Vol. 314, No. 2, 2021

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 314: 1  2
Vol. 313: 1  2
Vol. 312: 1  2
Vol. 311: 1  2
Vol. 310: 1  2
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Derivation Lie algebras of new $k$-th local algebras of isolated hypersurface singularities

Naveed Hussain, Stephen S.-T. Yau and Huaiqing Zuo

Vol. 314 (2021), No. 2, 311–331
Abstract

Let (V,0) = {(z1,,zn) n : f(z1,,zn) = 0} be an isolated hypersurface singularity with mult(f) = m. Let Jk(f) be the ideal generated by all k-th order partial derivative of f. For 1 k m 1, the new object k(V ) is defined to be the Lie algebra of derivations of the new k-th local algebra Mk(V ), where Mk(V ) := 𝒪n(f + J1(f) + + Jk(f)). Its dimension is denoted as δk(V ). This number δk(V ) is a new numerical analytic invariant. We compute 3(V ) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ3(V ). We also formulate a sharp upper estimate conjecture for the δk(V ) of weighted homogeneous isolated hypersurface singularities and verify this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture: δ(k+1)(V ) < δk(V ), k 1 and verify it for low-dimensional fewnomial singularities.

Keywords
isolated hypersurface singularity, Lie algebra, local algebra
Mathematical Subject Classification
Primary: 14B05, 32S05
Milestones
Received: 15 February 2021
Revised: 30 May 2021
Accepted: 16 July 2021
Published: 10 November 2021
Authors
Naveed Hussain
Guangzhou Huashang College
Guangzhou
China
Stephen S.-T. Yau
Tsinghua University
Beijing
China
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications
Beijing
China
Huaiqing Zuo
Tsinghua University
Beijing
China