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Register a new userA classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms
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We settle a conjecture by Migliore, Miro-Roig, and Nagel which gives a classification of the Weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms.
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We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.
Given an ideal $I=(f_1,ldots,f_r)$ in $mathbb C[x_1,ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $mathbb C[x_1,ldots,x_n]/I^k$ be? If $rle n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$th powers of the variables.
In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators.
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(ntimes n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+langle f rangle$ for an explicit polynomial $f$ constructed from the fibers of the Veronese grading map.
We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $dgeq 2$ and $ngeq 3$. We consider artinian ideals in the polynomial ring with $n$ variables generated by homogeneous polynomials of degree $d$ invariant under an action of the cyclic group $mathbb{Z}/dmathbb{Z}$, for any $ngeq 3$ and any $dgeq 2$. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.
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