Torified simplicial posets from Samuel-Rees degenerations

Abstract We associate a simplicial complex to a sequence of ring elements that generate the ring over R (or nearly so). This is motivated by a formula for the degree of R as the “volume” of the simplicial complex. Then we give a flat degeneration of the ring to one combinatorially defined from the simplicial complex – or almost; the simplicial complex sometimes has to be promoted to a simplicial poset, and usually has to be given a nontrivial lattice structure. This degeneration, based on Samuel-Rees filtrations, produces no nilpotents. While toric varieties provide the simplest case to visualize, flag manifolds are more interesting, giving a new proof of Lakshmibai-Seshadri conjecture (now Littelmann’s theorem), at least in high degree.