ON THE MULTIPLICITIES OF FAMILIES OF COMPLEX HYPERSURFACE-GERMS WITH CONSTANT MILNOR NUMBER

2013 ◽  
Vol 24 (03) ◽  
pp. 1350021 ◽  
Author(s):  
CAMILLE PLENAT ◽  
DAVID TROTMAN
Keyword(s):  
Tangent Cone ◽  
Milnor Number ◽  
Analytic Family ◽  
Singular Set ◽  
Analytic Hypersurface ◽  
Hypersurface Singularities

We show that the possible drop in multiplicity in an analytic family F(z, t) of complex analytic hypersurface singularities with constant Milnor number is controlled by the powers of t. We prove equimultiplicity of μ-constant families of the form f + tg + t2h if the singular set of the tangent cone of {f = 0} is not contained in the tangent cone of {h = 0}.

BRAID MONODROMY OF SOME BRIESKORN–PHAM SINGULARITIES

2010 ◽  
Vol 21 (08) ◽  
pp. 1047-1070 ◽  
Author(s):  
MICHAEL LÖNNE
Keyword(s):  
Fundamental Group ◽  
Braid Group ◽  
Milnor Number ◽  
Bifurcation Set ◽  
Hypersurface Singularities ◽  
Curve Singularities

For germs of isolated hypersurface singularities Lyashko and Looijenga introduced a map LL on the complement of the bifurcation set. It takes values in the space of simple polynomials of degree equal to the Milnor number μ of the singularity, and thus induces a map LL* from the fundamental group to the braid group on μ strands. For the case of curve singularities given by a polynomial x3 + yℓ+1 we give a uniform description of the image LL* and derive a presentation of the discriminant knot group, i.e. for the fundamental group of the complement to the discriminant divisor.

scholarly journals Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

2015 ◽  
Vol 218 ◽  
pp. 29-50
Author(s):  
Christophe Eyral ◽  
Maria Aparecida Soares Ruas
Keyword(s):  
Deformation Parameter ◽  
Topological Type ◽  
Type A ◽  
Milnor Number ◽  
Hypersurface Singularities ◽  
Special Case ◽  
Parameter Deformation

AbstractWe show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families withisolatedsingularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

STABLE MINIMAL HYPERSURFACES AND TANGENT CONE SINGULARITIES

1999 ◽  
Vol 10 (03) ◽  
pp. 407-413
Author(s):  
HELEN MOORE
Keyword(s):  
Tangent Cone ◽  
Volume Growth ◽  
Minimal Immersion ◽  
Minimal Hypersurface ◽  
Singular Set ◽  
Minimal Hypersurfaces ◽  
Bounded Volume

In this paper, I give an estimate on the dimension of the singular set of a tangent cone at infinity of a stable minimal hypersurface. Namely, let Mn ⊂ ℝn+1, n ≥ 2, be a complete orientable stable minimal immersion with bounded volume growth. Then n < 7 implies T∞(M) is smooth, and n ≥ 7 implies the singular set of T∞(M) has codimension at least seven.

scholarly journals Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

2015 ◽  
Vol 218 ◽  
pp. 29-50 ◽  
Author(s):  
Christophe Eyral ◽  
Maria Aparecida Soares Ruas
Keyword(s):  
Deformation Parameter ◽  
Topological Type ◽  
Type A ◽  
Milnor Number ◽  
Hypersurface Singularities ◽  
Special Case ◽  
Parameter Deformation

AbstractWe show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

scholarly journals 5d and 4d SCFTs: canonical singularities, trinions and S-dualities

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Cyril Closset ◽  
Simone Giacomelli ◽  
Sakura Schäfer-Nameki ◽  
Yi-Nan Wang
Keyword(s):  
String Theory ◽  
Field Theories ◽  
Flavor Symmetry ◽  
Crepant Resolution ◽  
Superconformal Field Theories ◽  
Hypersurface Singularities ◽  
Canonical Singularities ◽  
Type Iib String Theory ◽  
M Theory

Abstract Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of ‘trinion’ singularities which exhibit these properties. In Type IIB, they give rise to 4d $$ \mathcal{N} $$ N = 2 SCFTs that we call $$ {D}_p^b $$ D p b (G)-trinions, which are marginal gaugings of three SCFTs with G flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $$ \mathcal{N} $$ N = 4 theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as ‘ugly’ components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver ‘bad’. We propose a way to redeem the badness of these quivers using a class $$ \mathcal{S} $$ S realization. We also discover new S-dualities between different $$ {D}_p^b $$ D p b (G)-trinions. For instance, a certain E8 gauging of the E8 Minahan-Nemeschansky theory is S-dual to an E8-shaped Lagrangian quiver SCFT.

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