On monomial curves obtained by gluing

Abstract

We study arithmetic properties of tangent cones associated to affine monomial curves, using the concept of gluing. In particular we characterize the Cohen-Macaulay and Gorenstein properties of tangent cones of some families of monomial curves obtained by gluing. Moreover, we provide new families of monomial curves with non-decreasing Hilbert functions. introductionA monomial curve C in the affine space A d k over a field k consists on the set of points defined parametrically by X 1 = t m1 , . . . , X d = t m d , for some positive integers m 1 < · · · < m d . In order to be sure that different parameterizations give rise to different monomial curves, we may assume that gcd(m 1 , . . . , m d ) = 1.In fact, it is known that the set C is an affine variety whose coordinate ring issee for instance E. Reyes, R. H. Villarreal and L. Zárate [17]. The set S = {r 1 m 1 + · · · + r d m d ; r i ≥ 0} is a subset of the non-negative integers N ∪ {0} which is closed under addition, and the condition gcd(m 1 , . . . , m d ) = 1 is equivalent to the property # N \ S < ∞. In other words, S =< m 1 , . . . , m d > is a numerical semigroup minimally generated by the unique minimal system of generators {m 1 , . . . , m d }. The coordinate ring R is called the numerical semigroup ring associated to S and it is denoted by k[S]. Since we are interested in the arithmetical properties at the origin, which is the only singular point of the curve C, we will consider the ring R = k[[t m1 , . . . , t m d ]] = k[[S]]. Note that R is a complete one-dimensional local domain with maximal ideal m = (t m1 , . . . , t m d ).We also consider the tangent cone associated to k[[S]]; that is the graded ring G(S) := n≥0 m n /m n+1 .

Homogeneous numerical semigroups

“…If S is a nice gluing of S 1 and S 2 , then qm 1 = an 1 m 1 ≤ ord S 1 (p)n 1 m 1 ≤ pn 1 and so m(S) = qm 1 . This is also the case when S is a specific gluing, from [16,Corollary 3.14]. Proposition 5.4.…”

On semigroup rings with decreasing Hilbert function

“…in case e = 12 the cases to be veryfied are (n i , n j ) ∈ {(2, n j = 3)(2, 9), (4, −), (6, −), (8, −)}, which are clearly incompatible with the assumptions (for (2,9) : 2n j ≡ 3n i ).…”

“…(n i , n j ) = (1,4) : S 1 =< 13, 14, 17, 29, 32, 33, 35, 36, 37, 38 > k ′ = 1 c = 26 (n i , n j ) = (2,8) : S 2 =< 13, 15, 21, 32, 38, 40, 44, 46, 48, 50 > k ′ = 1 c = 38 (n i , n j ) = (3, 12), : S 3 =< 13, 16, 25, 35, 44, 47, 53, 56, 59, 62 > k ′ = 1 c = 50 (n i , n j ) = (4,3) : S 4 =< 13, 17, 16, 35, 36, 37, 38, 40, 41, 44 > k ′ = 0 c = 32 (n i , n j ) = (5,7) : S 5 =< 13, 18, 20, 41, 43, 45, 47, 48, 50, 55 > k ′ = 0 c = 43 (n i , n j ) = (6, 11) : S 6 =< 13, 19, 24, 44, 49, 54, 55, 59, 60, 66 > k ′ = 0 c = 54 (n i , n j ) = (7,2) : S 7 =< 13, 20, 28, 47, 55, 62, 63, 70, 71, 77 > k ′ = 0 c = 65 (n i , n j ) = (8,6) : S 8 =< 13, 21, 19, 44, 46, 48, 50, 54, 56, 62 > k ′ = −1 c = 50 (n i , n j ) = (9, 10) : S 9 =< 13, 22, 23, 53, 54, 55, 56, 63, 64, 73 > k ′ = −1 c = 61 (n i , n j ) = (10, 1) : S 10 =< 13, 23, 27, 56, 60, 64, 68, 70, 74, 84 > k ′ = −1 c = 72 (n i , n j ) = (11, 5) : S 11 =< 13, 24, 31, 59, 66, 73, 77, 80, 84, 95 > k ′ = −1 c = 83 (n i , n j ) = (12,9)…”

“…If R = k[[S]] is a semigroup ring, many authors proved that H R is non-decreasing in several cases: • S is generated by an almost arithmetic sequence (if the sequence is arithmetic, then G is Cohen-Macaulay) [17], [13] • S belongs to particular subclasses of four-generated semigroups, which are symmetric [1], or which have Buchsbaum tangent cone [3] • S is balanced [12], [3] • S is obtained by techniques of gluing numerical semigroups [2], [9] • S satisfies certain conditions on particular subsets of S (see below) [ [14] (here recalled in Example 1.6). When G is not Cohen-Macaulay, a useful method to describe H R can be found in some recent papers (see [12], [3], [5]): it is based on the study of certain subsets of S, called D k and C k , (k ∈ N).…”

“…Let S =< 13, 19, 24, 44, 49, 54, 55, 59, 60, 66 >, with e = 13, v = 10, Ap = { 0 19, 24, 38, 43, 44, 48, 49, 54, 55, 59, 60, 66 }. Then H R = [1,10, 9, 11, 12, 13 →], H R ′ =[1,9, 3]. Further Supp(D 2 + e) = Supp C 2 = {19, 24}: D…”