Monomial Curves Research Articles

On the Factorial Closure of Certain Monomial Modules

Let R = K[x, y, z] denote the polynomial ring in three variables over an arbitrary field K. We study the factorial closure B(E) of certain R-modules E of projective dimension 1, called monomial modules. The module E is—in a certain sense—related to the defining ideal of a monomial affine space curves in In general, we prove that B(E) is different from Sym R (E). This extends a particular case of an R-module E with this property studied by Samuel [7]. As the main result, we characterize those monomial R-modules E such that B(E) is an R-algebra of finite type and generated in degree 1 and 2 over the symmetric algebra Sym R (E).

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n−space for any suitably chosen n − 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).

On the equality of ordinary and symbolic powers of ideals

We consider the following question concerning the equality of ordinary and symbolic powers of ideals. In a regular local ring, if the ordinary and symbolic powers of a one-dimensional prime ideal are the same up to its height, then are they the same for all powers? We provide supporting evidence of a positive answer for classes of prime ideals defining monomial curves or rings of low multiplicities.

Macaulay's theorem for some projective monomial curves

Macaulay's theorem for some projective monomial curves

Smoothability and order bound for AS semigroups

In this paper we consider numerical semigroups S generated by arithmetic sequences m 0,…,m n (AS-semigroups). First we state some results on the module $T^{1}_{k[S]}$ ; further in the cases m 0≡1 and m 0≡n (modulo n), we prove these semigroups are Weierstrass by showing that the associated monomial curves $X=\operatorname {Spec}{k[S]}$ are smoothable. Finally for each semigroup S generated by an arithmetic sequence we evaluate the so-called “order bounds”: when S is Weierstrass, these invariants are good approximations for the minimum distance of the related one-point codes.

Castelnuovo-Mumford regularity and reduction number of smooth monomial curves in $\mathbb{P}^{5}$

Castelnuovo-Mumford regularity and reduction number of smooth monomial curves in $\mathbb{P}^{5}$

Tropical secant graphs of monomial curves

We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$.

Bases and Ideal Generators for Projective Monomial Curves

In this article we study bases for projective monomial curves and the relationship between the basis and the set of generators for the defining ideal of the curve. We understand this relationship best for curves in ℙ3 and for curves defined by an arithmetic progression. We are able to prove that the latter are set theoretic complete intersections.

Scarf lattice ideals

This paper deals with the Scarf property of lattice ideals initiated by Peeva and Sturmfels[10], [11]. We will present a Scarf lattice ideal that is neither generic nor of codimension 2 and show that this property gives rise to several algebraic and combinatorial properties. In particular, we prove that for monomial curves, this property coincides with the notion of genericity, and that certain Scarf lattice ideals can have certain Scarf initial ideals.

On the smoothness of blowups for certain monomial curves

Let K be a field and let \({m_{0}<\cdots < m_{p+1}}\) be a sequence of positive integers such that they form a minimal arithmetic progression. Let \({\wp}\) denote the defining ideal of the monomial curve \({\mathcal{C}}\) in \({\mathbb{A}_{K}^{p+2}}\) , defined parametrically by \({X_{0}=T^{m_{0}},\ldots,X_{p}=T^{m_{p}},Y=T^{m_{p+1}}}\) . Let \({R=K[X_{0},\ldots,X_{p},Y]}\) . In this article, we prove that for p ≥ 2, the blowup scheme \({{\rm Proj}\,R[\wp t]}\) is not smooth. This proves Francia’s conjecture in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.

Gevrey solutions of irregular hypergeometric systems in two variables

We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine plane monomial curve. We also describe the irregularity complex of such a system with respect to its singular support and in particular we prove, using elementary methods, that this irregularity complex is a perverse sheaf as assured by a theorem of Z. Mebkhout.

