Ideals of general forms and the ubiquity of the Weak Lefschetz property

By Juan Migliore, R. Mir'o-Roig. 2002

83 citations recorded by the source corpus.

Abstract Let d 1 ,…, d r be positive integers and let I =( F 1 ,…, F r ) be an ideal generated by forms of degrees d 1 ,…, d r , respectively, in a polynomial ring R with n variables. With no further information virtually nothing can be said about I , even if we add the assumption that R / I is Artinian. Our first object of study is the case where the F i are chosen generally, subject only to the degree condition. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of R / I has been conjectured by Froberg. In a previous work the authors showed that in many situations the minimal free resolution of R / I must have redundant terms which are not forced by Koszul (first or higher) syzygies among the F i (and hence could not be predicted from the Hilbert function), but the only examples came when r = n +1. Our second main set of results in this paper show that when n +1⩽ r ⩽2 n −2, there are again situations where there must be redundant terms. Finally, we show that if Froberg's conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Closely connected to the Froberg conjecture is the notion of Strong Lefschetz property, and slightly less closely connected is the Weak Lefschetz property. We also study an intermediate notion, the Maximal Rank property. We continue the description of the ubiquity of these properties, especially the Weak Lefschetz property. We show that any ideal of general forms in k [ x 1 , x 2 , x 3 , x 4 ] has the Weak Lefschetz property. Then we show that for certain choices of degrees, any complete intersection has the Weak Lefschetz property and any almost complete intersection has the Weak Lefschetz property. Finally, we show that most of the time Artinian “hypersurface sections” of zeroschemes in P 2 have the Weak Lefschetz property.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/309651b98ea59736cb3973513fc5ef0638a9ea1d

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Citations in the corpus

On the Weak Lefschetz Property for Powers of Linear Forms (2010, background): “This latter fact gave a negative answer to a question in [Migliore and Miró-Roig 2003].”
Generic sequences of polynomials (2010, background): “To prevent further delays, I have not attempted to bring this paper up-to-date with the current literature, but have expanded the bibliography to include some more recent relevant papers, notably [1,4,5,7,16,18–22].”
Survey Article: A tour of the weak and strong Lefschetz properties (2013, background): “See [38, 58] for some connections between these two properties.”
The Hilbert Function of a Level Algebra (2007, background): “The study of this property has also been taken up by several other authors for Gorenstein algebras (see [35], [32], [55], [54], [71]).”