Resurgence number of matroid configuration
Haoxi Hu
TL;DR
This work tackles the resurgence problem for symbolic powers in the context of matroidal configurations by leveraging $C$-matroidal ideals and their associated simplicial complexes. It develops a combinatorial framework connecting matroid theory to containment problems, and proves a binomial expansion for symbolic powers $I^{(m)}=\sum_{i=1}^{c} I^{(i)}I^{(m-i)}$ for $m>c$, enabling explicit resurgence computations. The paper provides an exact formula for the resurgence of generalized uniform matroidal configurations (and thus star configurations) as $\rho(I)=\frac{c(n-c+1)}{n}$, along with an improved upper bound $\rho(I)\le c-1$ under certain height and peaked conditions on the associated simplicial complex. The results extend the understanding of containment relations in symbolic powers, linking matroidal and star configurations to computable resurgence values and highlighting the role of peaked simplicial complexes in tightening bounds.
Abstract
This article gives a new upper bound for the resurgence number of symbolic powers of matroidal configuration in the following situations: the height of the matroidal configuration is big, or the height is small, and the corresponding simplicial complex of the matroidal configuration is peaked. The Peaked simplicial complex is a generalization of bipartite graph. Furthermore, the article also gives a clean formula to compute the resurgence number and the strict containment of generalized uniform matroidal configuration which includes case of star configuration of hypersurfaces.