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

Abstract

Let and be integers such that Let be the defining ideal of the monomial curve in parametrized by where for all In this paper, we describe the symbolic powers for all As a consequence, we show that the symbolic blowup algebras and are Cohen–Macaulay. This gives a positive answer to a question posed by S. Goto (1994). We also discuss when these blowup algebras are Gorenstein. Moreover, for d = 3, considering as a weighted homogeneous ideal, we compute the resurgence, the Waldschmidt constant and the Castelnuovo–Mumford regularity of for all The techniques of this paper for computing are new, and we hope that these will be useful to study the symbolic powers of other prime ideals.