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

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 Fröberg. 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 Fröberg'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 Fröberg 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.