Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb{P}}^3$

Abstract

In this paper we explicitly describe the symbolic powers of curves $\mathcal{C}(q,m)$ in ${\mathbb{P}}^3$ parametrized by $(x^{d+2m},x^{d+m}{y}^m,x^d{y}^{2m},y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd (d,m)=1$. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal $\mathfrak{p}:={\mathfrak{p}}_{\mathcal{C}(q,m)}$ defining the curve $\mathcal{C}(q,m)$ is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers ${\mathfrak{p}}^{(n)}$ for all $n\ge 1$.

to appear in Communications in Algebra

References in corpus (1)

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

Cited by in corpus (2)

• Symbolic Rees algebras
• Symbolic powers, set-theoretic complete intersection and certain invariants