Symbolic powers and free resolutions of generalized star configurations of hypersurfaces

TL;DR

This paper explores generalized star configuration ideals of hypersurfaces, analyzing their algebraic properties, symbolic powers, and invariants, especially focusing on cases with uniform multiplicities and large parameters.

Contribution

It introduces a new class of ideals generalizing star configurations, studies their resolutions, symbolic powers, and key invariants, providing new insights into their algebraic structure.

Findings

Ideal has complete intersection quotients for same degree forms

Describes resurgence and symbolic defect for large a

Analyzes invariants for minimal-component symbolic powers

Abstract

As a generalization of the ideals of star configurations of hypersurfaces, we consider the $a$-fold product ideal $I_a(f_1^{m_1}\cdots f_s^{m_s})$ when ${f_1,\dots,f_s}$ is a sequence of generic forms and $1\le a\le m_1+\cdots+m_s$. 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 $m_1=\cdots=m_s$. 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.