Algebraic invariants of certain projective monomial curves

Masoumeh Javanbakhat, Masoumeh Javanbakhat, Leila Sharifan

Abstract:

Let 𝕂
be an infinite field and consider the following sequence of positive integers: 𝑎0=𝑚
, 𝑎1=𝑚+𝑑
, 𝑎2=𝑚+3𝑑
and 𝑎3=𝑚+6𝑑
where gcd(𝑚,𝑑)=1
. We study the projective monomial curve ̃ ⊂ℙ4
parametrically defined by

[𝑠𝑎3:𝑠𝑎3−𝑎0𝑡𝑎0:𝑠𝑎3−𝑎1𝑡𝑎1:𝑠𝑎3−𝑎2𝑡𝑎2:𝑡𝑎3].
We prove that the homogeneous coordinate ring 𝕂[̃ ]
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.