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

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 ${\mathbb A}_{ \Bbbk{k}}^d$parametrizedby$(t^{n_1}, \ldots, t^{n_d})$where$n_i = d + (i-1)m$forall$i = 1, \ldots, d$. 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 $n \geq 1$\mathrm{.}Thetechniquesofthispaperforcomputing${\mathfrak p}^{(n)}$ are new and we hope that these will be useful to study the symbolic powers of other prime ideals.