On the Lex-plus-powers Conjecture

Abstract

Let S be a polynomial ring over a field and I ⊆ S a homogeneous ideal containing a regular sequence of forms of degrees d 1 , . . . , dc. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that d i ≥ i−1 j=1 (d j − 1) + 1 for each i; that is, we show that the Betti table of I is bounded above by the Betti table of the lex-plus-powers ideal of I.

F. S. Macaulay: From plane curves to Gorenstein rings

“…Interesting modern extensions of this result can be found in the works of Caviglia and many others; see for example [CS18]. The related case of ideals in an exterior algebra is the subject of the Kruskal-Katona theorem, fundamental in algebraic combinatorics; see for example [GK78].…”

Syzygies in Hilbert schemes of complete intersections

“…Note that both Exp HP R/E η + 1, R and Exp HP R/E χ − 1, R exist by induction, since the corresponding Hilbert schemes are nonempty. For the former, by Proposition 2.5 (10) there exists some ideal J ⊆ E h with HP R/J = HP R/E h + 1. For the latter, the Hilbert scheme of q(ζ) is nonempty by assumption.…”

“…Now let I = Lex( E) ⊆ R and decompose I = ⊕ dn−1 ℓ=0 I ℓ x ℓ n ⊆ R. The recursive criterion for lex ideals in Clements-Lindström rings proved in [8, Proof of Theorem 3.3, Lemma 3.7, Lemma 3.8], cf. also [10,Remark 3.1], implies that I ℓ is lex for every ℓ and that the following inequalities hold for every ρ, τ ≥ 0 (6.1)…”

A Survey on the Eisenbud-Green-Harris Conjecture

“…In [CS18], when characteristic of K is 0, it was shown that the LPP conjecture holds for the homogeneous ideals containing a regular sequence with degrees 2 ≤ a 1 ≤ . .…”