Generic hypersurface singularities

1997 ◽  
Vol 107 (2) ◽  
pp. 139-154 ◽  
Author(s):  
James Alexander ◽  
André Hirschowitz
Keyword(s):  
Generic Hypersurface ◽  
Hypersurface Singularities

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.

Characterization of isolated homogeneous hypersurface singularities in ℂ4

2006 ◽  
Vol 49 (11) ◽  
pp. 1576-1592 ◽  
Author(s):  
Kepao Lin ◽  
Zhenhan Tu ◽  
Stephen S. T. Yau
Keyword(s):  
Hypersurface Singularities ◽  
Homogeneous Hypersurface

scholarly journals On the semicontinuity of the mod 2 spectrum of hypersurface singularities

2015 ◽  
Vol 24 (2) ◽  
pp. 379-398 ◽  
Author(s):  
Maciej Borodzik ◽  
András Némethi ◽  
Andrew Ranicki
Keyword(s):  
Hypersurface Singularities

scholarly journals The Noether–Lefschetz Theorem Via Vanishing of Coherent Cohomology

2008 ◽  
Vol 51 (2) ◽  
pp. 283-290 ◽  
Author(s):  
G. V. Ravindra
Keyword(s):  
Picard Number ◽  
Generic Hypersurface ◽  
Lefschetz Theorem

AbstractWe prove that for a generic hypersurface in ℙ2n+1 of degree at least 2 + 2/n, the n-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

Alternative algorithms for computing generic μ∗-sequences and local Euler obstructions of isolated hypersurface singularities

2019 ◽  
Vol 18 (08) ◽  
pp. 1950156
Author(s):  
Katsusuke Nabeshima ◽  
Shinichi Tajima
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Rational Functions ◽  
Isolated Singularity ◽  
Euler Obstruction ◽  
Standard Bases ◽  
Hypersurface Singularities ◽  
Previous Algorithm ◽  
Computation Speed

A new algorithm is introduced for computing [Formula: see text]-sequences of isolated hypersurface singularities. It is shown that the new algorithm results in better performance, compared to our previous algorithm that utilizes parametric local cohomology systems, in computation speed. Furthermore, it can be used to compute local Euler obstruction of a hypersurface with an isolated singularity. The key idea of the new algorithm is computing standard bases in a local ring over a field of rational functions.

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