Quasi-homogeneous Singularities Research Articles

A characterization and solvability of quasihomogeneous singularities

A characterization and solvability of quasihomogeneous singularities

Free curves, eigenschemes, and pencils of curves

AbstractLet . A reduced plane curve is free if its associated module of tangent derivations is a free ‐module, or equivalently if the corresponding sheaf of vector fields tangent to splits as a direct sum of line bundles on . In general, free curves are difficult to find, and in this paper, we describe a new method for constructing free curves in . The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves.

Hodge Diamonds of the Landau-Ginzburg Orbifolds

Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi-Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of $(f,G)$ form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.

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

𝒬-conic arrangements in the complex projective plane

We study the geometry of Q \mathcal {Q} -conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such Q \mathcal {Q} -conic arrangements are never free. Moreover, we provide combinatorial constraints of the weak combinatorics of such arrangements.

Algebraic curves and foliations

Consider a field k ${\mathsf {k}}$ of characteristic 0 $\hskip.001pt 0$ , not necessarily algebraically closed, and a fixed algebraic curve f = 0 $f=0$ defined by a tame polynomial f ∈ k [ x , y ] $f\in {\mathsf {k}}[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane A k 2 $\mathbb {A}^2_{\mathsf {k}}$ having f = 0 $f=0$ as a fixed invariant curve is generated as k [ x , y ] ${\mathsf {k}}[x,y]$ -module by at most four elements, three of them are the trivial foliations f d x , f d y $fdx,fdy$ and d f $df$ . Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen–Suslin theorem, we show that for a suitable field extension K ${\mathsf {K}}$ of k ${\mathsf {k}}$ such a module over K [ x , y ] ${\mathsf {K}}[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, K = k ${\mathsf {K}}={\mathsf {k}}$ .

The integral monodromy of isolated quasihomogeneous singularities

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom-Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of ${\mathbb Z}$-lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminants.

The combinatorics of weight systems and characteristic polynomials of isolated quasihomogeneous singularities

A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the characteristic polynomial. Here, the conjecture is proved, and some of the other problems are solved, too. In the cases where also an old conjecture of Orlik on the integral monodromy holds, this has implications on the automorphism group of the Milnor lattice. The combinatorics used in the proof of the conjecture consists of tuples of orders on sets {0,1,ldots ,n} with special properties and may be of independent interest.

Seven combinatorial problems around isolated quasihomogeneous singularities

This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of an isolated quasihomogeneous hypersurface singularity. One of them is a new conjecture on the characteristic polynomial. It is an amendment to an old conjecture of Orlik on the integral monodromy of an isolated quasihomogeneous singularity. The search for a combinatorial proof of the new conjecture led us to the seven purely combinatorial problems.

Non-degenerate jumps of Milnor numbers of quasihomogeneous singularities

Let $f_0: (\mathbb {C}^n,0)\rightarrow (\mathbb {C},0)$ be a holomorphic function germ having an isolated critical point at $0\in \mathbb {C}^n$ and let $[f_0]$ be the singularity generated by $f_0$, i.e. the equivalence class of $f_0$ with respect t

Motivic Milnor Fibers of Plane Curve Singularities

We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the Hodge–Steenbrink spectrum of such a singularity in terms of that of a quasi-homogeneous singularity.

Purity for graded potentials and quantum cluster positivity

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

On smooth separatrices of foliations on singular surfaces

We consider foliations on a germ of an isolated surface singularity. We give criteria for the existence of smooth separatrices in terms of the maximal cycle of a desingularization of the surface. We apply these criteria for some examples of quasi-homogeneous singularities, in particular rational double points; moreover, for these singularities we show for which cases there exist a smooth separatrix for foliations defined by sufficiently general derivations.

Equivalence of real Milnor fibrations for quasi-homogeneous singularities

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in order to do that, we explicitly characterize the critical points of projection $\frac{f}{\|f\|}:S_{\epsilon}^{m}\setminus K_{\epsilon}\to S^{1}$, where $K_{\epsilon}$ is the link of singularity.

Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Let X be a surface with an isolated singularity at the origin, given by the equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C) finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson homology of Y, as a graded vector space with respect to the weight grading. In the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth Hochschild homology of the n-th symmetric power of the algebra of G-invariant differential operators on C. That is, the Brylinski spectral sequence degenerates in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, for the parameter equal to all but countably many points of the elliptic curve.

On Arnold’s 14 ‘exceptional’ $ \mathcal{N} = 2 $ superconformal gauge theories

We study the four-dimensional superconformal $ \mathcal{N} = 2 $ gauge theories engineered by the Type IIB superstring on Arnold’s 14 exceptional unimodal singularities (a.k.a. Arnold’s strange duality list), thus extending the methods of arXiv:1006.3435 to singularities which are not the direct sum of minimal ones. In particular, we compute their BPS spectra in several ‘strongly coupled’ chambers. From the TBA side, we construct ten new periodic Y-systems, providing additional evidence for the existence of a periodic Y-system for each isolated quasi-homogeneous singularity with ĉ < 2 (more generally, for each $ \mathcal{N} = 2 $ superconformal theory with a finite BPS chamber whose chiral primaries have dimensions of the form $ {{\mathbb{N}} \left/ {\ell } \right.} $ ).

Quantum Singularity Theory for A_{(r - 1)} and r-Spin Theory

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r-spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W-curves is canonically isomorphic to the stack of r-spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an r-spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the Witten Integrable Hierarchies Conjecture for r-spin curves applies to our theory for A-type singularities; that is, the total descendant potential function of our theory for A-type singularities satisfies the corresponding Gelfand-Dikii integrable hierarchy.

Minimal resolutions of geometric [formula omitted]-modules

In this paper, we study minimal free resolutions for modules over rings of linear differential operators. The resolutions we are interested in are adapted to a given filtration, in particular to the so-called V -filtrations (see Oaku and Takayama (2001) [18] and Granger and Oaku (2004) [9]). We are interested in the module D x , t f s associated with germs of functions f 1 , … , f p , which we call a geometric module, and it is endowed with V -filtration along t 1 = ⋯ = t p = 0 . The Betti numbers of the minimal resolution associated with this data lead to analytical invariants for the germ of space defined by f 1 , … , f p . For p = 1 , we show that, under some natural conditions on f , the computation of the Betti numbers is reduced to a commutative algebra problem. This includes the case of an isolated quasi-homogeneous singularity, for which we give the Betti numbers explicitly. Moreover, for an isolated singularity, we characterize the quasi-homogeneity in terms of the minimal resolution.

The geometric genus, the casson invariant conjecture and splice type singularities

AbstractThis is a survey of some results on splice-quotient singularities which are a natural and broad generalization of quasihomogeneous surface singularities with rational homology sphere links. From its topology (i.e., the link or the resolution graph), we can write down the “leading terms” of equations of a splice-quotient singularity, and compute the geometric genus. Applying the formula for the geometric genus, we can verify the Casson invariant conjecture for splice-quotient singularities.

Another proof of the end curve theorem for normal surface singularities

Neumann and Wahl introduced the notion of splice-quotient singularities, which is a broad generalization of quasihomogeneous singularities with rational homology sphere links, and proved the End Curve Theorem that characterizes splice-quotient singularities. The purpose of this paper is to give another proof of the End Curve Theorem. We use combinatorics of “monomial cycles” and some basic ring theory, whereas they applied their theory of numerical semigroups.