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

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, by Stanley (1989). Of course even less is known about Betti numbers. There are some general results on the Betti table by Pardue and Richert (2009), and by Diem (2015). Then there are results on Betti numbers in the case $n+1$ relations in $n$ variables, described below, by Migliore and Miró-Roig (2003). In this paper we consider the same case as  Migliore and Miró-Roig did, $n+1$ 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 $\sum_{i=1}^{n+1}\deg(g_i)-n$ is even, often in more cases; see Theorem 4.5. Thus, given {\em any} set $\{ e_1,\ldots,e_n\}$, $e_i\ge2$ for all $i$, such that $\deg(g_i)=e_i$, $i=1,\ldots,n$, we get many numbers $e_{n+1}$, so that we can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $\deg(g_i)=e_i$, $1\le i\le n+1$. When we say that the Betti numbers could be determined, we don't claim that we can give formulas for them. We mean that the Betti numbers can be calculated from the Hilbert series, which is known, and the Koszul complex on $g_1,\ldots,g_{n+1}$, which is well known and easy to get hold of. The main ingredients of the proof is a theorem of Pardue and Richert (2009), and of Diem (2015), together with a theorem on Hilbert function of artinian complete intersections by Reid, Roberts, and Roitman (1991). A second goal of this article is to give a short elementary proof of the theorem by Reid, Roberts, and Roitman; see Theorem 3.3.