Square-free Gröbner degenerations

Abstract

Let I be a homogeneous ideal of S = K[x 1 , . . . , x n ] and let < be a term order. We prove that if the initial ideal J = in < (I) is radical then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I) = depth(S/J) and reg(S/I) = reg(S/J).

Simplicial complexes of small codimension

“…So, somehow Buchbaum-ness plays the role of 'smooth-ness' for simplicial complexes. This way of thinking is also supported from the results in the recent paper [2], which imply that, if the ideal defining a smooth projective variety has a square-free Gröbner degeneration, then the associated simplicial complex is Buchsbaum. With this definition in mind, the same philosophy that led Hartshorne to make his conjecture brings one to expect the following: If ∆ is a Buchbaum simplicial complex with small codimension, then K[∆] should have large depth.…”

“…The following corollary is in the spirit of Hartshorne's conjecture and goes in the direction of a question raised in [2,Question 4.2]. Corollary 3.6.…”

Singularities and radical initial ideals

“…We have two reasons for doing this. On the one hand, having a square‐free initial ideal is a desirable property which better preserves homological invariants under degeneration [10]. For example, extremal Betti numbers stay constant (in value and position), and thus also depth and regularity, so Cohen–Macaulayness is passed on.…”

Freiman ideals

“…The concept of Freiman ideals appeared the first time in [15]. Based on a famous theorem from additive number theory, due to Freiman [7], it was proved in [14] that if I ⊂ S is an equigenerated monomial ideal, then µ(I 2 ) ≥ ℓ(I)µ(I) − ℓ(I) 2 . Here µ(I) denotes the least number of generators of the ideal I and ℓ(I) the analytic spread of I.…”