On the minimal free resolution of 𝑛+1 general forms

Source abstract

Let R = k [ x 1 , … , x n ] R = k[x_1,\dots ,x_n] and let I I be the ideal of n + 1 n+1 generically chosen forms of degrees d 1 ≤ ⋯ ≤ d n + 1 d_1 \leq \dots \leq d_{n+1} . We give the precise graded Betti numbers of R / I R/I in the following cases: n = 3 n=3 ; n = 4 n=4 and ∑ i = 1 5 d i \sum _{i=1}^5 d_i is even; n = 4 n=4 , ∑ i = 1 5 d i \sum _{i=1}^{5} d_i is odd and d 2 + d 3 + d 4 > d 1 + d 5 + 4 d_2 + d_3 + d_4 > d_1 + d_5 + 4 ; n n is even and all generators have the same degree, a a , which is even; ( ∑ i = 1 n + 1 d i ) − n (\sum _{i=1}^{n+1} d_i) -n is even and d 2 + ⋯ + d n > d 1 + d n + 1 + n d_2 + \dots + d_n > d_1 + d_{n+1} + n ; ( ∑ i = 1 n + 1 d i ) − n (\sum _{i=1}^{n+1} d_i) - n is odd, n ≥ 6 n \geq 6 is even, d 2 + ⋯ + d n > d 1 + d n + 1 + n d_2 + \dots +d_n > d_1 + d_{n+1} + n and d 1 + ⋯ + d n − d n + 1 − n ≫ 0 d_1 + \dots + d_n - d_{n+1} - n \gg 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 n and the socle degree) when n 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.