Subsets of Complete Intersections and the EGH Conjecture

Abstract

We give a "Macaulay-type" characterization of the Hilbert functions of finite sets of distinct points which are subsets of complete intersections in projective space P 2 and a family of complete intersections in P 3. Doing so, we prove that the Hilbert functions of subsets of complete intersections of type ¹d 1 ; d 2 º; ¹2; d; d º and ¹3; d; d º are the same as those for rectangular complete intersections of the same type. This verifies special cases of the Eisenbud-Green-Harris Conjecture, which is a statement that lex-plus-powers ideals exhibit extremal conditions among all homogeneous ideals containing a homogeneous regular sequence in fixed degrees. The characterization is then applied to give a family of points which has the Cayley-Bacharach Property.

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…”

“…It is worth noticing that, although the two statements are apparently independent of each other, Conjecture 1.1 is actually equivalent to the special case i = 0 of Conjecture 1.2, see e.g. [ [1,3,8,10,17,20,27] for other special cases. On the other hand, much less is known about Conjecture 1.2, cf.…”