Some new Betti numbers of ideals generated by n+1 generic forms in n variables

Very little is known on the Hilbert series of graded algebras Cx₁, , xₙ/ (g₁, , gᵣ), r>n, gᵢ generic form of degree eᵢ, in general. One instance when the series is known, is for n+1 forms in n variables, St. Of course even less is known about Betti numbers. There are some general results on the Betti table by Pardue and Richert in Pa-Ri, Pa-Ri1, and by Diem in Di. Then there are results on Betti numbers in the case n+1 relations in n variables, described below, by Migliore and Mirò-Roig in Mi-Mi, and more partial results in the general case by the same authors in Mi-Mi1. In this paper we consider the same case as in Mi-Mi, n+1 forms in n variables. Our results can be described as follows. We can determine all graded Betti numbers of Cx₁, , xₙ/ (g₁, , g₍+₁), gᵢ generic, at least if ₈=₁^n+1° (gᵢ) -n is even, often in more cases. Thus, given any set \ e₁, , eₙ\, eᵢ2 for all i, such that ° (gᵢ) =eᵢ, i=1, , n, we get many numbers Dⱼ, so that we can determine all graded Betti numbers of Cx₁, , xₙ/ (g₁, , g₍+₁), ° (gᵢ) =eᵢ, 1 i n, ° (g₍+₁) =Dⱼ. The main ingredients of the proof is a theorem by Pardue and Richert, Pa-Ri, Pa-Ri1, and later by Diem, Di, and a new short proof of a theorem on Hilbert series of artinian complete intersections by Reid, Roberts, and Roitman, R-R-R. We also give examples of algebras with many so called "ghost terms" in the minimal resolution.