BIMSA > Seminar on Algebraic, Complex Geometry and Singularities On $T$-maps and Ideals of Antiderivatives of Hypersurface Singularities
On $T$-maps and Ideals of Antiderivatives of Hypersurface Singularities
Organizer
Shing Toung Yau
Speaker
Quan Shi
Time
Saturday, May 6, 2023 10:30 AM - 11:30 AM
Venue
Online
Abstract
Mather-Yau theorem leads to the massive 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 to 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 the Milnor number of its principal part. Finally, we use two conditions extit{T-fullness} and extit{T-dependence} to determine whether an ideal is a $T$-principal ideal and provide a constructive way of giving a generator of a $T$-principal ideal. Thus, we solve the problem about reconstruction of a hypersurface singularity from its generalized moduli algebras.