Hilbert Scheme and Multiplet Matter Content

Development of the concept of Euler characteristic, from the Euclidean geometry to the algebraic geometry is considered. A singular toric variety is studied within the framework of the algebraic geometry. Procedure of the blowing up of its singularities in terms of cones is represented by Hilbert scheme. Special cases of the blowing up of orbifold singularities of using Nakamura’s algorithm are performed. Hilbert schemes and their physical interpretation in terms of Euler characteristic are presented.