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

Quan Shi ; Stephen S.-T. Yau ; Huaiqing Zuo

Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1185-1220.

Résumé

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.

Détail

Keywords: isolated singularities, local rings, Kähler differential, semi quasi-homogeneous singularities, Tjurina ideals.

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