On the weak Lefschetz property for powers of linear forms (Q442437)

Let \(A=R/I\) be a standard graded Artinian algebra, where \(R=k[x_1,\dots,x_r]\) and \(k\) is a field. One of the most studied property for \(A\) is the weak Lefschetz property (WLP). The authors study the WLP for ideals \(I\) generated by powers of linear forms. They give answers in the following cases (\(k\) is a field of characteristic zero):NEWLINENEWLINE1)\, \(I=(L_1^{a_1},\dots,L_5^{a_5})\subset k[x_1,x_2,x_3,x_4],\) where the \(L_i\) are general linear forms and \(2\leq a_1\leq\dots\leq a_5.\) In this case they give a fairly complete answer.NEWLINENEWLINE2)\, \(I=(x_1^{d},\dots,x_5^{d},L^{e})\subset k[x_1,x_2,x_3,x_4,x_5],\) where \(e\geq d\) and \(L\) is a general linear form.NEWLINENEWLINE3)\, \(I=(x_1^{d},\dots,x_{2n}^{d},L^{d})\subset k[x_1,\dots,x_{2n}],\) where \(L\) is a general linear form (even number of variables).