Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userVeronesean almost binomial almost complete intersections
106
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
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.
Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by $Bbbk[K_{r_1,ldots,r_n}]$ with $1 leq r_1 leq cdots leq r_n$. We give the complete characterization of which $Bbbk[K_{r_1,ldots,r_n}]$ is level in terms of $n$ and $r_1,ldots,r_n$. Similarly, we also give the complete characterization of which $Bbbk[K_{r_1,ldots,r_n}]$ is almost Gorenstein in terms of $n$ and $r_1,ldots,r_n$.
We settle a conjecture by Migliore, Miro-Roig, and Nagel which gives a classification of the Weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
