Resurgence and Castelnuovo-Mumford Regularity of Certain Monomial Curves in A 3 $\mathbb {A}^{3}$
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Resurgence and Castelnuovo-Mumford Regularity of Certain Monomial Curves in A3 Clare D’Cruz1 Received: 28 March 2019 / Revised: 26 August 2019 / Accepted: 11 March 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract Let p be the defining ideal of the monomial curve C (2q + 1, 2q + 1 + m, 2q + 1 + 2m) in the affine space A3k parameterised by (x 2q+1 , x 2q+1+m , x 2q+1+2m ), where gcd(2q + 1, m) = 1. In this paper we compute the resurgence of p, the Waldschmidt constant of p and the Castelnuovo-Mumford regularity of the symbolic powers of p. Keywords Resurgence · Waldschmidt constant · Regularity Mathematics Subject Classification (2010) Primary 13A30 · 1305 · 13H15 · 13P10
1 Introduction Let R = k[x1 , x2 , x3 ] and S = k[x] be a polynomial rings over a field k of characteristic zero. Let q and m be positive integers, d = 2q + 1 and gcd(d, m) = 1. Consider the homomorphism φ : R −→ S defined by φ(xi ) = x d+(i−1)m , where 1 ≤ i ≤ 3. Throughout this paper p := pC (d,d+m,d+2m) = ker(φ). For q = 1, the resurgence ρ(p), the Waldschmidt constant γ (p) and the Castelnuovo-Mumford regularity of the symbolic powers of p have been computed in [9]. In this paper we generalise these results for all q ≥ 1. We also verify that certain conjectures posed in [13] hold true for p. Before describing our main results, we will give some background on these quantities. For any ideal I in a Noetherian ring A of positive dimension with no embedded components, the nth symbolic power of I is defined by I (n) := ∩p∈Ass(R/I ) I n Ap ∩ A. In general, the generators of I (n) are hard to describe. Hence, in order to have a more precise relation between symbolic powers and ordinary powers of ideals, Harbourne posed the following conjecture: Let I ⊆ k[x1 , . . . , xt ] be an homogeneous ideal. Then I (m) ⊆ I r if m ≥ r(t − 1) − (t − 2) [1, Conjecture 8.4.2]. In the same paper, the authors give evidence to show that this conjecture is true if char k > 0. Later, Bocci and Harbourne introduced an Clare D’Cruz
[email protected] 1
Chennai Mathematical Institute, Plot H1 SIPCOT IT Park, Siruseri, Kelambakkam, Tamil Nadu 603103, India
C. D’Cruz
asymptotic quantity called resurgence which is defined as ρ(I ) := sup{m/r | I (m) ⊂ I r } [3]. This supremum exists and in fact 1 ≤ ρ(I ) ≤ t − 1 [3, Lemma 2.3.2]. Since resurgence in general is hard to compute, in [3] the authors define another invariant which they call the Waldschmidt constant. The Waldschmidt constant was first introduced by Waldschmidt in [14]. We use the definition as in [3]. Let α(I ) := min{n | In = 0}. The Waldschmidt (n) constant is defined as γ (I ) = limn→∞ α(In ) . Bocci and Harbourne showed that if I is a homogenous ideal, then α(I )/γ (I ) ≤ ρ(I ), and in addition if I is a zero dimensional subscheme in a projective space, then α(I )/γ (I ) ≤ ρ(I ) ≤ reg(I )/γ (I ), where reg(I ) is the Castelnuovo-Mumford regularity of I [3, Theorem 1.2.1] . The resurgence and the Waldschmidt constant ha
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