Set-theoretic Complete Intersection Research Articles

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

Distinguished varieties in a family of domains associated with spectral interpolation and operator theory

We find characterization for the distinguished varieties in the symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize the work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on $\mathbb G_2$ by the author and Shalit. We show that a distinguished variety $\Lambda$ in $\mathbb G_n$ is a part of an algebraic curve, which is a set-theoretic complete intersection, and that $\Lambda$ can be represented by the Taylor joint spectrum of $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple of commuting Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ for which $\Gamma_n=\overline{\mathbb G_n}$ is a spectral set is called a $\Gamma_n$-contraction. To every $\Gamma_n$-contraction $(S_1, \dots ,S_{n-1},P)$ there is a unique operator tuple $(F_1, \dots , F_{n-1})$, called the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1},P)$, satisfying \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concrete functional model for the pure isometric-operator tuples associated with $\Gamma_n$ and by an application of that model we establish that the $\Gamma_n$-contractions $(S_1, \dots ,S_{n-1},P)$ and $(S_1^*, \dots , S_{n-1}^*,P^*)$ admit normal $\partial \overline{ \Lambda}_{\Sigma}-$dilations for a unique distinguished variety $\Lambda_{\Sigma}$ in $\mathbb G_n$, when $\Lambda_{\Sigma}$ is determined by the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1}, P)$. Further, we show that the dilation of $(S_1^*, \dots ,S_{n-1}^*,P^*)$ is minimal and acts on the minimal unitary dilation space of $P^*$. Also, we show interplay between the distinguished varieties in $\mathbb G_2$ and $\mathbb G_{3}$.

Hankel Edge Ideals of Trees and (Semi-)Hamiltonian Graphs

In this paper, we study the Hankel edge ideals of graphs. We determine the minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and semi-Hamiltonian graphs, and we investigate radicality, being a complete intersection, almost complete intersection and set-theoretic complete intersection for such graphs. We also consider the Hankel edge ideal of trees with a natural labeling, called rooted labeling. We characterize such trees whose Hankel edge ideal is a complete intersection, and moreover, we determine those whose initial ideal with respect to the reverse lexicographic order satisfies this property.

Quasi-cyclic modules and coregular sequences

We develop the theory of coregular sequences and codepth for modules that need not be finitely generated or artinian over a Noetherian ring. We use this theory to give a new version of a theorem of Hellus characterizing set-theoretic complete intersections in terms of local cohomology modules. We also define quasi-cyclic modules as increasing unions of cyclic modules, and show that modules of codepth at least two are quasi-cyclic. We then focus our attention on curves in \({\mathbb {P}}^3\) and give a number of necessary conditions for a curve to be a set-theoretic complete intersection. Thus an example of a curve for which any of these necessary conditions does not hold would provide a negative answer to the still open problem, whether every connected curve in \({\mathbb {P}}^3\) is a set-theoretic complete intersection.

On a torsion analogue of the weight-monodromy conjecture

We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.

CRITICAL BINOMIAL IDEALS OF NORTHCOTT TYPE

AbstractIn this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

SYMBOLIC ANALYTIC SPREAD: UPPER BOUNDS AND APPLICATIONS

AbstractThe symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article, we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$. Our methods also work for more general systems of ideals. As applications, we provide bounds for the (local) Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity. We also show sufficient conditions for an ideal to be a set-theoretic complete intersection.

PROPERTIES OF $T$-SPREAD PRINCIPAL BOREL IDEALS GENERATED IN DEGREE TWO

In this paper, we have studied the stability of $t$-spread principal Borel ideals in degree two. We have proved that $\Ass^\infty(I) =\Min(I)\cup \{\mathfrak{m}\}$ , where $I=B_t(u)\subset S$ is a $t$-spread Borel ideal generated in degree $2$ with $u=x_ix_n, t+1\leq i\leq n-t.$ Indeed, $I$ has the property that $\Ass(I^m)=\Ass(I)$ for all $m\geq 1$ and $i\leq t,$ in other words, $I$ is normally torsion free. Moreover, we have shown that $I$ is a set theoretic complete intersection if and only if $u=x_{n-t}x_n$. Also, we have derived some results on the vanishing of Lyubeznik numbers of these ideals.

