Given a sequence of degrees A = {a(1), ..., a(n)} with 1 <= a(1) <= a(2) <= ... <= a(n), an ideal L is called an A lex-plus-powers ideal if L minimally contains (x(1)(a1), ..., x(n)(an)) and is otherwise as lex as possible. Such ideals are conjectured by Eisenbud, Green, and Harris, to exhibit extremal Hilbert function growth (among all ideals also containing a regular sequence in degrees A), and by Charalambous and Evans to exhibit maximal graded Betti numbers (among all ideals containing a regular sequence in degrees A and attaining the same Hilbert function). Of course, given a sequence of degrees A and a Hilbert function H, an A lex-plus-powers ideal with Hilbert function H may or may not exist. Evans has conjectured that the existence of such ideals is governed by a certain convexity property: that is, given a Hilbert function H and three sequences of degrees A <= B <= L where L gives the degrees of the pure powers in the lex ideal attaining H, if there exists an A lex-plus-powers ideal attaining H, then there exists a B lex-plus-powers ideal attaining H. In this paper we give a proof of Evans' Convexity Conjecture. As an application, we use Convexity to show that the conjectures of Eisenbud, Green, and Harris, and Charalambous and Evans are equivalent to similar statements utilizing a definition of lex-plus-powers ideals that does not require minimal containment of x(1)(a1), ..., x(n)(an) ).
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