= Resolution of the (2,5) cusp by two blowups
{title2=$(x,y)=(t^5,t^2)\longmapsto(u,y)=(t^3,t^2)\longmapsto(v,y)=(t,t^2)$}
Blow up the origin and use the chart $x=uy$, giving the <strict transform> $u^2-y^3=0$. Blow up its remaining singular point and use $u=vy$, giving $v^2-y=0$, which is smooth because its derivative in $y$ is $-1$. The complementary chart at each stage has no point of the <strict transform> on its exceptional divisor. These two <blowups of a smooth algebraic surface> resolve the branch, although the smooth transform is still tangent to an exceptional divisor.
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