A nonsingular formal arc on a normal Noetherian surface becomes regular

Let $P=\ker (A\to V)$ and let $t\mapsto$ a uniformizer of $V$; minimally generate $P$ by $u_2,\dots ,u_r$, so that $\mathfrak{m}=(t,u_2,\dots ,u_r)$. The codimension formula for the Cohen--Macaulay local ring $A$ makes $P$ of height one, so $A_P$ is a discrete valuation ring; a generator of its maximal ideal occurs among the $u_i$, and after relabelling we may take it to be $u_2$.

The finite module $P\mathrm{/}(P^2+(u_2))$ vanishes at $P$ and is a module over $A\mathrm{/}P=V$; being torsion over the discrete valuation ring, it is killed by a power of $t$, so for every $i>2$ there are $n_i\ge 0$ and $a_i\in A$ with $t^{n_i}{u}_i-a_i{u}_2\in P^2$.

If some $n_i=0$, the relation expresses $u_i$ in terms of $u_2$ and $P^2$, so Nakayama removes $u_i$ from the kernel generators. If some $a_i$ is a unit, it instead removes $u_2$; the relation at $A_P$, where $t$ is a unit, then shows that $u_i$ can serve as the new uniformizer. Otherwise, when $a_i\in P$ absorb $a_i{u}_2$ into $P^2$. For $a_i\mathrm{\notin }P$, write its residue in $V$ as a unit times $t^{m_i}$, with $m_i>0$, and lift that unit to $A$; the difference contributes to $P^2$. The relations thus have the form $t^{n_i}{u}_i\in P^2$ or $t^{n_i}{u}_i-c_i{t}^{m_i}{u}_2\in P^2$, with $n_i,m_i>0$ and $c_i$ a unit. In the $t$-chart set $v_i=u_i\mathrm{/}t$ and divide by $t^2$: the exponents become $n_i-1,m_i-1$, and the right sides lie in $(v_2,\dots ,v_r{)}^2$. At the arc centre the induced map to $V$ has kernel generated by these $v_i$, with the same uniformizer $t$.

Termination: if the minimal number of generators of the kernel drops, restart with that smaller number. Each successor local ring is Noetherian and maps onto $V$ with the same residue field. Its kernel is a nonzero, nonmaximal prime of its two-dimensional local domain, hence has height one, and its localization is the original DVR $A_P$, since $t$ is outside $P$ and the blowups are isomorphisms there; otherwise the nonnegative exponents decrease at each step until one becomes zero, which forces a generator drop by Nakayama's lemma. Hence there are only finitely many drops and the process terminates.

Every arc centre is a closed point of the integral modification of $\mathrm{Spec}A$, since its residue field is the original residue field. By [F4] its local dimension is two. At termination the kernel is principal and the maximal ideal of the local ring at the arc centre is generated by two elements; that local ring has dimension two by the dimension computation for local normal surface modifications, hence is regular, so the blowups become regular at the arc centre after finitely many steps. The Axiom of Choice and the Axiom of Dependent Choice are inherited from the cited suppliers.