Lefschetz properties for higher order Nagata idealizations

Abstract

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.