Symbolic Blowup algebras and invariants of certain monomial curves in an affine space

Abstract

Let $d\ge 2$ and $m\ge 1$ be integers such that $\gcd (d,m)=1.$ Let $\mathfrak{p}$ be the defining ideal of the monomial curve in ${\mathbb{A}}_{\mathbb{k}k}^d$ parametrized by $(t^{n_1},\dots ,t^{n_d})$ where $n_i=d+(i-1)m$ for all $i=1,\dots ,d$. In this paper, we describe the symbolic powers ${\mathfrak{p}}^{(n)}$ for all $n\ge 1$. As a consequence we show that the symbolic blowup algebras ${\mathcal{R}}_s(\mathfrak{p})$ and $G_s(\mathfrak{p})$ are Cohen-Macaulay. This gives a positive answer to a question posed by S.~Goto in \cite{goto}. We also discuss when these blowup algebras are Gorenstein. Moreover, for $d=3$, considering $\mathfrak{p}$ as a weighted homogeneous ideal, we compute the resurgence, the Waldschmidt constant and the Castelnuovo-Mumford regularity of ${\mathfrak{p}}^{(n)}$ for all $n\ge 1$. The techniques of this paper for computing ${\mathfrak{p}}^{(n)}$ are new and we hope that these will be useful to study the symbolic powers of other prime ideals.

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 Blowup algebras of monomial curves in ${\mathbb{A}}^3$ defined by arithmetic sequence
• Symbolic powers, set-theoretic complete intersection and certain invariants