Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces

Abstract

As a generalization of the ideals of star configurations of hypersurfaces, we consider the a-fold product ideal ${\mathit{I}}_{\mathit{a}}({\mathit{f}}_1^{{\mathit{m}}_1}\cdots {\mathit{f}}_{\mathit{s}}^{{\mathit{m}}_{\mathit{s}}})$ when ${\mathit{f}}_1,\dots ,{\mathit{f}}_{\mathit{s}}$ is a sequence of n-generic forms and $1\le \mathit{a}\le {\mathit{m}}_1+\cdots +{\mathit{m}}_{\mathit{s}}$. Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then, we study its symbolic powers while focusing on the uniform case with ${\mathit{m}}_1=\cdots ={\mathit{m}}_{\mathit{s}}$. For large a, we describe its resurgence and symbolic defect. And for general a, we also investigate the corresponding invariants for meeting-at-the-minimal-components version of symbolic powers.