The goal of the present paper is study some algebraic invariants Stanley–Reisner rings Cohen–Macaulay simplicial complexes dimension d−1. We prove that inequality d≤reg(Δ)⋅type(Δ) holds for any (d−1)-dimensional complex Δ satisfying Δ=core(Δ), where reg(Δ) (resp. type(Δ)) denotes Castelnuovo–Mumford regularity type) ring k[Δ]. Moreover, given integers d,r,t r,t≥2 and r≤d≤rt, we construct a Δ(G)...

Let $V$ be the weighted projective variety defined by a homogeneous ideal $J$ and $C$ maximal cone in Gr\"obner fan of with $m$ rays. We construct flat family over $\mathbb A^m$ that assembles degenerations associated all faces $C$. This is multi-parameter generalization classical one-parameter degeneration to weight. explain how our can constructed from Kaveh-Manon's recent work on classificat...

Given an infinite field k and a simplicial complex Δ, common theme in studying the f- h-vectors of Δ has been consideration Hilbert series Stanley–Reisner ring k[Δ] modulo generic linear system parameters Θ. Historically, these computations have restricted to special classes complexes (most typically triangulations spheres or manifolds). We provide compact topological expression hd−1a(Δ), dimen...