On the minimal free resolution of $n+1$ generic forms

Abstract

We consider the minimal free resolution of a generic set of n+1 forms (not necessarily of the same degree) in a polynomial ring of n variables. The Hilbert function for such an ideal is known, thanks to a result of Stanley and of Watanabe. (For more than n+1 forms the Hilbert function is conjectured by Froberg.) However, in all but a few trivial cases, the resolution is not known. A conjecture of Iarrobino says that apart from Koszul syzygies, there should be no overlaps in the twists occurring in the free modules in the resolution. We are able to find the precise minimal free resolutions in many cases (depending on n and on the particular degrees) and give good bounds in many other cases. As a result, we see that Iarrobino's conjecture does not hold and we begin to have an understanding of what causes this failure. We conjecture that there are no such overlaps when all n+1 generators have the same degree. In the process of achieving these results, we extend a result of Boij by giving the graded Betti numbers of a generic Gorenstein algebra when n is even and the socle degree is large.