Modular Jordan type for $\mathbb k [x, y] / (x^m,y^n)$ for $m = 3, 4$
J. Korean Math. Soc. 2020 Vol. 57, No. 2, 283-312
Published online March 1, 2020
Jung Pil Park, Yong-Su Shin
Seoul National University; and
Abstract : A sufficient condition for an Artinian complete intersection quotient $S={\mathbb k}[x,y]/(x^m,y^n)$, where $\mathbb k$ is an algebraically closed field of a prime characteristic, to have the strong Lefschetz property (SLP) was proved by S. B. Glasby, C. E. Praezer, and B. Xia in \cite{GPX}. In contrast, we find a necessary and sufficient condition on $m$, $n$ satisfying $3 \le m \le n$ and $p > 2m-3$ for $S$ to fail to have the SLP. Moreover we find the Jordan types for $S$ failing to have SLP for $m \le n$ and $m = 3, 4$.
Keywords : Jordan types, strong Lefschetz property, weak Lefschetz property, Hilbert function
MSC numbers : Primary 13A02; Secondary 16W50
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