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Seminar on Algebraic, Complex Geometry and Singularities
On the $k$-th Tjurira number of weighted homogeneous singularities
On the $k$-th Tjurira number of weighted homogeneous singularities
Organizer
Speaker
Time
Wednesday, January 31, 2024 9:00 PM - 10:00 PM
Venue
Online
Abstract
Let $(X,0)$ be an isolated singularity defined by a weighted homogeneous polynomial $f$. We consider the $k$-t Tjurira algebra $A_k(f): = \mathcal{O} / \left( f , m^k J(f) \right)$ and the corresponding dimension called $k$-th Tjurina numbers. It is well-known that the zeroth Tjurina algebra represents the tangent space of the base space of the semi-universal deformation of $(X, 0)$. Inspired by this fact, we generated the deformation of $(X,0)$ to the one associated with the fixed $k$-residue point and consequently the tangent space of the corresponding deformation functor is exactly the $k$-th Tjurina algebra $A_k(f)$. Computing the $k$-th Tjurina numbers explicitly plays a distinguished role in understanding these deformations. From the result of Milnor and Orlik, the zeroth Tjurina number is expressed by the weights of variables of $f$ explicitly. However, for the case when $k$ is larger than the multiplicity of $X$, we find that the $k$-Tjurina number is more complicated and not only decided by the weights of variables. In this talk, we develope a new complex from the classical Koszul complex and derive a computable formula of $k$-th Tjurina numbers for all $k geqslant 0$. As applications, we calculate the $k$-th number of Brieskorn-Pham singularities and all weighted homogeneous singularities in there variables.
Speaker Intro
Hu chuangqiang joined Bimsa in the autumn of 2021. The main research fields include: coding theory, function field and number theory, singularity theory. In recent years, he has made a series of academic achievements in the research of quantum codes, algebraic geometric codes, Drinfeld modules, elliptic singular points, Yau Lie algebras and other studies. He has published 13 papers in famous academic journals such as IEEE Trans. on IT., Final Fields and their Applications, Designs, Codes and Cryptography. He has been invited to attend domestic and international academic conferences for many times and made conference reports.