Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences

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• 作者：Isabel Bermejo^a ; ^{ibermejo@ull.es} ; Eva Garcí ; a-Llorente^a ; ^{evgarcia@ull.es} ; Ignacio Garcí ; a-Marco^b ; ^{iggarcia@ull.es} ; ^{ignacio.garcia-marco@lif.univ-mrs.fr}
• 关键词：Generalized arithmetic sequence ; Projective monomial curve ; Computational commutative algebra
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：July-August 2017
• 年：2017
• 卷：81
• 期：Complete
• 页码：1-19
• 全文大小：511 K
• 卷排序：81

文摘

Let K   be an infinite field and let m1<⋯<mnm1<⋯<mn be a generalized arithmetic sequence of positive integers, i.e., there exist h,d,m1∈Z+h,d,m1∈Z+ such that mi=hm1+(i−1)dmi=hm1+(i−1)d for all i∈{2,…,n}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]K[C] of CC. Whenever K[C]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 GG of the vanishing ideal of CC with respect to the degree reverse lexicographic order. From GG we derive formulas for the Castelnuovo–Mumford regularity, the Hilbert series and the Hilbert function of K[C]K[C] in terms of the sequence.