Monomial Space Curves Research Articles

Betti Numbers of the Tangent Cones of Monomial Space Curves

Betti Numbers of the Tangent Cones of Monomial Space Curves

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.

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.

A stable version of Harbourne's Conjecture and the containment problem for space monomial curves

The symbolic powers I(n) of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(n)⊆Im holds. When I is an ideal of height 2 in a regular ring, I(3)⊆I2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curves (ta,tb,tc). More generally, given a radical ideal I of big height h, while the containment I(hn−h+1)⊆In conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee I(hn−h+1)⊆In for n≫0.

Expected resurgences and symbolic powers of ideals

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.

Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves

In this paper, we study finite generation of symbolic Rees rings of the defining ideal p of the space monomial curve (ta,tb,tc) for pairwise coprime integers a, b, c. Suppose that the base field is of characteristic 0, and the ideal p is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element ξ of the minimal degree in p is a negative curve, we determine the minimal degree of an element η such that the pair {ξ,η} satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring Rs(p) is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of p in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of p in which the homogeneous element of the minimal degree in p(2) is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.

Deforming monomial space curves into set-theoretic complete intersection singularities

We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.

A short note on the multiplier ideals of monomial space curves

Thompson (2014) exhibits a formula for the multiplier ideal with multiplier λ of a monomial curve C with ideal I as an intersection of a term coming from the I-adic valuation, the multiplier ideal of the term ideal of I, and terms coming from certain specified auxiliary valuations. This short note shows it suffices to consider at most two auxiliary valuations. This improvement is achieved through a more intrinsic approach, reduction to the toric case.

Multiplicities of Monomial Space Curves with Non-Noetherian Symbolic Blowups

We study the multiplicities of space monomial curves with non-Noetherian symbolic blowups. We extend the celebrated examples of Goto, Nishida, and Watanabe, as well as introduce a new family of examples. In particular, there are such curves with multiplicity 11, and curves having any given multiplicity over 25 (with 12 possible exceptions).

Multiplier ideals of monomial space curves

This paper presents a formula for the multiplier ideals of a monomial space curve. The formula is obtained from a careful choice of log resolution. We construct a toric blowup of affine space in such a way that a log resolution of the monomial curve may be constructed from this toric variety in a well controlled manner. The construction exploits a theorem of González Pérez and Teissier (2002).

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).

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.

Log canonical thresholds of binomial ideals

We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming.

On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves

In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves ( t a , t b , t c ) for pairwise coprime integers a, b, c such that ( a , b , c ) ≠ ( 1 , 1 , 1 ) . If such a ring is not finitely generated over a base field, then it is a counterexample to the Hilbert's fourteenth problem. Finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces. We study a sufficient condition (Definition 3.6) for existence of a negative curve. Using it, we prove that, in the case of ( a + b + c ) 2 > a b c , a negative curve exists. Using a computer, we shall show that there exist examples in which this sufficient condition is not satisfied.

On the H -Polynomial of Certain Monomial Curves

Let n1, . . . , ne be an increasing sequence of positive integers with gcd (n1, . . . , ne) = 1 and let A be the coordinate ring of the algebroid monomial curve in the affine algebroid e-space AK over a field K, defined parametrically by X1 = tn1 , . . . , Xe = tne . In this article assuming that some e − 1 terms of n1, . . . , ne form an arithmetic sequence, we compute (under some mild additional assumptions, see Theorem (2.7) for more precise assumptions) the h-polynomial (and hence the Hilbert function) of A explicitly in terms of the standard basis of the semi-group generated by n1, . . . , ne. Our special assumptions are satisfied in the case e = 3; in particular, for the class of algebroid monomial space curves, we can write down the h-polynomial and hence the Hilbert function explicitly.

ON THE TYPE OF THE ASSOCIATED GRADED RING OF CERTAIN MONOMIAL CURVES

ABSTRACT Let be positive integers, , and let n be positive integer such that and . Let A be the coordinate ring of the algebroid monomial curve in the affine algebroid -space over a field K, defined parametrically by In this article assuming that the associated graded ring of A is Cohen-Maucaulay (and some more mild additional assumptions, see (2.4)), we give an explicit formula for the type of in terms of the standard basis of the semigroup generated by the almost arithmetic progression . Our special assumptions are satisfied if , that is, for the class of algebroid monomial space curves.

Non-Cohen-Macaulay Symbolic Blow-Ups for Space Monomial Curves and Counterexamples to Cowsik's Question

Non-Cohen-Macaulay Symbolic Blow-Ups for Space Monomial Curves and Counterexamples to Cowsik's Question

Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question

Let A = k [ [ X , Y , Z ] ] A = k[[X,Y,Z]] and k [ [ T ] ] k[[T]] be formal power series rings over a field k k , and let n ⩾ 4 n \geqslant 4 be an integer such that n ≢ 0 mod 3 n\not \equiv 0\;\bmod \;3 . Let φ : A → k [ [ T ] ] \varphi :A \to k[[T]] denote the homomorphism of k k -algebras defined by φ ( X ) = T 7 n − 3 , φ ( Y ) = T ( 5 n − 2 ) n \varphi (X) = {T^{7n - 3}},\;\varphi (Y) = {T^{(5n - 2)n}} , and φ ( Z ) = T 8 n − 3 \varphi (Z) = {T^{8n - 3}} . We put p = Ker φ {\mathbf {p}} = \operatorname {Ker} \,\varphi . Then R s ( p ) = ⊕ i ⩾ 0 p ( i ) {R_s}({\mathbf {p}}) = { \oplus _{i \geqslant 0}}{{\mathbf {p}}^{(i)}} is a Noetherian ring if and only if ch k > 0 \operatorname {ch} \,k > 0 . Hence on Cowsik’s question there are counterexamples among the prime ideals defining space monomial curves, too.

The Gorensteinness of the Symbolic Blow-Ups for Certain Space Monomial Curves

The Gorensteinness of the Symbolic Blow-Ups for Certain Space Monomial Curves

The Gorensteinness of the symbolic blow-ups for certain space monomial curves

Let p = p ( n 1 , n 2 , n 3 ) {\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3}) denote the prime ideal in the formal power series ring A = k [ [ X , Y , Z ] ] A = k[[X,Y,Z]] over a field k k defining the space monomial curve X = T n 1 X = {T^{{n_1}}} , Y = T n 2 Y = {T^{{n_2}}} , and Z = T n 3 Z = {T^{{n_3}}} with GCD ( n 1 , n 2 , n 3 ) = 1 {\text {GCD}}({n_1},{n_2},{n_3}) = 1 . Then the symbolic Rees algebras R s ( p ) = ⊕ n ≥ 0 p ( n ) {R_s}({\mathbf {p}}) = { \oplus _{n \geq 0}}{{\mathbf {p}}^{(n)}} are Gorenstein rings for the prime ideals p = p ( n 1 , n 2 , n 3 ) {\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3}) with min { n 1 , n 2 , n 3 } = 4 \min \{ {n_1},{n_2},{n_3}\} = 4 and p = p ( m , m + 1 , m + 4 ) {\mathbf {p}} = {\mathbf {p}}(m,m + 1,m + 4) with m ≠ 9 , 13 m \ne 9,13 . The rings R s ( p ) {R_s}({\mathbf {p}}) for p = p ( 9 , 10 , 13 ) {\mathbf {p}} = {\mathbf {p}}(9,10,13) and p = p ( 13 , 14 , 17 ) {\mathbf {p}} = {\mathbf {p}}(13,14,17) are Noetherian but non-Cohen-Macaulay, if ch k = 3 \operatorname {ch}\,k = 3 .