Asymptotic resurgence via integral closures

Abstract

Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed).For a monomial ideal the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Hà, and Hoefel. Using this intuition we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.

Steiner Configurations Ideals: Containment and Colouring

“…This result improves the bounds found in Corollary 4.8 in [1]. We also point out that Lemma 1 is similar to results from the work in [28,29]. As a consequence, in Corollary 4 we show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height (see in [30] for the first definition).…”

“…It is clear from the definition that 1 ≤ ρ a (I) ≤ ρ(I). As pointed out in [27], DiPasquale, Francisco, Mermin and Schweig showed that ρ a (I) = sup{m/r : I (m) I r }, where I r is the integral closure of I r (see also [28] Corollary 4.14) .…”

Expected resurgences and symbolic powers of ideals

“…For squarefree monomial ideals , [42, Corollary 3.6] or [9, Theorem 3.17] shows that the resurgence is bounded above by the maximal degree of a minimal generating set of the ideal .…”

“…The resurgence of an ideal can be bounded by other invariants [4, Theorem 1.2.1], and it has been explicitly computed for certain ideals [1, 5, 11]. Related invariants have also been studied, such as the asymptotic resurgence [23], which can be computed via integral closures [9].…”

“…or[9, Theorem 3.17] shows that the resurgence is bounded above by the maximal degree of a minimal generating set of the ideal I.Recall that if the resurgence is expected, then the stable version of Harbourne's conjecture must hold. It is natural to ask [20, Conjecture 4.2] if Harbourne's conjecture holds for a single value, does it hold in general?…”