Monomial Curves Research Articles

On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

In this article we solve the following problem: “The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing.” We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation.

The strongly robust simplicial complex of monomial curves

To every simple toric ideal IT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_T$$\\end{document} one can associate the strongly robust simplicial complex ΔT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _T$$\\end{document}, which determines the strongly robust property for all ideals that have IT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_T$$\\end{document} as their bouquet ideal. We show that for the simple toric ideals of monomial curves in As\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {A}^{s}$$\\end{document}, the strongly robust simplicial complex ΔT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _T$$\\end{document} is either {∅}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\emptyset \\}$$\\end{document} or contains exactly one 0-dimensional face. In the case of monomial curves in A3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {A}^{3}$$\\end{document}, the strongly robust simplicial complex ΔT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _T$$\\end{document} contains one 0-dimensional face if and only if the toric ideal IT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_T$$\\end{document} is a complete intersection ideal with exactly two Betti degrees. Finally, we provide a construction to produce infinitely many strongly robust ideals with bouquet ideal the ideal of a monomial curve and show that they are all produced this way.

Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves

Let C be a Gorenstein noncomplete intersection monomial curve in the 4-dimensional affine space with defining ideal I(C). In this article, we use the minimal generating set of I(C) to give a criterion for determining whether the tangent cone of C is Cohen-Macaulay. We also show that if the tangent cone of C is Cohen-Macaulay, then the minimal number of generators of the ideal I(C)⁎ is either 5 or an even integer of the form 2d+2, for a suitable integer d. Additionally, we provide a family of Gorenstein noncomplete intersection monomial curves C whose tangent cone is Cohen-Macaulay and the minimal number of generators of I(C)⁎ is large.

On the symbolic powers of defining ideals of monomial curves associated to generalized arithmetic sequences

Let s , a , n , d be positive integers such that n ≥ 2 and GCD ( a , d ) = 1 . Let P denote the defining ideal of the monomial curve associated to the sequence a , sa + d , … , sa + nd . In this survey, we investigate the symbolic powers of P. We first show that for each i > 1 the equality P ( i ) = P i holds if and only if either n = 2 and a even or n = 3 , i = 2 and a ≡ 2 ( mod 3 ) . The finitely generated property of the symbolic Rees algebra R S ( P ) are also explored.

Convex hulls of monomial curves, and a sparse positivstellensatz

AbstractConsider the closed convex hull K of a monomial curve given parametrically as $$(t^{m_1},\ldots ,t^{m_n})$$ ( t m 1 , … , t m n ) , with the parameter t varying in an interval I. We show, using constructive arguments, that K admits a lifted semidefinite description by $$\mathcal {O}(d)$$ O ( d ) linear matrix inequalities (LMIs), each of size $$\left\lfloor \frac{n}{2} \right\rfloor +1$$ n 2 + 1 , where $$d= \max \{m_1,\ldots ,m_n\}$$ d = max { m 1 , … , m n } is the degree of the curve. On the dual side, we show that if a univariate polynomial p(t) of degree d with at most $$2k+1$$ 2 k + 1 monomials is non-negative on $${\mathbb {R}}_+$$ R + , then p admits a representation $$p = t^0 \sigma _0 + \cdots + t^{d-k} \sigma _{d-k}$$ p = t 0 σ 0 + ⋯ + t d - k σ d - k , where the polynomials $$\sigma _0,\ldots ,\sigma _{d-k}$$ σ 0 , … , σ d - k are sums of squares and $$\deg (\sigma _i) \le 2k$$ deg ( σ i ) ≤ 2 k . The latter is a univariate positivstellensatz for sparse polynomials, with non-negativity of p being certified by sos polynomials whose degree only depends on the sparsity of p. Our results fit into the general attempt of formulating polynomial optimization problems as semidefinite problems with LMIs of small size. Such small-size descriptions are much more tractable from a computational viewpoint.

