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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: J. Migliore and R. M. Miró-Roig
Journal: Trans. Amer. Math. Soc. 355 (2003), 1-36
MSC (2000): Primary 13D02, 13D40; Secondary 13P10, 13C40, 13H10
Published electronically: August 28, 2002
MathSciNet review: 1928075
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Abstract: 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:

  • $n=3$;
  • $n=4$ and $\sum_{i=1}^5 d_i$ is even;
  • $n=4$, $\sum_{i=1}^{5} d_i$ is odd and $d_2 + d_3 + d_4 < d_1 + d_5 + 4$;
  • $n$ is even and all generators have the same degree, $a$, which is even;
  • $(\sum_{i=1}^{n+1} d_i) -n$ is even and $d_2 + \dots + d_n < d_1 + d_{n+1} + n$;
  • $(\sum_{i=1}^{n+1} d_i) - n$ is odd, $n \geq 6 $ is even, $d_2 + \dots+d_n < d_1 + d_{n+1} + n$ and $d_1 + \dots + d_n - d_{n+1} - n \gg 0$.
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.


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Additional Information

J. Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: Juan.C.Migliore.1@nd.edu

R. M. Miró-Roig
Affiliation: Facultat de Matemàtiques, Departament d’Algebra i Geometria, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: miro@cerber.mat.ub.es

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03092-1
PII: S 0002-9947(02)03092-1
Received by editor(s): October 1, 2001
Received by editor(s) in revised form: March 10, 2002
Published electronically: August 28, 2002
Additional Notes: The first author was partially supported by the University of Barcelona
The second author was partially supported by BFM2001-3584
Article copyright: © Copyright 2002 American Mathematical Society