Abstract: 

Polytopes are frequently used in the study of both theoretical and applied areas of mathematics. Much progress has been made in recent years in order to understand their combinatorial properties. Subdivisions of polytopes, on the other hand, have been receiving extensive attention over the past few years. For a finite poset, we are studying a class of simplicial complexes which are subdivisions of the ordered complex of r-multichains of a given poset. The vertices of the associated ordered complex of r-multichains are in bijection with the ideal I, generated by the monomial of degree r in a polynomial ring. We are interested in, whether the given simplicial complex in this class supports a free resolution I. Also, we are interested in the study of the relation between the Stanley Reisner ideal of the subdivided simplicial complex and the Veronese ideal associated to I.