On $T$-maps and ideals of antiderivatives of hypersurface singularities

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Mather–Yau's theorem leads to an extensive study about moduli algebras of isolated hypersurface singularities. In this paper, the Tjurina ideal is generalized as $T$-principal ideals of certain $T$-maps for Noetherian algebras. Moreover, we introduce the ideal of antiderivatives of a $T$-map, which creates many new invariants. Firstly, we compute two new invariants associated with ideals of antiderivatives for ADE singularities and conjecture a general pattern of polynomial growth of these invariants.Secondly, the language of $T$-maps is applied to generalize the well-known theorem that the Milnor number of a semi quasi-homogeneous singularity is equal to that of its principal part. Finally, we use the $T$- fullness and $T$-dependence conditions to determine whether an ideal is a $T$-principal ideal and provide a constructive way of giving a generator of a $T$-principal ideal. As a result, the problem about reconstruction of a hypersurface singularitiy from its generalized moduli algebras is solved. It generalizes the results of Rodrigues in the cases of the $0$th and $1$st moduli algebra, which inspired our solution.Bibliography: 24 titles.

作者简介

Quan Shi

Department of Mathematical Sciences, School of Sciences, Tsinghua University; Tsinghua University

编辑信件的主要联系方式.
Email: shiq20@mails.tsinghua.edu.cn

Stephen Yau

Beijing Institute of Mathematical Sciences and Applications; Department of Mathematical Sciences, School of Sciences, Tsinghua University

Email: yau@uic.edu

PhD, Professor

Huaiqing Zuo

Department of Mathematical Sciences, School of Sciences, Tsinghua University

Email: hqzuo@mail.tsinghua.edu.cn

参考

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