Intuitively, in the Nash
blowing-up process each singular point of an algebraic (or analytic) variety is
replaced by the limiting positions of tangent spaces (at non-singular points). The
following properties of this process are shown: 1) It is, locally, a monoidal transform;
2) in characteristic zero, the process is trivial if and only if the variety is nonsingular.
Examples show that this is not true in characteristic p > 0; that, in general, the
transform of a hypersurface is not locally a hypersurface; and that this process does
not give, in general, minimal resolutions.
