ON TJURINA IDEALS OF HYPERSURFACE SINGULARITIES

Abstract

The Tjurina ideal of a germ of a holomorphic function $f$ is the ideal of ${\mathcal{𝒪}}_{{ℂ}^n,0}$ — the ring of those germs at $0\in {ℂ}^n$ — generated by $f$ itself and by its first-order partial derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the structure of closed subscheme of $({ℂ}^n,0)$ to the singular set of the hypersurface defined by $f$, being an object of central interest in singularity theory. In this note we introduce $T$-fullness and $T$-dependence, two easily verifiable properties for arbitrary ideals of germs of holomorphic functions. These two properties allow us to give necessary and sufficient conditions on an ideal $I\subset {\mathcal{𝒪}}_{{ℂ}^n,0}$ for the equation $I=T(f)$ to admit a solution $f$. As a result we characterize closed subschemes of $({ℂ}^n,0)$ arising as singularities of germs of hypersurfaces.