Lefschetz Properties for Higher Order Nagata Idealizations

TL;DR

This paper explores the Lefschetz properties of higher order Nagata idealizations, revealing conditions under which the Weak Lefschetz Property holds and providing a detailed algebraic description in specific cases.

Contribution

It generalizes Nagata idealization for level algebras with explicit dual generators and analyzes Lefschetz properties for these higher order constructions.

Findings

WLP holds if d1 ≥ d2.

Geometry of Nagata hypersurface of order e resembles the original.

Complete algebraic description in monomial square-free case.

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.