Almost complete intersections and the Lex-Plus-Powers Conjecture

We prove the almost complete intersection case of the Lex-Plus-Powers Conjecture on graded Betti numbers. We show that the resolution of a lex-plus-powers almost complete intersection provides an upper bound for the graded Betti numbers of any other ideal with regular sequence in the same degrees and the same Hilbert function. A key ingredient is finding an explicit comparison map between two Koszul complexes. Finally, we obtain bounds on the Hilbert function of an almost complete intersection, including a special case of a conjecture of Eisenbud–Green–Harris.