Hypersurface Singularities Research Articles

Equivariant characteristic classes of singular hypersurfaces

In this paper, we introduce definitions for the integrated equivariant Milnor number [Formula: see text] and the equivariant Milnor class [Formula: see text], for singular hypersurfaces. We prove that the [Formula: see text] are constant on the strata in a Whitney stratification of [Formula: see text], along with the correlation [Formula: see text] for hypersurfaces hosting isolated singularities [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th equivariant Milnor class of [Formula: see text]. We also introduce the equivariant Fulton–Johnson class of singular hypersurfaces. We give an equivariant version of Verdier’s specialization morphism in homology, and also for constructible functions. This is used for finding a relation between equivariant Fulton–Johnson and Schwartz–MacPherson classes.

Rings of differential operators on (k,s)-th Tjurina algebras of singularities

Abstract In this paper, we give a description of differential operators on tensor products A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 𝕂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B, respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , 𝟎 ) = ( V ⁢ ( f ) , 𝟎 ) ⊆ ( ℂ r , 𝟎 ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n.

Theta Pairing of Hypersurface Rings

In this article, we prove a conjecture on the positive definiteness of the Hochster Theta pairing over a general isolated hypersurface singularity, namely: Let R be an admissible isolated hypersurface singularity of dimension n. If n is odd, then is positive semi-definite on . The conjecture is expected to be true for the polynomial ring over any field. We prove this conjecture over any field of arbitrary characteristic. We also provide two different proofs of the above conjecture overusing the Hodge theory of isolated hypersurface singularities and structural facts about the category of matrix factorizations. The first proof overis a more complete and developed version of a former work of the author. We have extended some of the former results in this article. The second proof overis quite direct and uses a former result of the author on Riemann-Hodge bilinear relations for Grothendieck residue pairing of isolated hypersurface singularities.

Colocalizing subcategories of singularity categories

Utilizing previously established results concerning costratification in relative tensor-triangular geometry, we classify the colocalizing subcategories of the singularity category of a locally hypersurface ring and then we generalize this classification to singularity categories of schemes with hypersurface singularities.

Rational points on a class of cubic hypersurfaces

Abstract Let r ⩾ 3 r\geqslant 3 be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 .

A Deterministic Method for Computing Bertini Type Invariants of Parametric Ideals

A new framework is proposed for computing general numerical invariants, called Bertini type invariants, associated to a family of polynomial ideals depending on deformation parameters. A deterministic algorithm is introduced for computing parameter dependency of Bertini type invariants. As an application, an algorithm for computing Chern–Schwartz–MacPherson classes of a family of singular projective hypersurfaces is obtained.

Jet scheme and simple isolated hypersurface singularities

Let m≥2 be a positive integer. Let (H(f),o) be the analytic germ of hypersurface attached to a weighted-homogeneous polynomial f∈C[x1,…,xm]. In this article, we mainly characterize the simpleness of any isolated analytic quasi-homogeneous complex hypersurface singularity (H,o) in Cm in terms of the associated jet schemes. We also compute the dimension of the jet scheme Ln(H)o, formed by the n-jets centered at o, for every positive integer n>>0, which is the crucial technical part of our simpleness criterion.

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Abstract We study the relation between isolated hypersurface singularities (e.g., ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m {A_{m}} -singularity, (2) if the symplectic manifold contains an A m {A_{m}} -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.

A note on the semistability of singular projective hypersurfaces

A note on the semistability of singular projective hypersurfaces

On maximal operators associated with singular hypersurfaces

Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. We prove the boundedness of these operators and define a critical exponent in the space of summable functions, when hypersurfaces are given by parametric equations.

On Maximal Operators Associated with Singular Hypersurfaces

On Maximal Operators Associated with Singular Hypersurfaces

Deformations of Calabi–Yau varieties with k-liminal singularities

Abstract The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$ . For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$ -liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

Milnor versus Tjurina

This note, based on the talk given by the author at the Conference in Analytic and Algebraic Geometry, Lodź 2023, is an overview of the results on the invariants of hypersurface singularities in positive characteristic.

On pseudo-Riemannian quartics in Finsler geometry

Finsler geometry usually describes an extension of Riemannian geometry into a direction-dependent geometric structure. Historically, the well-known Riemann quartic length element example served as the inspiration for this construction. Surprisingly, the same quartic expression emerges as a fundamental dispersion relationcovariant Fresnel equationin solid-state electrodynamics. As a result, it is possible to conceive of the Riemann quartic length expression as a mathematical representation of a well-known physical phenomenon. This paper provides a number of Riemann quartic examples that show Finsler geometry to be overly constrictive for many applications, even when the signature space is positive definite in the Euclidean sense. The strong axioms of Finsler geometry are broken down on many more singular hypersurfaces for the spaces having an indefinite (Minkowski) signature. We suggest a more flexible definition of a Finsler structure that only has to hold for open subsets of a manifold’s tangent bundle. We demonstrate the distinctive singular hypersurfaces connected to the Riemann quartic and discuss the potential physics explanations for them. As an illustration of the pseudo-Riemannian quartic, we took into consideration the dispersion relation that appears in electromagnetic wave propagation in uniaxial crystal. Our analysis suggests that the signature of the Finsler measure may be altered for large anisotropy factors.

On $\mu$-Zariski pairs of links

The notion of Zariski pairs for projective curves in $\mathbb{P}^{2}$ is known since the pioneer paper of Zariski. In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski) pair of curves $C = \{f(x,y,z) = 0\}$ and $C' = \{g(x, y, z) = 0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a Milnor number. We give new examples of weak Zariski pairs which have same $\mu^{*}$ sequences and same zeta functions but two functions belong to different connected components of $\mu$-constant strata (Theorem 14). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology, which implies the Jordan forms of their monodromies are different (Theorem 24). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that the hypersurface pair constructed from a Zariski pair of irreducible plane curves with simple singularities give a diffeomorphic links (Theorem 25).

Multiplier ideals of plane curve singularities via Newton polygons

We give a description of the multiplier ideals and jumping numbers associated with a plane curve singularity in a smooth surface in terms of Newton polygons. Our approach is inspired by a theorem of Howald about multiplier ideals of Newton non-degenerate hypersurfaces and our results provide a generalization of it to the case of plane curve singularities. We use toroidal embedded resolutions, which can be applied to the case of quasi-ordinary hypersurface singularities.

Asymptotically Kasner-like singularities

abstract: We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $$ {}^{(4)}g = -dt^2 + \sum_{i,j=1}^3 a_{ij}t^{2 p_{\max\{i,j\}}}\,{\rm d} x^i\,{\rm d} x^j $$ on $(0,T]_t\times\Bbb{T}^3_x$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $t\to 0^+$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' $\{t=0\}$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$t$ hypersurfaces.

Moving plane method for varifolds and applications

abstract: In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.

On the Boundedness of the Maximal Operators Associated with Singular Hypersurfaces

On the Boundedness of the Maximal Operators Associated with Singular Hypersurfaces