I first got concious of the notion of normal variety varieties around 3 years ago and despite the fact that by now I can more-less manipulate with it a bit, this notion still puzzles me a lot. What really One thing that strikes me is that this the definition of normality is so entirely algebraic.
From my pedestrian geometric point of view it looks that common sense understanding the notion of normal varieties should restrict restricts the class of spaces that we consider to more-less reasonable ones. It looks to me that this definition is analogous to the definition of pseudo-manifold. At least the obvious similarity is that in both cases the set of non-singular points is connected.
This notion is obviously efficient (maybe even beautiful in
Normality pops up everywhere and its simplicity)definition is very short. But it is hard for me t to imagine that a differential topologist or differential geometer could come up with such a definition. Why is the notion of normatilty is so efficientomnipresent? What is geometric "geometric" meaning of normality?
Maybe a more concrete question would be like this. Suppose $X$ is an irreducible algebraic subvariety in $\mathbb C^n$ with singularities in co-dimension $2$. Can one somehow looking on singularities(or maybe on , their stratification ) say if and the way $X$ lies in $\mathbb C^n$ say if it is normal or not?
Added. Who was the person who invented this notion?
I would like to thank everybody for useful comments and links.