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On the Weak Lefschetz Property for Powers of Linear Forms

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 $R:=k[x_1,\dots,x_r]$. Assume $1< a_1 \leq \dots \leq a_{r+1}$. In this paper, we concentrate our attention on almost complete intersection ideals $I = \langle L_1^{a_1}, \dots ,L_r^{a_r},L_{r+1}^{a_{r+1}} \rangle \subset R$ generated by powers of general linear forms $L_{i}$. 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 $a_1 = \cdots = a_5 \leq a_6$. When $r \geq 6$ is even we solve the problem for uniform powers $a_1 = \cdots = a_{r+1} = d$; 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 $r \geq 9$ is odd and all powers have the same degree, say $d$, then the WLP fails for all $d>1$.