Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$ Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and $k$-points $x$ and $y\in X(\mathcal {O})$ such that $x=y ~[t^{n}]$. We also assume that over $X(F)$, the points factor through the smooth part of $X$.
In particular, it defines for example an element $t^{n}z\in T_{x}X(\mathcal{O}/t^{2n})$, under which conditions could we lift this element to an element in $T_{x}X(\mathcal{O})$?
