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...

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi’s extension of Nakai’s conjecture to arbitrary characteristic. This gives a new proof of Nakai’s conjecture for curves and Stanley-Reisner rings.

Given a constant weight linear code, we investigate its weight hierarchy and the Stanley-Reisner resolution of its associated matroid regarded as a simplicial complex. We also exhibit conditions on the higher weights sufficient to conclude that the code is of constant weight.

We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz’s ...