Homogeneous Singularities Research Articles

Construction of Free Curves by Adding Lines to a Given Curve

In the present note we construct new families of free plane curves starting from a curve C and adding high order inflectional tangent lines of C, lines joining the singularities of the curve C, or lines in the tangent cone of some singularities of C. These lines L have in common that the intersection C∩L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\cap L$$\\end{document} consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by Beorchia and Miró-Roig in arXiv:2303.04665

K-th Milnor numbers and k-th Tjurina numbers of weighted homogeneous singularities

k-th Milnor numbers and k-th Tjurina numbers of weighted homogeneous singularities

Deformations of Hypersurfaces with Nonconstant Alexander Polynomial

Abstract Let $X \subset \mathbf {P}^n$ be an irreducible hypersurface of degree $d\geq 3$ with only isolated semi-weighted homogeneous singularities, such that $\exp (\frac {2\pi i}{k})$ is a zero of its Alexander polynomial. Then we show that the equianalytic deformation space of $X$ is not $T$-smooth except for a finite list of triples $(n,d,k)$. This result captures the very classical examples by B. Segre of families of degree $6m$ plane curves with $6m^2$, $7m^2$, $8m^2$, and $9m^2$ cusps, where $m\geq 3$. Moreover, we argue that many of the hypersurfaces with nontrivial Alexander polynomial are limits of constructions of hypersurfaces with not $T$-smooth deformation spaces. In many instances, this description can be used to find candidates for Alexander-equivalent Zariski pairs.

Northern Protestants: On Shifting Ground by Susan McKay

Reviewed by: Northern Protestants: On Shifting Ground by Susan McKay Chris Arthur Susan McKay Northern Protestants: On Shifting Ground Belfast. The Blackstaff Press. 2021. 346 pages. war photographer Robert Capa famously said, "If your pictures aren't good enough, you aren't close enough." The outstanding quality of the word-pictures drawn in Northern Protestants: On Shifting Ground relies on the author's practice of getting close enough to people to see the detail of their lives, rather than presenting us with distant generalizations. But proximity alone would not guarantee a book of this caliber. For that, you also need the author's skill as an empathic listener and lucid writer. Susan McKay expertly coaxes a variety of voices to tell their stories and presents these to readers with impressive clarity. The book is based on interviews conducted between 2019 and 2021. McKay traveled all over Northern Ireland to take soundings from one sector of the country's populace. She wanted to get a sense of how Protestants think of themselves and what they want, in the context of a milieu where old allegiances have been strained by the legacy of the Troubles, Brexit, the pandemic, renewed worries about a possible united Ireland, and loss of faith in politicians caused by the Renewable Heat Incentive scandal. (The Renewable Heat Incentive scandal was a shockingly mismanaged green-energy scheme that lost a colossal amount of public money and ultimately led to the collapse of the Northern Ireland Assembly—the devolved legislature for Northern Ireland. An account of what happened—uncovering political and administrative incompetence and duplicity on a scale that's hard to credit—is given in Sam McBride's masterpiece of investigative journalism, Burned: The Inside Story of the "Cash-for-Ash" Scandal and Northern Ireland's Secretive New Elite [2019].) Northern Protestants shows the poverty of those simplifying stereotypes that are too often the currency of media reports. Commentators who lack McKay's skills, and her intrepid determination to seek out enlightening face-to-face encounters, tend to present Northern Ireland in crudely dualistic terms: Protestant-Unionist-Loyalist versus Catholic-Nationalist-Republican. The impression given is that within each tribal bloc there is an obdurate uniformity of opinion. McKay repeatedly demonstrates how fatuous such a picture is. "Northern Protestant" is far from being a homogeneous singularity. Thinking within this group is neither fixed nor predictable. Her book, together with its older sibling, Northern Protestants: An Unsettled People (published in 2000, with a new edition brought out last year to coincide with the second volume's appearance), provides a sensitive mapping of how Protestant mind-sets have changed, and responded to change, over the last several decades. The books compile a record of the raw oral literature of the moment. Susan McKay offers an always sympathetic, often surprising, but not uncritical portrait of her people. The books have more than local relevance. Reading them at this juncture in European history gives a particular edge to her insights about the impact of violence. She talks to many survivors, relatives, and descendants of those killed and maimed in Ulster's thirty years of sectarian conflict. Long after the immediate lethality of terrorist bombings and shootings, lives are still disfigured by these acts of savagery. Over 3,500 people lost their lives in Northern Ireland's Troubles. This is a tiny number compared to casualties in Ukraine. It is terrible to think how far into the future this war will cast its blighting shadows. As millions of distraught, displaced, bereaved, and injured people struggle to pick up the pieces of their lives, one hopes that those who listen to, record, and tell their stories will be as skillful in this art as Susan McKay is. She provides a master class in how to listen and has a fine sense of when to put people's own words on the page, when to paraphrase, what is best juxtaposed with what, when the author herself needs to comment and when to stand back. As a result, readers receive a powerful sense of the variety of ways, some admirable, some decidedly not, in which individuals cope—or fail to cope—with challenging, often desperate circumstances. We...

Homological mirror symmetry for Milnor fibers via moduli of A∞$A_\infty$‐structures

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A ∞ $A_\infty$ -structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n $n$ -dimensional affine hypersurface of degree n + 2 $n+2$ and the double cover of the n $n$ -dimensional affine space branched along a degree 2 n + 2 $2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest.

