Simplicial ideals, 2-linear ideals and arithmetical rank

TL;DR

This paper investigates the algebraic and geometric properties of linearly joined varieties, introduces simplicial ideals, and provides algorithms for computing arithmetical rank, with applications to Cohen--Macaulay varieties and monomial ideals.

Contribution

It characterizes linearly joined ideals, introduces simplicial ideals, and develops algorithms for computing arithmetical rank, extending understanding of these algebraic structures.

Findings

Computed depth and cohomological dimension of linearly joined ideals

Provided an effective algorithm to find equations and compute arithmetical rank

Characterized ideals defining unions of linear spaces using Ferrer tableaux

Abstract

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying the fiber cone of of a codimension two toric ideals. Let $I\subset S$ be an ideal defining a linearly joined arrangement of varieties: - We compute the depth, and the cohomological dimension. is the connectedness dimension. - We characterize sets of generators of $I$, and give an effective algorithm to find equations, as an application we compute arithmetical rank. in the case if $I$ defines a union of linear spaces, (ara =projective dimension), in particular this applies to any square free monomial ideal having a $2-$ linear resolution. - In the case where $V$ is a union of linear spaces, the ideal $I$, can be characterized by a tableau, which is an extension of a Ferrer (or Young) tableau. - We introduce a new class of ideals called simplicial ideals, ideals defining linearly-joined varieties are a particular case of simplicial ideals.