathcal{G})$, where $\mathcal{G}$ is a $1$-dimensional foliation on $ {\mathbb P^{n-1}}$ and $f:{\mathbb P^n}--->{\mathbb P^{n-1}}$ a non-linear generic rational map. We use local stability results…

The dual complex associated to a resolution of singularities generalizes the notion of a resolution graph of a surface singularity to any dimension. We show that homotopy type of the dual complex is an invariant of an isolated singularity.

Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations…

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence…

In this note, we describe the possible singularities on a stable surface which is in the boundary of the moduli space of surfaces isogenous to a product. Then we use the $\mathbb Q$-Gorenstein deformation theory to get some connected…

We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the…

We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S. A normal surface X with…

We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a…

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and…