Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences

Abstract

Let K be an infinite field and let m1<⋯<mn be a generalized arithmetic sequence of positive integers, i.e., there exist h,d,m1∈Z+ such that mi=hm1+(i−1)d for all i∈{2,…,n}. We consider the projective monomial curve C⊂PKn parametrically defined byx1=sm1tmn−m1,…,xn−1=smn−1tmn−mn−1,xn=smn,xn+1=tmn. In this work, we characterize the Cohen–Macaulay and Koszul properties of the homogeneous coordinate ring K[C] of C. Whenever K[C] is Cohen–Macaulay we also obtain a formula for its Cohen–Macaulay type. Moreover, when h divides d, we obtain a minimal Gröbner basis G of the vanishing ideal of C with respect to the degree reverse lexicographic order. From G we derive formulas for the Castelnuovo–Mumford regularity, the Hilbert series and the Hilbert function of K[C] in terms of the sequence.