On the minimal free resolution of n+1 general forms

• Published in: Transactions of the American Mathematical Society 2003-01, Vol.355 (1), p.1-36
• Main Authors: Migliore, J., Miró-Roig, R. M.
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic topology; Commutative rings and algebras; Computer algebra systems; Counterexamples; Exact sciences and technology; Integers; Mathematical functions; Mathematical minima; Mathematical rings; Mathematical theorems; Mathematics; Polynomials; Sciences and techniques of general use; Topologia algebraica; Àlgebra
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-02-03092-1

Let R = k[x_1,\dots,x_n] and let I be the ideal of n+1 generically chosen forms of degrees d_1 \leq \dots \leq d_{n+1}. We give the precise graded Betti numbers of R/I in the following cases: \begin{itemize} \item n=3; \item n=4 and \sum_{i=1}^5 d_i is even; \item n=4, \sum_{i=1}^{5} d_i is odd and d_2 + d_3 + d_4 < d_1 + d_5 + 4; \item n is even and all generators have the same degree, a, which is even; \item (\sum_{i=1}^{n+1} d_i) -n is even and d_2 + \dots + d_n < d_1 + d_{n+1} + n; \item (\sum_{i=1}^{n+1} d_i) - n is odd, n \geq 6 is even, d_2 + \dots+d_n < d_1 + d_{n+1} + n and d_1 + \dots + d_n - d_{n+1} - n \gg 0. \end{itemize} 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 {\em no} redundant summands, and we present some evidence for this conjecture.