Lefschetz properties for higher order Nagata idealizations

We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1, d). We consider the algebra associated to polynomials of the same type of bidegree (d1, d2). We prove that the geometry of the Nagata hypersurface of order e is very similar to the geometry of the original hypersurface. We study the Lefschetz properties for Nagata idealizations of order d1, proving that WLP holds if d1 ≥ d2. We give a complete description of the associated algebra in the monomial square free case.