A node is resolved by one point blowup

The parametrization of [F2] identifies $R=k[x,y]\mathrm{/}(y^2-x^2(x+1))$ with $k[t^2-1,t(t^2-1)]$. Indeed every polynomial reduces uniquely to $a(x)+yb(x)$, and its image $a(t^2-1)+t(t^2-1)b(t^2-1)$ is zero only when both polynomials vanish: the first term has even powers of $t$, the second odd powers, and $k[t]$ is a domain. Thus $R$ is a domain, finite integral over $k[x]$, and has dimension one by Injective integral extensions preserve Krull dimension. At the origin the local maximal ideal has the independent classes of $x,y$ modulo its square, since the defining equation has no linear term; the embedding dimension is two, so this closed point of the integral curve is not regular. Elsewhere $x$ is invertible, because $x=0$ on the curve forces $y=0$; putting $t=y\mathrm{/}x$ gives $R[1\mathrm{/}x]=k[t,1\mathrm{/}(t^2-1)]$, a localization of the regular affine line (localisation and polynomial extension of regular rings). Therefore the origin is the unique singular point over every field of characteristic different from two, including characteristic three.

The completed ambient local ring is the ring of formal power series in $x,y$: compatible residues modulo $(x,y{)}^n$ specify its coefficients, and denominators with nonzero constant term are inverted by formal geometric series. Construct the formal power series $u=1+{\sum }_{n\ge 1}{a}_n{x}^n$ recursively by $u^2=1+x$. The coefficient equation gives $a_1=1\mathrm{/}2$, and for each $n>1$ determines $a_n$ by $2a_n+{\sum }_{1\le i<n}{a}_i{a}_{n-i}=0$, so only powers of two need to be inverted. Thus this construction works in every allowed characteristic. In the ring of formal power series in $x$ and $y$, the equation factors as $(y-xu)(y+xu)$. Each factor has nonzero linear term $y-x$ or $y+x$, and its quotient is the formal power-series ring in $x$, a DVR with uniformizer $x$; thus it defines a regular formal branch; their ideal together is $(x,y)$ because two and $u$ are units. Hence the branches have distinct tangent directions and intersection length one. They are formal branches of the single integral global curve established in step 1.1.

On the $x$-chart $y=xt$, the exceptional curve is $E=V(x)$ and the strict transform is $V(t^2-x-1)$, a regular curve with parameter $t$. Its exceptional intersection is $V(x,t^2-1)$, the two reduced points $t=1,-1$ because two is invertible; each has intersection length one. On the other chart $x=ys$, the strict-transform equation is $1-s^2-y{s}^3=0$. It makes $s$ invertible, since $s^2(1+ys)=1$, so this entire portion belongs to the overlap with the $x$-chart, where $t=1\mathrm{/}s$. Thus the $x$-chart describes the entire strict transform, including all points above the origin, and its two transverse exceptional contacts are precisely the two formal tangent directions of step 2.1.

The strict transform is ${\mathbf{A}}_k^1$ with $x=t^2-1$ and $y=t(t^2-1)$, so its map to $Z$ is the finite birational map of [F2]. The normality invoked there follows directly from the UFD assertion: for an integral reduced fraction $a\mathrm{/}b$, a monic equation implies $b\mid a^n$ after clearing denominators, and coprimality forces $b$ to be a unit. Hence the map is the normalization. The original curve has multiplicity two at the origin, from its lowest-degree term $y^2-x^2$; the strict transform is regular and has curve multiplicity one at each of the two points above the origin. The contact calculation of step 3.1 is direct and does not apply the regular-source multiplicity lemma to the singular original curve or to nonexistent global branch components.

By [F1] the surface $S_1$ is regular and $E$ is a regular curve. The support of the total transform of $Z$ is $Z^{\mathrm{\prime }}\cup E$: two regular curves meeting exactly in the two points of step 3.1, each with multiplicity $1$. At every closed point at most two components pass, and when two pass they meet transversally, so by [F3] the support is a strict normal crossings divisor; therefore the resolution sequence of Embedded strict-normal-crossings resolution of a reduced curve on a regular surface terminates after this single blowup, and no further blowups are needed.