Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

Abstract

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; see for example [BH1, Cu, ELS, HaHu, HoHu, Hu1, Hu2] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [BH1, GHvT].There have been exciting new developments in this area recently. It had been expected for several years that I Nr−N+1 ⊆ I r should hold for the ideal I of any finite set of points in P N for all r > 0, but in the last year various counterexamples have now been constructed (see [DST, HS, C. et al]), 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.

On the containment hierarchy for simplicial ideals

“…This is a delicate invariant and the family of ideals for which it is known is growing slowly, see e.g. [5]. Here we expand this knowledge a little bit.…”

A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in P2

“…More precisely, it is proven therein that removing any single point from among the 13 points of the finite projective plane P 2 F 3 together with all the lines passing through the removed point yields on the remaining 12 points the same incidence structure exhibited by the dual Hesse configuration of [8]. It is worth noting, however, that although they share the same combinatorial data and the property that I (3) I 2 , the dual Hesse configuration of [8] and the characteristic 3 counterexample of [2] behave very differently when viewed from the perspective of algebraic-geometric invariants associated to them (see [7] for a detailed account of the differences and computations of the resurgence for these counterexamples).…”

“…Following [7], which uses terminology introduced by Urzuá [24], we refer to these configurations of n 2 + 3 points as Fermat configurations. It is proven in [13,Proposition 2.1] that, if I is the defining ideal of a Fermat configuration of points, then I (3) I 2 .…”

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

“…Let A = P 1 P 2 P 3 Q 1 Q 2 Q 3 T be a presentation matrix for the module of syzygies on I. By the proof of [DHNSST,Theorem 2.1], we have deg P i = 2 and deg Q i = n − 1. As quotients of the polynomial ring S = R[T 1 , T 2 , T 3 ] the symmetric and Rees algebra of I are then defined by…”

“…The minimal free resolution of I (the case r = 1) can be found in the proof of [DHNSST,Theorem 2.1]. The minimal free resolutions for the higher powers (r ≥ 2) follow by setting d = n + 1, d 0 = 2, d 1 = n − 1 in Theorem 2.5.…”