Maximal minors and linear powers

Abstract

ABSTRACT. An ideal I in a polynomial ring S has linear powers if all the powers I k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.

Blowup algebras of rational normal scrolls

“…This confirms the fact that the Veronese surface exhibits an exceptional behavior in various contexts, see for instance [26,Section 3.4], [9,Example 3.5]. It also gives the first known example of a prime ideal with linear powers [4] whose Rees ring is not Koszul.…”

“…. We combine the results on ∆ with those from [4,10] to prove the main theorem of this section. , and they form a squarefree quadratic Gröbner basis with respect to the term order ≺.…”

“…, f s ] ⊆ R, known as the special fiber ring of I. They are also known as blowup algebras of I, since BiProj(R(I)) is the blowup of P n along the subscheme defined by I. Blowup algebras are central objects in the study of rational maps [13,18,27,29], Castelnuovo-Mumford regularity [4,8,12,25], geometric modeling [6,11], and several other areas such as linkage, residual intersections, multiplicity theory, integral dependence, singularities, and combinatorics.…”

When does depth stabilize early on?

“…(see [BCV,Remark 3.2] for the explicit proof). So, accordingly to the characteristic, the remaining assumption of Theorem 3.4 has to fail.…”

Note on the Regularity of the Radical of Ideals

“…It is proven in [3] that the ideal I(V ) of an irreducible variety V of minimal degree has linear powers. By the Nullstellensatz, we are assuming that √ I = I(V ) for a minimal variety V .…”