Buchstaber invariant, minimal non-simplices and related

Abstract

Buchstaber invariant is a numerical characteristic of a simplicial complex (or a polytope), measuring the degree of freeness of the torus action on the corresponding moment-angle complex. Recently an interesting combinatorial theory emerged around this invariant. In this paper we answer two questions, considered as conjectures in [2], [11]. First, Buchstaber invariant of a convex polytope $P$ equals $1$ if and only if $P$ is a pyramid. Second, there exist two simplicial complexes with isomorphic bigraded $\mathrm{Tor}$-algebras, which have different Buchstaber invariants. In the proofs of both statements we essentially use the result of N. Erokhovets, relating Buchstaber invariant of simplicial complex $K$ to the distribution of minimal non-simplices of $K$. Gale duality is used in the proof of the first statement. Taylor resolution of a Stanley--Reisner ring is used for the second.