Projective closure of Gorenstein monomial curves and the Cohen–Macaulay property

Projective closure of Gorenstein monomial curves and the Cohen–Macaulay property

Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Let A={a_0,ldots ,a_{n-1}} be a finite set of nge 4 non-negative relatively prime integers, such that 0=a_0<a_1<cdots <a_{n-1}=d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve mathcal {C}_A parametrized by A, CA={(vd:ua1vd-a1:⋯:uan-2vd-an-2:ud)∣(u:v)∈Pk1}⊂Pkn-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\quad \\mathcal {C}_A=\\{(v^d:u^{a_1}v^{d-a_1}:\\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \\mid (u:v)\\in \\mathbb {P}^{1}_k\\}\\subset \\mathbb {P}^{n-1}_k. \\end{aligned}$$\\end{document}The exponents in the previous parametrization of mathcal {C}_A define a homogeneous semigroup mathcal {S}subset mathbb {N}^2. We provide several results relating the Castelnuovo–Mumford regularity of mathcal {C}_A to the behavior of the sumsets of A and to the combinatorics of the semigroup mathcal {S} that reveal a new interplay between commutative algebra and additive number theory.

Normal bundle of monomial curves: an application to rational curves

In this note, we give an application to the study of general rational curves in \mathbb{P}^s(\mathbb{C}) of the calculation of the splitting type of the normal bundle of any smooth monomial rational curve (i.e., embedded by monomial functions).

Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves

In this article, we study minimal graded free resolutions of Cohen-Macaulay tangent cones of some monomial curves associated to 4-generated pseudo symmetric numerical semigroups. We explicitly give the matrices in these minimal free resolutions.

Betti sequence of the projective closure of affine monomial curves

We introduce the notion of star gluing of numerical semigroups and show that this preserves the arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure. Next, we give a sufficient condition involving Gröbner basis for the matching of Betti sequences of the affine curve and its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct numerical semigroups by gluing, such that for every positive integer n, the last Betti number of the corresponding affine curve and its projective closure are both n.

On the stable Harbourne conjecture for ideals defining space monomial curves

For the ideal p \mathfrak {p} in k [ x , y , z ] k[x, y, z] defining a space monomial curve, we show that p ( 2 n − 1 ) ⊆ m p n \mathfrak {p}^{(2 n - 1)} \subseteq \mathfrak {m} \mathfrak {p}^{n} for some positive integer n n , where m \mathfrak {m} is the maximal ideal ( x , y , z ) (x, y, z) . Moreover, the smallest such n n is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that p ( 3 ) ⊆ m p 2 \mathfrak {p}^{(3)} \subseteq \mathfrak {m} \mathfrak {p}^2 if k k is a field of characteristic not 3 3 ; however, the stable Harbourne conjecture holds for space monomial curves as they claimed.

On the Hilbert series of the tangent cones for some 4-generated pseudosymmetric monomial curves

In this article, we study Hilbert Series of non-Cohen-Maculay tangent cones for some 4-generated pseudo symmetric monomial curves. We show that the Hilbert Function is nondecreasing by explicitly computing it. We also compute standard bases of these toric ideals.

Real analytic functions and monomial curves

Real analytic functions and monomial curves

On the boundedness of the type of an almost Gorenstein monomial curve in 𝔸5

We prove that the Cohen-Macaulay type of an almost Gorenstein monomial curve is bounded.

The Poincaré Index on Singular Varieties

In this paper, we discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. They are based on properties of regular meromorphic and logarithmic differential forms, of the dualizing (canonical) module and related constructions. In particular, we show how to compute the index on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with known traditional approaches, we use neither computers, nor integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics.

Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.

Affine monomial curves

Affine monomial curves

On restriction of exponential sums to hypersurfaces with zero curvature

We prove essentially sharp bounds for the $$L^p$$ restriction of weighted Gauss sums to monomial curves. Getting the $$L^2$$ upper bound combines the $$TT^*$$ method for matrices with the first and second derivative test for exponential sums. The matching lower bound follows via constructive interference on short blocks of integers, near the critical point of the phase function. This method is used to make the broader point that restriction to hypersurfaces is really sensitive to curvature. Our results here complement those in [4].

On the reduction numbers and the Castelnuovo–Mumford regularity of projective monomial curves

This paper gives explicit formulas for the reduction number and the Castelnuovo–Mumford regularity of projective monomial curves.

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Let a,b and n&gt;1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n&gt;3, the ideal I has a unique minimal system of generators if and only if a&lt;b−1.