Resurgences for ideals of special point configurations in ${\mathbf{P}}^N$ coming from hyperplane arrangements

Abstract

Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence. There have been exciting new developments in this area recently. It had been expected for several years that $I^{Nr-N+1}\subseteq I^r$ should hold for the ideal $I$ of any finite set of points in ${\mathbf{P}}^N$ for all $r>0$, but in the last year various counterexamples have now been constructed, all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

9 pages, 1 figure

References in corpus (2)

• On the behavior of Castelnuovo-Mumford regularity with respect to some functors
• Containments of symbolic powers of ideals of generic points in $\PP^3$

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• Resurgence and Castelnuovo-Mumford regularity of certain monomial curves in ${\mathbb{A}}^3$
• On Yoshinaga's arrangement of lines and the containment problem