Hypersurface Singularities Research Articles

On the Pinsky Phenomenon for B-Elliptic Operators

Necessary conditions for the summability of spectral expansions in eigenfunctions of an elliptic operator with a Bessel operator in one of the variables in an arbitrary-dimensional domain adjacent to the singularity hypersurface are obtained. It is proved that if the spectral expansion of an arbitrary function at some point of this hypersurface is summable by Riesz means, then its average value over the half-ball centered at the specified point has generalized smoothness.

On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities

It is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras Lkl(V) associated to isolated hypersurface singularities defined to be the Lie algebra of derivations of the local Artinian algebra Mnl(V):=Ol/⟨F,Jn⟩, i.e., Lkl(V)=Der(Mnl(V)). In this paper, we construct a new conjecture for the complete characterization of simple hypersurface singularities using the Lie algebras Lkl(V) under certain condition and prove it true for Lkl(V) when k,l=2.

On $$L^2$$ extension from singular hypersurfaces

In $$L^2$$ extension theorems from a singular hypersurface in a complex manifold, important roles are played by certain measures such as the Ohsawa measure which determine when a given function can be extended. We show that the singularity of the Ohsawa measure can be identified in terms of singularity of pairs from algebraic geometry. Using this, we give an analytic proof of the inversion of adjunction in this setting. Then these considerations enable us to compare various positive and negative results on $$L^2$$ extension from singular hypersurfaces. In particular, we generalize a recent negative result of Guan and Li which places limitations on strengthening such $$L^2$$ extension by employing a less singular measure in the place of the Ohsawa measure.

Efficient representation of spatio-temporal data using cylindrical shearlets

Efficient representations of multivariate functions are critical for the design of state-of-the-art methods of data restoration and image reconstruction. In this work, we consider the representation of spatio-temporal data such as temporal sequences (videos) of 2- and 3-dimensional images, where conventional separable representations are usually very inefficient, due to their limitations in handling the geometry of the data. To address this challenge, we define a class E(A)⊂L2(R4) of functions of 4 variables dominated by hypersurface singularities in the first three coordinates that we apply to model 4-dimensional data corresponding to temporal sequences (videos) of 3-dimensional objects.To provide an efficient representation for this type of data, we introduce a new multiscale directional system of functions based on cylindrical shearlets and prove that this new approach achieves superior approximation properties with respect to conventional multiscale representations. We illustrate the advantages of our approach by applying a discrete implementation of the new representation to a challenging problem from dynamic tomography. Numerical results confirm the potential of our novel approach with respect to conventional multiscale methods.

K-th singular locus moduli algebras of singularities and their derivation Lie algebras

In this paper, we introduce a series of new invariants for singularities. A new conjecture about the non-existence of negative weight derivations of the new k-th singular locus moduli algebras for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture in some cases.

Computing Holonomic D-Modules Associated to a Family of Non-isolated Hypersurface Singularities via Comprehensive Gröbner Systems of PBW Algebra

Computing Holonomic D-Modules Associated to a Family of Non-isolated Hypersurface Singularities via Comprehensive Gröbner Systems of PBW Algebra

Categories for Grassmannian Cluster Algebras of Infinite Rank

Abstract We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\textrm {Ext}^{1}$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

Skew graded (A ∞ ) hypersurface singularities

For a skew version of a graded (A ∞ ) hypersurface singularity A, we study the stable category of graded maximal Cohen-Macaulay modules over A. As a consequence, we see that A has countably infinite Cohen–Macaulay representation type and is not a noncommutative graded isolated singularity.

DERIVATIONS OF LOCAL k-TH HESSIAN ALGEBRAS OF SINGULARITIES

In our previous work, we introduced a series of new derivation Lie algebras Lk(V) associated to an isolated hypersurface singularity (V,0). These are new analytic invariants of singularities. Here, we investigate L2(V) for fewnomial isolated singularities and obtain the formula of λk(V) (i.e., the dimension of Lk(V)) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for L2(V). This is a continuation of our previous work (Math. Z. 298:3-4 (2021), 1813–1829). We proposed two new conjectures for τk(V) and λk(V) and we prove these conjectures for a large class of singularities.

