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

Abstract

O-ROIG ⁄⁄ Abstract. Let R = k(x1;:::;xn) and let I be the ideal of n + 1 generically chosen forms of degrees d1 • ¢¢¢ • dn+1. We give the precise graded Betti numbers of R=I in the following cases: † n = 3. † n = 4 and P5=1 di is even. † n = 4, P 5=1 di is odd and d2 + d3 + d4 < d1 + d5 + 4. † n is even and all generators have the same degree, a, which is even. † ( Pn+1 i=1 di) i n is even and d2 + ¢¢¢ + dn < d1 + dn+1 + n. † ( P n+1 i=1 di) i n is odd, n ‚ 6 is even, d2 + ¢¢¢ + dn < d1 + dn+1 + n and d1 + ¢¢¢ + dn i dn+1 i n ? 0. We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e. one for which the Hilbert function is maximal given n and the socle degree) when n is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree then there are no redundant summands, and we present some evidence for this conjecture.