An application of liaison theory to the Eisenbud–Green–Harris conjecture

A Cayley–Bacharach theorem for points in Pn

“…One more case in which the conjecture is known is for the class of minimally licci ideals, defined by Chong [8]. Chong proves that, if g ⊆ S is an ideal of height three which contains a regular sequence of degrees d among its minimal generators, and such that S/g is Gorenstein, then g satisfies EGH d,n .…”

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.…”

“…Let Ω ⊆ P 3 be a 0-dimensional complete intersection of degrees(4,4,8). Let Γ ⊆ Ω be a closed subscheme with Hilbert function HF(A/I Γ ) =(1, 4, 10, 20, 32, 44, 56, 68, 79, 88, 94, 96, 96, 96, .…”

An Application of Liaison Theory to Zero-dimensional Schemes

“…This idea has been also generalized by allowing links by arithmetically Gorenstein schemes (see, e.g., [22]). Currently, Liaison theory is an area of active research [2-5, 7, 12, 23-25], and has many useful applications, for instance, constructing interesting projective varieties [2,24,25], or computing invariants and establishing properties of projective varieties [3][4][5]9].…”