Lefschetz properties and the Veronese construction

TL;DR

This paper explores Lefschetz properties of Veronese subalgebras, showing that for large r, these subalgebras exhibit near Lefschetz properties, leading to new insights into their h- and g-polynomials.

Contribution

The paper demonstrates that high-degree Veronese subalgebras of Cohen-Macaulay algebras possess 'almost' Lefschetz properties, extending understanding of their algebraic and combinatorial structure.

Findings

Veronese subalgebras have 'almost weak' Lefschetz properties for large r.

New results on h-polynomials of Veronese subalgebras.

New results on g-polynomials of Veronese subalgebras.

Abstract

In this paper, we investigate Lefschetz properties of Veronese subalgebras. We show that, for a sufficiently large $r$, the $r$\textsuperscript{th} Veronese subalgebra of a Cohen-Macaulay standard graded $K$-algebra has properties similar to the weak and strong Lefschetz properties, which we call the `almost weak' and `almost strong' Lefschetz properties. By using this result, we obtain new results on $h$- and $g$-polynomials of Veronese subalgebras.