CM defect and Hilbert functions of monomial curves

In this article we consider a semigroup ring R = K 〚 Γ 〛 of a numerical semigroup Γ and study the Cohen–Macaulayness of the associated graded ring G ( Γ ) ≔ gr m ( R ) ≔ ⊕ n ∈ N m n / m n + 1 and the behaviour of the Hilbert function H R of R . We define a certain (finite) subset B ( Γ ) ⊆ Γ and prove that G ( Γ ) is Cohen–Macaulay if and only if B ( Γ ) = 0̸ . Therefore the subset B ( Γ ) is called the Cohen–Macaulay defect of G ( Γ ) . Further, we prove that if the degree sequence of elements of the standard basis of Γ is non-decreasing, then B ( Γ ) = 0̸ and hence G ( Γ ) is Cohen–Macaulay. We consider a class of numerical semigroups Γ = ∑ i = 0 3 N m i generated by 4 elements m 0 , m 1 , m 2 , m 3 such that m 1 + m 2 = m 0 + m 3 —so called “balanced semigroups”. We study the structure of the Cohen–Macaulay defect B ( Γ ) of Γ and particularly we give an estimate on the cardinality | B ( Γ , r ) | for every r ∈ N . We use these estimates to prove that the Hilbert function of R is non-decreasing. Further, we prove that every balanced “unitary” semigroup Γ is “2-good” and is not “1-good”, in particular, in this case, G ( Γ ) is not Cohen–Macaulay. We consider a certain special subclass of balanced semigroups Γ . For this subclass we try to determine the Cohen–Macaulay defect B ( Γ ) using the explicit description of the standard basis of Γ ; in particular, we prove that these balanced semigroups are 2-good and determine when exactly G ( Γ ) is Cohen–Macaulay.

Ideals of Herzog–Northcott type

AbstractThis paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

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

Starting from [1] we compute the Groebner basis for the defining ideal, P, of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P, inP. The first result of this article introduces a procedure for generating infinite families of Ratliff–Rush ideals, in polynomial rings with multivariables, from a Ratliff–Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff–Rush. The proof is through applying the first result of this article combined with Corollary 12 in [2]. This generalizes the work of [4] (or [5])\\ for the case of arithmetic sequences. Finally, we apply the main result of [3] to give the necessary and sufficient conditions for the integral closedness of any power of inP.

Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve

Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve

The Ratliff-Rush Closure of Initial Ideals of Certain Prime Ideals

We prove that the initial ideal of the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers is Ratliff-Rush closed.

On the regularity of certain projective monomial curves

In this paper we present a method to find the regularity of projective monomial curves in terms of an ordering of monoids associated to them. We use this result to find the regularity of certain monomial curves and investigate where regularity is attained in their minimal graded free resolutions.

Tropical secant graphs of monomial curves

We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$. On construit et on étude un graphe plongé dans $\mathbb{R}^{n+1}$ paramétrisé par une suite strictement croissante de $n$ nombres entiers $\{ i_1, \ldots, i_n\}$, premiers entre eux. Ce graphe s'appelle $\textit{graphe tropical surface sécante}$. On montre que ce graphe est la tropicalisation d'une surface dans $\mathbb{C}^{n+1}$ paramétrisé par des binômes. On utilise ce graphe pour construire la tropicalisation de la première sécante d'une courbe monomiale ayant comme vecteur d'exponents $(0, i_1, \ldots, i_n)$. On représente cette variété tropicale pour un graphe balancé (le $\textit{graphe tropical sécante}$). La combinatoire qu'on utilise pour le calcul du degré de ces variétés sécantes classiques n'est pas triviale, et a été developpée par K. Ranestad. En utilisant des techniques de la géométrie tropicale, on donne des algorithmes qui calculent le degré (même le multidegré) et le polytope de Newton de la première sécante d'une courbe monomiale de $\mathbb{P}^4$.

Generation in Degree Four of Symbolic Blowups of Self-Linked Monomial Space Curves

We study the symbolic blowup ring of a prime ideal defining a monomial curve in the power series ring in 3 variables over a field. We characterize the conditions required to have the symbolic blowup generated in degree 4 when the monomial curve is self-linked.

Castelnuovo-Mumford regularity and the reduction number of some monomial curves

We compare the Castelnuovo-Mumford regularity and the reduction number of some classes of monomial projective curves with at most one singular point. Furthermore, for smooth monomial curves we prove an upper bound on the regularity which is stronger than the one given by L'vovsky.