Lefschetz Properties for Higher Order Nagata Idealizations

Armando Cerminara; Rodrigo Gondim; Giovanna Ilardi; Fulvio Maddaloni

paper

Lefschetz Properties for Higher Order Nagata Idealizations

arXiv:1807.06415

Abstract

We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicity as a bigraded polynomial of bidegree $(1, d)$ . We consider the algebra associated to polynomials of the same type of bidegree $(d_{1}, d_{2})$ . 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 $e$ , proving that WLP holds if $d_{1} \geq d_{2}$ . We give a complete description of the associated algebra in the monomial square free case.

16 pages, 4 figures. To appear in Advances in Applied Mathematics