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 $\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(g_1,\dots ,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, is for $n+1$ forms in $n$ variables, \cite{St}. Of course even less is known about Betti numbers. There are some general results on the Betti table by Pardue and Richert in \cite{Pa-Ri,Pa-Ri1}, and by Diem in \cite{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 \cite{Mi-Mi}, and more partial results in the general case by the same authors in \cite{Mi-Mi1}. In this paper we consider the same case as in \cite{Mi-Mi}, $n+1$ forms in $n$ variables. Our results can be described as follows. We can determine all graded Betti numbers of $\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(g_1,\dots ,g_{n+1})$, $g_i$ generic, at least if ${\sum }_{i=1}^{n+1}\mathrm{°}(g_i)-n$ is even, often in more cases. Thus, given {\em any} set $\{e_1,\dots ,e_n\}$, $e_i\ge 2$ for all $i$, such that $\mathrm{°}(g_i)=e_i$, $i=1,\dots ,n$, we get many numbers $D_j$, so that we can determine all graded Betti numbers of $\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(g_1,\dots ,g_{n+1})$, $\mathrm{°}(g_i)=e_i$, $1\le i\le n$, $\mathrm{°}(g_{n+1})=D_j$. The main ingredients of the proof is a theorem by Pardue and Richert, \cite{Pa-Ri,Pa-Ri1}, and later by Diem,\cite{Di}, and a new short proof of a theorem on Hilbert series of artinian complete intersections by Reid, Roberts, and Roitman, \cite{R-R-R}. We also give examples of algebras with many so called “ghost terms” in the minimal resolution.