Cohomological dimension and top local cohomology modules

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this paper, we first determine a condition under which a given integer $t$ is a lower bound for the cohomological dimension $\operatorname {cd}(I,M)$, and use this to conclude that non-catenary Noetherian domains contain prime ideals that are not set-theoretic complete intersection. We also show the existence of a descending chain of ideals with successive diminishing cohomological dimensions. We then resolve the Artinianness of top local cohomology modules over local unique factorization domains of Krull dimension at most three, and obtain several related results on the top local cohomology modules for much more general cases.

On a conjecture of Murthy

This article concerns Murthy's conjecture on complete intersections, made in 1975. The sole breakthrough on this conjecture has still been the result proved by Mohan Kumar in 1978. The conjecture is open in general. In this article we improve Mohan Kumar's bound when the base field is F‾p. As an application, we prove that any local complete intersection surface in the affine space AF‾pd is a set-theoretic complete intersection, generalizing a result of Bloch–Murthy–Szpiro.

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.

Almost Gorenstein monomial curves in affine four space

In this paper, we classify the almost symmetric semigroups of embedding dimension four. And we give minimal systems of generators and minimal free resolutions for the defining ideals of almost Gorenstein monomial curves in affine four space. Finally we conclude that they are set-theoretic complete intersections.

The arithmetical rank of the edge ideals of cactus graphs

ABSTRACTWe prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases, it can be improved. Moreover, we show that the edge ideal of a Cohen–Macaulay graph that contains exactly one cycle or is chordal or has no cycles of length 4 and 5 is a set-theoretic complete intersection.

A Note on Cohen–Macaulay Graphs

We show that the edge ideal of a Cohen–Macaulay graph on 2n non-isolated vertices, whose height is n, is always a set-theoretic complete intersection. This result, in particular, applies to all Cohen–Macaulay bipartite graphs.

A Result on Macaulay's Curve

It is not known whether Macaulay's curve is a set-theoretic complete intersection or not in characteristic zero. There are known (weak) indications that the answer is negative; clearly, a negative such answer would provide the first example of an irreducible non-set-theoretic complete intersection curve, i.e., of a curve in affine or projective n-space that cannot be cut out by n − 1 polynomial equations.We prove new necessary conditions for two (assumed) homogeneous polynomials cutting out C4 set-theoretically. We use local cohomology and an idea from Thoma.

Star configurations are set-theoretic complete intersections

Let $\mathcal A\subset\mathbb P^{k-1}$ be a rank $k$ arrangement of $n$ hyperplanes, with the property that any $k$ of the defining linear forms are linearly independent (i.e., $\mathcal A$ is called $k-$generic). We show that for any $j=0,\ldots,k-2$, the subspace arrangement with defining ideal generated by the $(n-j)-$fold products of the defining linear forms of $\mathcal A$ is a set-theoretic complete intersection, which is equivalent to saying that star configurations have this property.

New Classes of Set-theoretic Complete Intersection Monomial Ideals

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δχ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δχ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δχ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.

Equations Defining Recursive Extensions as Set Theoretic Complete Intersections

Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.

Arithmetical rank of Gorenstein squarefree monomial ideals of height three

We prove that a squarefree monomial ideal of height 3 whose quotient ring is Gorenstein is a set-theoretic complete intersection.

SET-THEORETIC COMPLETE INTERSECTIONS ON BINOMIALS, THE SIMPLICIAL TORIC CASE

Let V be a simplicial toric variety of codimension r over a field of any characteristic. We completely characterize the implicial toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that: 1. In characteristic zero, V is a set-theoretic complete intersection on binomials if and only jf V is a. complete intersection. Moreover, if F 1 ,…,F r ; are binomials such that I(V)= rad( F 1 , . .. ,F r ), th en I(V) = (F 1 , ... ,F r ) . We also get a geometric proof of some of the results in [9] characterizing complete intersections by gluing; semigroups. 2. In positive characteristic p, V is a set-theoretic complete intersection on binomials if and only if V is complete 1y p-glued. These results improve and complete all known results on these topics.