Some new Betti numbers of ideals generated by n+1 generic forms in n variables
Ralf Fröberg
TL;DR
The paper investigates Betti numbers for ideals generated by $n+1$ generic forms in $n$ variables, showing that in many cases the entire graded Betti table is determined from the Hilbert series and the Koszul complex, especially when the socle degree $\sum_{i=1}^{n+1} e_i - n$ is even. It leverages results of Pardue–Richert and Diem, along with a new short proof of a Reid–Roberts–Roitman theorem, to describe when the Hilbert series suffices to fix Betti numbers. It also demonstrates ghost-term phenomena with explicit examples where additional syzygies beyond the Koszul complex occur. The work broadens understanding of the relationship between Hilbert series, Koszul data, and Betti numbers for generic complete intersections and highlights subtleties in predicting Betti tables from Hilbert data alone.
Abstract
Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ 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 $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $g_i$ generic, at least if $\sum_{i=1}^{n+1}°(g_i)-n$ is even, often in more cases. Thus, given {\em any} set $\{ e_1,\ldots,e_n\}$, $e_i\ge2$ for all $i$, such that $°(g_i)=e_i$, $i=1,\ldots,n$, we get many numbers $D_j$, so that we can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $°(g_i)=e_i$, $1\le i\le n$, $°(g_{n+1})=D_j$. 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.