Nash blowing-up

In algebraic geometry, Nash blowing-up is a process in which, roughly speaking, each singular point is replaced by all limiting positions of the tangent spaces at the non-singular points. More formally, let ${\displaystyle X}$ be an algebraic variety of pure dimension r embedded in a smooth variety ${\displaystyle Y}$ of dimension n, and let ${\displaystyle X_{\text{reg}}}$ be the complement of the singular locus of ${\displaystyle X}$. Define a map ${\displaystyle \tau :X_{\text{reg}}\rightarrow X\times G_{r}(TY)}$, where ${\displaystyle G_{r}(TY)}$ is the Grassmannian of r-planes in the tangent bundle of ${\displaystyle Y}$, by ${\displaystyle \tau (a):=(a,T_{X,a})}$, where ${\displaystyle T_{X,a}}$ is the tangent space of ${\displaystyle X}$ at ${\displaystyle a}$. The closure of the image of this map together with the projection to ${\displaystyle X}$ is called the Nash blow-up of ${\displaystyle X}$.

Although the above construction uses an embedding, the Nash blow-up itself is unique up to unique isomorphism.