RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

04. definition of singularities

Definition 4.1. An subvariety V = V of an affine space 𝔸^{N} of codimension k is non-singular at its point P if V is locally defined by k polynomials f_{1}, …, f_{k} such that df_{1}, … df_{k} is linearly independent at P.

The dimension and codimensions are defined by using the transcendence degree of the extension of the function field k(V) = Q(k[V]). The definition of regularity (singularity) is more naturally defined by using theory of commutative algebras.