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Register a new userOn almost revlex ideals with Hilbert function of complete intersections
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In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field $K$, we give a new constructive proof of the existence of the almost revlex ideal $Jsubset K[x_1,dots,x_n]$, with the same Hilbert function as a complete intersection defined by $n$ forms of degrees $d_1leq dots leq d_n$. Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We also detect several cases in which an almost revlex ideal having the same Hilbert function as a complete intersection corresponds to a singular point in a Hilbert scheme. This second result is the outcome of a more general study of lower bounds for the dimension of the tangent space to a Hilbert scheme at stable ideals, in terms of the number of minimal generators.
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The Hilbert function of standard graded algebras are well understood by Macaulays theorem and very little is known in the local case, even if we assume that the local ring is a complete intersection. An extension to the power series ring $R$ of the theory of Gr{o}bner bases (w.r.t. local degree orderings) enable us to characterize the Hilbert function of one dimensional quadratic complete intersections $A=R/I$, and we give a structure theorem of the minimal system of generators of $I$ in terms of the Hilbert function. We find several restrictions for the Hilbert function of $A$ in the case that $I$ is a complete intersection of type $(2,b). $ Conditions for the Cohen-Macaulyness of the associated graded ring of $A$ are given.
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(ntimes n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+langle f rangle$ for an explicit polynomial $f$ constructed from the fibers of the Veronese grading map.
For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.
We give an explicit formula for the Hilbert-Poincar{e} series of the parity binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal generated by all $2times 2$-permanents of a $2times n$-matrix. It follows that the depth and Castelnuovo-Mumford regularity of these ideals are independent of $n$.
Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the binomial edge ideal of $G$ in the polynomial ring $S = mathbb{K}[x_1, ldots, x_n, y_1, ldots, y_n].$ In this article, we compute the second graded Betti numbers of $J_G$, and we obtain a minimal presentation of it when $G$ is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a $d$-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.
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