Castelnuovo–Mumford regularity of projective monomial varieties of codimension two (original) (raw)
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On Castelnuovo-Mumford regularity of codimension two monomial varieties
ACM Sigsam Bulletin, 1999
Let K be an algebraically closed field, and let V ⊂ P n+1 K be a projective monomial variety of codimension two with n ≥ 2, i.e., a projective toric variety of codimension two whose homogeneous coordinate ring is a simplicial semigroup ring. We give an explicit formula for the Castelnuovo-Mumford regularity of V, reg(V), in terms of the reduced Gröbner basis of I (V) with respect to the reverse lexicographic order. As a consequence, we show that reg (V) ≤ deg V − 1, where deg V is the degree of V, and characterize when equality holds.
On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals
Mathematische Zeitschrift, 1998
In the first part of this paper we show that the Castelnuovo-Mumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Gröbner basis.
Journal of Pure and Applied Algebra, 2001
Given a homogeneous ideal I ⊂ K[x0; : : : ; xn] deÿning a subscheme X of projective n-space P n K , we provide an e ective method to compute the Castelnuovo-Mumford regularity of X in the following two cases: when X is arithmetically Cohen-Macaulay, and when X is a not necessarily reduced projective curve. In both cases, we compute the Castelnuovo-Mumford regularity of X by means of quotients of zero-dimensional monomial ideals.
Journal of Pure and Applied Algebra, 2001
Given a homogeneous ideal I ⊂ K[x0; : : : ; xn] deÿning a subscheme X of projective n-space P n K , we provide an e ective method to compute the Castelnuovo-Mumford regularity of X in the following two cases: when X is arithmetically Cohen-Macaulay, and when X is a not necessarily reduced projective curve. In both cases, we compute the Castelnuovo-Mumford regularity of X by means of quotients of zero-dimensional monomial ideals.
Castelnuovo–Mumford regularity, postulation numbers, and reduction numbers
Journal of Algebra, 2007
We establish a bound for the Castelnuovo-Mumford regularity of the associated graded ring G I (A) of an m-primary ideal I of a local Noetherian ring (A, m) in terms of the dimension of A, the relation type and the number of generators of I. As a consequence, we obtain that the existence of uniform bounds for the regularity of the associated graded ring, and the relation type of parameter ideals in A, are equivalent conditions. In addition, we establish an equation for the postulation number and the Castelnuovo-Mumford regularity of the associated graded ring G q (A) of a parameter ideal q, which holds under certain conditions on the depths of the occurring rings. We also show, that the regularity of the ring G q (A) is bounded in terms the dimension of A, the length of A/q and the postulation number of G q (A).
Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals
Mathematical Proceedings of the Cambridge Philosophical Society, 2010
Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits limntoinftyai(R/(capjoverlineI1jn+cdots+capjoverlineIpjn))/n\lim_{n\to\infty} a_i(R/(\cap_j \overline{I_{1j}^n} + \cdots + \cap_j \overline{I_{pj}^n}))/n limntoinftyai(R/(capjoverlineI1jn+cdots+capjoverlineIpjn))/n also exist.As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.
A computation of the Castelnuovo-Mumford regularity of certain two-dimensional unmixed ideals
Communications in Algebra, 2020
In this article, we consider ideals of the form I ¼ \ 1 i<j n P wi, j i, j of a polynomial ring R ¼ K½x 1 , :::, x n over a field, where P i, j is an ideal generated by variables fx 1 , :::, x n g n fx i , x j g and w i, j is a non-negative integer for all i, j. We will give explicit formulas for computing the a i-invariants a i ðR=IÞ, i ¼ 1, 2, and the Castelnuovo-Mumford regularity regðIÞ in the case w i, j takes a value a or b, where a > b > 0: ARTICLE HISTORY