Singular point of an algebraic variety

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of non-local flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non singular or smooth.