Notes on Nash modification

The Nash blowing-up (or modification) of an algebraic variety $X$ is a canonical process that produces a proper, birational morphism $\pi :X^{\mathrm{\prime }}\to X$ of varieties. It is expected that the singularities of $X^{\mathrm{\prime }}$ will be better than those of $X$. In the mid-1970’s, it was proved that in characteristic zero, $\pi$ is an isomorphism if and only if $X$ is nonsingular, which is false in positive characteristic. The focus of this article is on several subsequent studies on this subject. Topics covered include: (a) the extension of the mentioned theorem to the case where $X$ is normal, in any characteristic, (b) the introduction and study of Nash modifications of higher order, © the case where the variety $X$ is toric, where more precise results can be obtained and (d) desingularization properties of the Nash process.