On the Type of the Associated Graded Ring of Certain Monomial Curves

Abstract

Let p; m; d be positive integers, m i : m id, 0 i p and let n be positive integer such that gcdm; d; n 1 and m 0 < n. Let A be the coordinate ring of the algebroid monomial curve in the ane algebroid p 2-space A p2 K over a ®eld K, de®ned parametrically by X 1 T m 0 ; X 2 T m 1 ; . . . ; X p T m p ; X p1 T n : In this article assuming that the associated graded ring gr m A of A is Cohen-Maucaulay (and some more mild additional assumptions, see (2.4)), we give an explicit formula for the type of gr m A in terms of the standard basis of the semigroup generated by the almost arithmetic progression m 0 ; m 1 ; . . . ; m p ; n. Our special assumptions are satis®ed if p 1, that is, for the class of algebroid monomial space curves.

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

“…One might be able to prove Corollary 3.2 with their method. We avoid this and prove it directly by a very easy comparison between elements of S and S. It is worthy to remark that in [15], the authors computed the Hilbert function for the case of almost arithmetic sequences just for m k > m 0 .…”

“…, a n is an arithmetic sequence and a 0 an arbitrary positive integer. In this case, the associated graded ring was studied in several papers by Molinelli, Patil and Tamone (see [13,15,16]) where a very technical machinery has been introduced in order to give some conditions to control the Cohen-Macaulayness property of G and to compute the Hilbert function of A and Cohen-Macaulay type of G. Even if a priori these criteria could be used in order to prove that G is Cohen-Macaulay in our class, the easy and new method presented in this paper (see for example Lemma 3.1 and Corollary 3.2) will be useful to produce a larger class of monomial curve with Cohen-Macaulay associated graded ring and to give information on a free resolution of G.…”

CM defect and Hilbert functions of monomial curves

“…(a) If Γ is generated by 7,9,17,19, [24]) and {0, 9,17,18,19,27, 36} is the standard basis of Γ with respect to 7, but the sequence…”

Results on the Ratliff–Rush Closure and the Integral Closedness of Powers of Certain Monomial Curves

“…For the remaining of this paper, we let p = n−1 and the parameters q and r will have the same meaning assigned to them by the above remark, that is, m 0 = qn + r with 1 ≤ r ≤ n. Note that q ≥ 1 since m 0 > n. Now by Theorem (3), Lemma (4), and the remarks above we may state the following proposition which is the beginning step towards proving two of the main theorems of this paper, namely, Theorem (18) and Theorem (22). Proposition 8 Let P be the defining ideal of the monomial curve that corresponds to the arithmetic sequence m 0 , ..., m n with m 0 a positive integer, m i = m 0 +id, and gcd(m 0 , ..., m n ) = 1, where d is a positive integer with gcd(m 0 , d) = 1.…”

“…The theory of toric varieties plays an important role at the crossroads of geometry, algebra and combinatorics. The initial ideals, inP , of the monomial curves that correspond to an (almost) arithmetic sequence have been studied by many authors such as [4], [15], [16], [17], [18], and [23]. In this paper we are interested in studying the Ratliff-Rush and the integral closedness of powers of inP for the case when the sequence m 0 , ..., m n is arithmetic.…”