Some cases of the Eisenbud-Green-Harris conjecture

Abstract

The Eisenbud-Green-Harris conjecture states that a homogeneous ideal in k[x 1 , . . . , x n ] containing a homogeneous regular sequence f 1 , . . . , f n with deg(f i ) = a i has the same Hilbert function as an ideal containing x ai i for 1 ≤ i ≤ n. In this paper we prove the Eisenbud-Green-Harris conjecture when a j > j−1 i=1 (a i − 1) for all j > 1. This result was independently obtained by the two authors.

On a conjecture by Kalai

“…. , x 2 g ) with the same Hilbert function of I ′ by [Ab,Corollary 4.3] and [CM,Proposition 10]. Clearly pd(J ′ ) ≤ p − 1, but we can actually choose J ′ such that pd(J ′ ) = p − 1 by [CS,Theorem 4.4].…”

“…. , r ( [CM]) and when each f i factors as product of linear forms ( [Ab,Corollary 4.3] for the case r = n, and [Ab] together with [CM,Proposition 10] for the general case).…”

A Proof of Evans’ Convexity Conjecture

“…A version of EGH regarding weakly lex-plus-powers ideals (and the version which has received much recent attention, for example [1]) is as follows.…”

On the Lex-plus-powers Conjecture

“…where A = k[x 0 , x 1 , x 2 , x 3 ] is the homogeneous coordinate ring of P 3 . By going modulo a general linear form in A/I Γ , we reduce to considering Artinian algebras R = S/I where S = k[x 1 , x 2 , x 3 ] with HF(R) = (1,3,6,10,12,12,12,12,11,9,6,2) and I contains a regular sequence of degrees (4,4,8…”