2002/05/13 by Juan Migliore, J. Migliore, Migliore, J. +3 · 1 citation
Computer Science · Mathematics · #13C40 #13D02 #13D40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:13C40 #msc:13D02 #msc:13D40
paper · pdf · doi:10.48550/arxiv.math/0205133
24 pages
arxiv created 2002/05/13 · openalex publication_date 2002/05/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let d1,...,dr be positive integers and let I = (F1,...,Fr) be an ideal generated by general forms of degrees d1,...,dr, respectively, in a polynomial ring R with n variables. 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 Fi (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 further examples can be obtained when n+1 ≤ r ≤ 2n-2. We also 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. Related to the Fröberg conjecture is the notion of Weak Lefschetz property. We continue the description of the ubiquity of this property. We show that any ideal of general forms in k[x1,x2,x3,x4] has it. Then we show that for certain choices of degrees, any complete intersection has it and any almost complete intersection has it. Finally, we show that most of the time Artinian ``hypersurface sections'' of zeroschemes have it.