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 C[x₁,…,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 C[x₁,…,xₙ]/(g₁,…,gₙ₊₁), gᵢ generic, at least if ∑ᵢ₌₁ⁿ⁺¹°(gᵢ)-n is even, often in more cases. Thus, given { any} set \ e₁,…,eₙ\, eᵢ≥2 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 C[x₁,…,xₙ]/(g₁,…,gₙ₊₁), °(gᵢ)=eᵢ, 1≤ i≤ n, °(gₙ₊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.