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

In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$in${\mathbb P}^3$parametrizedby$( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$,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 ${\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) }$definingthecurve${\mathcal C}(q,m)$ is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers ${\mathfrak{p}}^{(n)}$ for all $n\ge 1$.