北京雁栖湖应用数学研究院

丘成栋

胡创强

2024年01月31日 21:00 至 22:00

Online

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.

胡创强，2021年秋季入职BIMSA。主要研究领域包括：编码理论，函数域及数论，奇点理论。近年来在量子码，代数几何码，Drinfeld模，椭圆奇点，丘-李代数等课题研究中取得了一系列学术成就。在《IEEE Trans. on IT.》《Finite Fields and Their Applications》《Designs, Codes and Cryptography》等著名学术期刊上发表论文13篇。先后多次应邀出席国内外学术会议并作大会报告。