Speaker: João Hélder Olmedo Rodrigues (UFF).
Date: 04 nov 2022, 16h.
Place: Room 407, Bloco H, Campus Gragoatá, UFF.
Abstract: The Tjurina ideal of a germ of a holomorphic function $f$ is the ideal of $\mathcal{O}_{\mathbb{C}^n,0}$ - the ring of those germs at $0\in\mathbb{C}^n$ - generated by $f$ itself and by its partial derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the structure of closed subscheme of $(\mathbb{C}^n,0)$ to the singular set of the hypersurface defined by $f$, being an object of central interest in Singularity Theory. In this talk we introduce \emph{$T$-fullness} and \emph{$T$-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 $I\subset \mathcal{O}_{\mathbb{C}^n,0}$ for the equation $I=T(f)$ to admit a solution $f$. As a result we characterize closed subschemes of $(\mathbb{C}^n,0)$ arising as singularities of germs of hypersurfaces. If time permits I will comment on some ongoing exploration in the case $n=2$.