A Fr\"{o}berg type theorem for higher secant complexes

TL;DR

This paper extends Froberg's theorem to higher secant complexes of simplicial complexes, linking algebraic properties to combinatorial and geometric phenomena in secant varieties.

Contribution

It generalizes Froberg's theorem to embedded joins of simplicial complexes and explores related combinatorial and geometric phenomena.

Findings

Generalization of Froberg's theorem to higher secant complexes

Connection between property N_{q+1,p} and secant complexes

Observation of combinatorial phenomena analogous to geometric secant varieties

Abstract

We generalize the celebrated Fr\"{o}berg's theorem to embedded joins of copies of a simplicial complex, namely higher secant complexes to the simplicial complex, in terms of property $N_{q+1,p}$ due to Green and Lazarsfeld. Furthermore, we investigate combinatorial phenomena parallel to geometric ones observed for higher secant varieties of minimal degree.