The method of fundamental solutions for analytic functions in complex analysis

<abstract><p>This paper extends the method of fundamental solutions (MFS) for solving the boundary value problems of analytic functions based on Cauchy-Riemann equations and properties of harmonic functions. The conformal mapping technique is applied to introduce the singularities of the approximate analytic functions and reconstruct the fundamental solutions. The presented method can naturally introduce the information of homogeneous boundary conditions and singularity properties, when the conformal mapping technique or the reconstructed fundamental solutions are used. The numerical examples show that the proposed method has the advantages of conciseness, reliability, efficiency, high accuracy and easy-using, respectively. The developed method can be used to solve the boundary value problems (BVPs) of analytic functions without considering single-valuedness, which simplify the numerical analysis.</p></abstract>

Analytic tangent cones of admissible Hermitian Yang–Mills connections

In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.

Torsion Type Invariants of Singularities

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on \(\mathbb {C}^{n}\) by a quasi-homogeneous polynomial f. Under some mild assumption to f, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time t. Then we prove a local index formula which expresses the Milnor number of f by a Gaussian type integration. The heat kernel expansion provides the spectral invariants of f. Furthermore, we can define the torsion type invariants associated to a homogeneous singularity. The spectral invariants provide another way to classify the singularity.

Milnor numbers in deformations of homogeneous singularities

Let f0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f0. We completely describe the set of these Milnor numbers for homogeneous singularities f0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.

METHODS FOR COMPUTING <i>b</i>-FUNCTIONS ASSOCIATED WITH <i>µ</i>-CONSTANT DEFORMATIONS: CASE OF INNER MODALITY TWO

New methods for computing parametric local b-functions are introduced for µ-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic D-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of b-functions associated with µ-constant deformations. In the case of inner modality two, local b-functions associated with µ-constant deformations are obtained by the resulting method and given the list of parametric local b-functions.

On rational cuspidal plane curves and the local cohomology of Jacobian rings

This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao’s conjecture on the freeness of a line arrangement being determined by its combinatorics.

The Abel map for surface singularities I: generalities and examples

Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions widetilde{X}rightarrow X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l'in H^2(widetilde{X},{mathbb {Z}}). With these data we define the variety mathrm{ECa}^{l'}(Z) of those effective Cartier divisors D supported on Z, which determine a line bundles {{mathcal {O}}}_Z(D) with first Chern class l'. Furthermore, we consider the affine space mathrm{Pic}^{l'}(Z)subset H^1({{mathcal {O}}}_Z^*) of isomorphism classes of holomorphic line bundles with Chern class l' and the Abel map c^{l'}(Z):mathrm{ECa}^{l'}(Z)rightarrow mathrm{Pic}^{l'}(Z). The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups H^1(Z,{mathcal {L}}), where we might vary the analytic structure (X, o) (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles {mathcal {L}}in mathrm{Pic}^{l'}(Z). The case of generic line bundles of mathrm{Pic}^{l'}(Z) and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill–Noether theory of reduced smooth projective curves.

Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves

We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve C. The relation to the zero set of Bernstein–Sato polynomial of the defining homogeneous polynomial for C is also discussed. When C has some non-weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.

Solvability of planar complex vector fields with homogeneous singularities

In this paper, we study the equation , where L is a -valued vector field in with a homogeneous singularity. The properties of the solutions are linked to the number theoretic properties of a pair of complex numbers attached to the vector field. As an application, we obtain results about an associated Riemann–Hilbert problem for the vector field L.

The jump of Milnor number for linear deformations of homogeneous singularities

The jump of Milnor number for linear deformations of homogeneous singularities

THE ŁOJASIEWICZ EXPONENT FOR WEIGHTED HOMOGENEOUS POLYNOMIAL WITH ISOLATED SINGULARITY

AbstractThe purpose of this paper is to give an explicit formula of the Łojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

Anisotropic matter in cosmology: locally rotationally symmetric Bianchi I and VIIo models

We examine the behaviour of homogeneous, anisotropic space-times, specifically the locally rotationally symmetric Bianchi types I and VIIo in the presence of anisotropic matter. By finding an appropriate constant of the motion, and transforming the equations of motion we are able to provide exact solutions in the presence of perfect fluids with anisotropic pressures. The solution space covers matter consisting of a single perfect fluid which satisfies the weak energy condition and is rich enough to contain solutions which exhibit behaviour which is qualitatively distinct from the isotropic sector. Thus we find that there is more ‘matter that matters’ close to a homogeneous singularity than the usual stiff fluid. Example metrics are given for cosmologies whose matter sources are magnetic fields, relativistic particles, cosmic strings and domain walls.

Syzygies of Jacobian ideals and weighted homogeneous singularities

Let V be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for V are exactly the syzygies with a generic first component vanishing on the singular locus subscheme of V. This yields in particular a positive answer in this setting to a question raised by Morihiko Saito and the first author. Finally we explain how our result can be used to improve the listing of Jacobian syzygies of a given degree by a computer algebra system such as Singular, CoCoA or Macaulay2.

Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras

Using the minors in Hessian matrices, we introduce new graded algebras associated to a homogeneous polynomial. When the associated projective hypersurface has isolated singularities, these algebras are related to some new local algebras associated to isolated hypersurface singularities, which generalize their Tjurina algebras. One consequence of our results is a new very rapid way to determine the number of weighted homogeneous singularities of such a hypersurface.

Some remarks on limit mixed Hodge structures and spectrum

Abstract In this note we clarify some subtle points on the limit mixed Hodge structures and on the spectrum. These are more or less well-known to the specialists, but do not seem to be stated explicitly in the literature. However, as they do not seem to be obvious to the beginners, we consider them to be worth writing down explicitly. The general constructions are exemplified by considering the isolated (weighted) homogeneous singularities in detail.