  The weak and strong Lefschetz properties are two basic properties that
Artinian algebras may have. Both Lefschetz properties may vary under small
perturbations or changes of the characteristic. We study these subtleties by
proposing a systematic way of deforming a monomial ideal failing the weak
Lefschetz property to an ideal with the same Hilbert function and the weak
Lefschetz property. In particular, we lift a family of Artinian monomial ideals
to finite level sets of points in projective space with the property that a
general hyperplane section has the weak Lefschetz property in almost all
characteristics, whereas a special hyperplane section does not have this
property in any characteristic.
