First blowup of the cusp y^2=x^3

In the first chart of [F2] write $s=T=y\mathrm{/}x$, so the chart ring is $k[x,s]$ with $y=xs$ and $E=V(x)$, and let $f=y^2-x^3$, a reduced equation of the cusp of multiplicity $m=2$ at the origin; substituting $y=xs$ gives $f(x,xs)=x^2{s}^2-x^3=x^2(s^2-x)$ with $s^2-x$ not divisible by $x$ because its reduction modulo $x$ is $s^2\mathrm{\ne }0$, so by [F1] (or [F4]) the strict transform is cut in this chart by $g=s^2-x$, and by [F3] the strict transform is the closure of the corresponding open part.

The curve $g=s^2-x=0$ is regular: its gradient $(-1,2s)$ never vanishes, so $C^{\mathrm{\prime }}$ is the smooth parabola $x=s^2$; its intersection with $E=V(x)$ in this chart is $V(x,s^2-x)=V(x,s^2)$, the single point $s=0$, and the local ring of $C^{\mathrm{\prime }}$ there is $k[s{]}_{(s)}$ with the equation of $E$ restricting to $s^2$, so the contact order is $\mathrm{length}(k[s{]}_{(s)}\mathrm{/}(s^2))=2$; both $C^{\mathrm{\prime }}$ and $E$ are regular at this point with the same tangent line, so the curves are tangent there. This agrees with [F5]: the leading form of $f$ is $f_2=y^2$, whose dehomogenization $f_2(1,s)=s^2$ vanishes only at $s=0$, with multiplicity $2$, so $C^{\mathrm{\prime }}\cap E$ is the single point of multiplicity $2$.

In the second chart of [F2] write $U=x\mathrm{/}y$, so the chart ring is $k[y,U]$ with $x=yU$ and $E=V(y)$; substituting gives $f=y^2-y^3{U}^3=y^2(1-y{U}^3)$, and the residual factor $1-y{U}^3$ is a unit at every point of $E$ (where $y=0$ it equals $1$), so the strict transform has no points of $E$ in this chart.

The total transform identity ${\pi }^{\ast }C=C^{\mathrm{\prime }}+2E$ of [F4] is visible in the two charts of steps 1.1 and 3.1: the pullback of $y^2-x^3$ factors as $x^2(s^2-x)$ and as $y^2(1-y{U}^3)$, the exceptional factor $x^2$ or $y^2$ contributing $2E$ and the residual factor the strict transform; no other component of $E$ appears, since the residual factors do not vanish along $E$.

The point $V(x,s)$ at which $C^{\mathrm{\prime }}$ is tangent to $E$ is a point at which the two distinct regular curves $C^{\mathrm{\prime }}$ and $E$ meet with contact order $n=2$; blowing up this point, [F6] applies with $n>1$ and shows that the strict transforms of $C^{\mathrm{\prime }}$ and of $E$ meet, at the point of the new exceptional curve corresponding to their common tangent direction, with contact order $n-1=1$, that is, transversally: the tangency is separated by the second blowup.

Therefore in the first chart the strict transform is the smooth parabola $s^2=x$, meeting $E=(x=0)$ at the single point $s=0$ with multiplicity $2$, while in the second chart the strict transform does not meet $E$; the cusp has become a regular curve tangent to $E$, and the second point blowup reduces the contact order of $C^{\mathrm{\prime }}$ with $E$ to one, separating the tangency.