Abstract
In [9], Migliore, Miró-Roig and Nagel proved that if
{R=\mathbb{K}[x,y,z]}
, where
{\mathbb{K}}
is a field of characteristic zero, and
{I=(L_{1}^{a_{1}},\dots,L_{4}^{a_{4}})}
is an ideal generated by powers of four general linear forms, then the multiplication by the square
{L^{2}}
of a general linear form L induces a homomorphism of maximal rank in any graded component of
{R/I}
. More recently, Migliore and Miró-Roig proved in [7] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjectured that the same holds for arbitrary powers. In this paper, we will prove that this conjecture is true, that is, we will show that if
{I=(L_{1}^{a_{1}},\dots,L_{r}^{a_{r}})}
is an ideal of R generated by arbitrary powers of any set of general linear forms, then the multiplication by the square
{L^{2}}
of a general linear form L induces a homomorphism of maximal rank in any graded component of
{R/I}
.
The weak Lefschetz property and powers of linear forms in $\mathbb {K}[x,y,z]$