Betti numbers of ideals generated by $n+1$ powers of general linear forms

TL;DR

This paper determines the Betti numbers of ideals generated by $n+1$ powers of general linear forms in a polynomial ring, showing they are level and exploring their algebraic properties, including Lefschetz properties.

Contribution

It generalizes previous work to compute Betti numbers for such ideals with at least one square generator and establishes their level property, also analyzing related Gorenstein algebras.

Findings

Betti numbers explicitly determined for ideals with at least one square generator

All such ideals are shown to be level

Identifies forms with the strong Lefschetz property, including elementary symmetric polynomials

Abstract

We study ideals generated by $n+1$ powers of general linear forms in $R= k[x_1,\dots,x_n]$. By generalizing the ideas in a recent paper of Diethorn et al., we determine the Betti numbers of such ideals when at least one generator is a square. It follows that all such ideals are level. As a consequence, we show that a generic ideal in $R$ generated by $n+1$ forms, with at least one quadric generator, is level. We also determine the Betti numbers of the Artinian Gorenstein algebras linked to these almost complete intersections. By describing the dual generators of these algebras, we obtain a family of forms, including the elementary symmetric polynomials, whose annihilator ideals have the strong Lefschetz property. Finally, we give explicit generators for the annihilator ideal of any elementary symmetric polynomial.