Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lefschetz property and powers of linear forms in $\mathbb{K}[x,y,z]$   In [9], Migliore, Mir\'o-Roig and Nagel, proved that if $R =
\mathbb{K}[x,y,z]$, where $\mathbb{K}$ is a field of characteristic zero, and
$I=(L_1^{a_1},\dots,L_r^{a_4})$ is an ideal generated by powers of 4 general
linear forms, then the multiplication by the square $L^2$ of a general linear
form $L$ induces an homomorphism of maximal rank in any graded component of
$R/I$. More recently, Migliore and Mir\'o-Roig proved in [8] that the same is
true for any number of general linear forms, as long the powers are uniform. In
addition, they conjecture that the same holds for arbitrary powers. In this
paper we will solve this conjecture and we will prove 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 an homomorphism of maximal
rank in any graded component of $R/I$.

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The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: econ