High-pliability Fano hypersurfaces

We show that five of Reid’s Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type $$cA_n$$ have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of $$\mathbb {P}^2 \times \mathbb {P}^2$$ -type having Picard rank 2.

Weighted Hodge ideals of reduced divisors

AbstractWe study the Hodge and weight filtrations on the localization along a hypersurface, using methods from birational geometry and theV-filtration induced by a local defining equation. These filtrations give rise to ideal sheaves called weighted Hodge ideals, which include the adjoint ideal and a multiplier ideal. We analyze their local and global properties, from which we deduce applications related to singularities of hypersurfaces of smooth varieties.

Higher Nash blow-up local algebras of singularities and its derivation Lie algebras

In this paper, we introduce new invariants to a singularity (V,0), i.e., the derivation Lie algebras Lk(V) of the higher Nash blow-up local algebra Mk(V). A new conjecture about the non-existence of negative weighted derivations of Lk(V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L2(V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of Lk(V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities.

Zariski’s dimensionality type of singularities. Case of dimensionality type 2

In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive “generic” corank 1 1 projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In this paper we completely settle the case of dimensionality type 2, by studying Zariski equisingular families of surfaces singularities, not necessarily isolated, in the three-dimensional space.

Enumeration of algebraic and tropical singular hypersurfaces

We develop a version of Mikhalkin’s lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a δ \delta dimensional linear space of degree d d real hypersurfaces containing 1 δ ! ( γ n d n ) δ + O ( d n δ − 1 ) \frac {1}{\delta !}(\gamma _nd^n)^{\delta }+O(d^{n\delta -1}) hypersurfaces with δ \delta real nodes, where γ n \gamma _n are positive and given by a recursive formula. This is asymptotically comparable to the number 1 δ ! ( ( n + 1 ) ( d − 1 ) n ) δ + O ( d n ( δ − 1 ) ) \frac {1}{\delta !} \left ( (n+1)(d-1)^n \right )^{\delta }+O\left (d^{n(\delta -1)} \right ) of complex hypersurfaces having δ \delta nodes in a δ \delta dimensional linear space. In the case δ = 1 \delta =1 we give a slightly better leading term.

Polar degree and vanishing cycles

We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which quantify local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.

Jet schemes of quasi-homogeneous hypersurfaces and motivic monodromy conjecture for isolated quasi-homogeneous hypersurface singularities

Let $k$ be a field. Let $m\in\mathbf{N}_{>0}$ be a positive integer. Let $f\in k[x_1,\ldots,x_m]$ be a polynomial with degree $d\geq 1$ and associated hypersurface $H:=H(f):=\mathrm{Spec}(k[x_1,\ldots,x_m]/\langle f\rangle)$. In this article, we firstly provide a structure property of the weighted-homogeneity of $f$ in terms of the jet schemes ${\mathscr{L}_{H}}$ of $H$. As a by-product, we deduce from this property a new and very \linebreak effective method for the computation of the motivic Poincaré power series $P_H(T):=\sum_{n\geq 0} [\mathscr{L}_{n}(H)]T^n\in K_0(\mathrm{Var}_k)[[T]]$ associated with a homogeneous hypersurface $H$ with a single isolated singularity at the origin $\frak o$ (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of $P_H(T)$ which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of $P_H(T)$; when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture.

First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere

In this paper, we study the first eigenvalue of the stability operator on an integral n n -varifold with constant mean curvature in the unit sphere S n + 1 \mathbb {S}^{n+1} . We find the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori.

On the number of rational points on a class of singular hypersurfaces

On the number of rational points on a class of singular hypersurfaces

Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras

In this article, we prove that for a finite quiver Q the equivalence class of a potential up to formal change of variables of the complete path algebra \(\widehat{{{\mathbb {C}}}Q}\), is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is a noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra is finite dimensional. These two theorems can be viewed as a first step towards the singularity theory of noncommutative power series. As an application, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential.

Hodge ideals and spectrum of isolated hypersurface singularities

We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal V-filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal V-filtration. As a consequence of a formula of Mustață and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-homogeneous cases where the defining function is semi-weighted-homogeneous or with non-degenerate Newton boundary in most cases. We also show that the convenience condition can be avoided in a formula of M. Zhang for the non-degenerate case, and present an example where the Hodge ideals are not weakly decreasing even modulo the Jacobian ideal.