particular, there is a canonical bijection $\nu'':C'(K)\tilde\lrar C(K)$ for any field extension $K/k$. The curve $C'$ has mildest possible singularities: for each singular point $s\in C'(\bar k)$ the branches of $\ovl C'$ through $s$ intersect like $n$ coordinate axes at $0\in {\mathbb{A}}^n_k$.

Let us define the following reduced $0$-dimensional schemes: $$\Lambda=\operatorname{Spec}(H^0(\wtl C,\mathcal{O}_{\wtl C})), \ \ 
\Pi=C_{\mathrm{sing}}, \ \ 
\Psi=\big(\Pi\times_C\wtl C\big)_{\mathrm{red}}.
\label{ee3}$$ Here $\Lambda$ is the $k$-scheme of geometric irreducible components of $C$ (or the geometric connected components of $\wtl C$); it is the disjoint union of closed points $\lambda=\operatorname{Spec}(k(\lambda))$ such that $k(\lambda)$ is the algebraic closure of $k$ in the function field of the corresponding irreducible component $k(C_\lambda)=k(\wtl C_\lambda)$. Next, $\Pi$ is the union of singular points of $C$, and $\Psi$ is the union of fibres of $\nu: \wtl C\to C$ over the singular points of $C$ with their reduced subscheme structure. The morphism $\nu''$ induces an isomorphism $\big(\Pi\times_C C'\big)_{\mathrm{red}}\tilde\lrar \Pi$, so we can identify these schemes. Let $i:\Pi\to C$, $i':\Pi\to C$ and $j:\Psi\to \wtl C$ be the natural closed immersions. We have a commutative diagram $$\xymatrix{
\wtl C \ar^{\nu'}[r]& C' \ar^{\nu''}[r]& C\\
\Psi \ar^{j}[u]\ar^{\nu'}[r] & \Pi \ar^{i'}[u]\ar_{i}[ur]&\\
}$$ The restriction of $\nu$ to the smooth locus of $C$ induces 