Higher singularities for hypersurfaces

Abstract

With an assumption on the codimension of the singular locus of a complex hypersurface $D$ in smooth variety $X$, we show that if $\underline{\hat{I}}{\text{c◯}}_D^m\cong \hat{I}{\text{c◯}}_D^m$, then $\underline{\hat{I}}{\text{c◯}}_D^i\cong \hat{I}{\text{c◯}}_D^i$ for all $0\le i\le m$. We also discuss an analogue of this statement in positive characteristic.

25 pages