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Deciding the presence of the weak Lefschetz property often is a challengingproblem. In this work an in-depth study is carried out in the case of Artinianmonomial ideals with four generators in three variables. We use a connection tolozenge tilings to describe semistability of the syzygy bundle of such anideal, to determine its generic splitting type, and to decide the presence ofthe weak Lefschetz property. We provide results in both characteristic zero andpositive characteristic.

Syzygy bundles and the weak Lefschetz property of almost complete intersections