2010/08/12 by Juan Migliore, Migliore, Juan, Rosa M. Miró‐Roig +3
Computer Science · Mathematics · #13C13. Secondary: 13E10 #13D40 #14C20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Primary: 13D02
paper · pdf · doi:10.48550/arxiv.1008.2149
openalex publication_date 2010/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a recent paper, Schenck and Seceleanu showed that in three variables, any ideal generated by powers of linear forms has the Weak Lefschetz Property (WLP). This result contrasts with examples, in our previous work, of ideals in four variables generated by powers of linear forms which fail the WLP. Set R:=k[x1,…,xr]. Assume 1< a1 ≤ … ≤ ar+1. In this paper, we concentrate our attention on almost complete intersection ideals I = ⟨ L1a1, … ,Lrar,Lr+1^ar+1 ⟩ ⊂ R generated by powers of general linear forms Li. Our approach is via the connection (thanks to Macaulay duality) to fat point ideals, together with a reduction to a smaller projective space. When r=4 we give an almost complete description of when such ideals have the WLP, leaving open only one case. When r=5 we solve the problem when a1 = ⋯ = a5 ≤ a6. When r ≥ 6 is even we solve the problem for uniform powers a1 = ⋯ = ar+1 = d; an asympotic version of this latter result was proven by Harbourne, Schenck and Seceleanu (see their simultaneous submission). As a special case, we prove half of their Conjecure 5.5.2, which deals with the case d=2. Other examples are analyzed, most notably when r=7, and we end up with a conjecture which says that if the number of variables r ≥ 9 is odd and all powers have the same degree, say d, then the WLP fails for all d>1.