Algebraic invariants of certain projective monomial curves

Abstract

Let ${\mathbb {K}}$ be an infinite field and consider the following sequence of positive integers: $a_0=m$, $a_1=m+d$, $a_2=m+3d$ and $a_3=m+6d$ where $\gcd (m,d)=1$. We study the projective monomial curve $\tilde{\mathcal {C}}\subset \mathbb {P}^{4}$ parametrically defined by

$$\begin{aligned} {[}s^{a_{3}}: s^{a_{3}-a_{0}} t^{a_{0}}: s^{a_{3}-a_{1}}t^{a_{1}}: s^{a_{3}-a_{2}}t^{a_{2}}: t^{a_{3}}]. \end{aligned}$$

We prove that the homogeneous coordinate ring ${\mathbb {K}}[\tilde{\mathcal {C}}]$ is Cohen–Macaulay. We will compute explicitly the Hilbert series. Taken together these two results we extract the Castelnuovo–Mumford regularity of this class of projective monomial curves. Finally we derive the H-basis of the underlying ideal.