Linearly presented perfect ideals of codimension 2 in three variables

Abstract

The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the common generic properties. One exception is a recent work of N. P. H. Lan which inspired the present work. Here we recover and extend his work. We strongly focus on the behavior of the ideals of minors of the corresponding so-called Hilbert-Burch matrix and on conjugation features of the latter. We apply the results to three important models: linearly presented ideals of plane fat points, reciprocal ideals of hyperplane arrangements and linearly presented monomial ideals.

Determinacy of determinantal varieties

“…Linear determinantal varieties have been intensively studied by many authors, see for instance [6], [8] and [13]. We discuss here a method to get examples of homogeneous EIDS of arbitrarily high degree from a special class of linear EIDS.…”

On ideals generated by fold products of linear forms

“…This result is also recovered in [10], by the means of Rees algebra, and special fiber related results. With this late approach, in the spirit of [5], there is the hope that one can prove that OT (2, A) is also Cohen-Macaulay, at least for the case when k = 3 (in three variables). Indeed, I n−2 (A) is linearly presented (from Theorem 2.4), and it is generated by the maximal minors of a (n − 2) × n matrix with linear forms entries (see the last paragraphs of the proof of [19,Proposition 2.1]).…”

“…Indeed, I n−2 (A) is linearly presented (from Theorem 2.4), and it is generated by the maximal minors of a (n − 2) × n matrix with linear forms entries (see the last paragraphs of the proof of [19,Proposition 2.1]). But the ideals I ⊂ A := K[x, y, z] considered in [5] are perfect ideals, causing for A/I to be Cohen-Macaulay. In our situation, if k = 3, R/I n−2 (A) is Cohen-Macaulay if and only if A is a rank 3 generic hyperplane arrangement (see Corollary 2.5).…”

“…This can be seen from the discussions at the end of the proof of [19,Proposition 2.1], and immediately after it; the point there is that the Eagon-Northcott complex becomes the desired linear graded free resolution of R/I a (Σ) (see [6,Theorem A2.60]). (5) If no two of the linear forms of Σ are proportional, then from part (4), I n−1 (Σ), so a = n − 1, has linear graded free resolution. (6) If Σ = ℓ , for some ℓ ∈ R 1 , then for any 1 ≤ a ≤ n, we have I a (Σ) = ℓ a , which, as any principal ideal, has linear graded free resolution.…”