Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences. (July 2017)

• Record Type: Journal Article
• Authors: Bermejo, Isabel; García-Llorente, Eva; García-Marco, Ignacio
• Is Part Of: Journal of symbolic computation. Volume 81(2017)
• Page Start: 1
• Page End: 19
• Publication Date: 2017-07
• Subjects: Generalized arithmetic sequence -- Projective monomial curve -- Computational commutative algebra; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.11.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Let K be an infinite field and let m 1 < ⋯ < m n be a generalized arithmetic sequence of positive integers, i.e., there exist h, d, m 1 ∈ Z + such that m i = h m 1 + ( i − 1 ) d for all i ∈ { 2, …, n } . We consider the projective monomial curve C ⊂ P K n parametrically defined by x 1 = s m 1 t m n − m 1, …, x n − 1 = s m n − 1 t m n − m n − 1, x n = s m n, x n + 1 = t m n . 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.