Woe is me! I'm again resorting to this forum to ask a silly question.

Here is the example I had in mind: observe the (complex) curve $y^3=x^2(x-1)$. In attempt to normalize this curve, I've begun by blowing it up once at the origin. Of the two affines resulting from the blow up, the $x=ys$ affine is the one that will meet the strict transform. Indeed the strict transform will be $\mathbb{C}[x,y,s]/x=ys,y=s^2(x-1)$. I've always assumed the natural morphism from the strict transform to the original curve is a finite one (otherwise using blow-ups to desingularize would be an odd concept!). This would imply that the above ring is integral over $\mathbb{C}[x,y]/y^3-x^2(x-1)$. But I've been staring at this, and staring at this, and by the life of me I can't come up with a monic polynomial that $s=\frac{x}{y}$ would satisfy over this ring.

Is the strict transform a finite morphism?

imagein the base curve $C$, not the entirety of $C$ necessarily (unless that is its image). Your error is that you have got to remove the image of the points outside of your chart, which amounts to removing the point $(1,0)$. This is the unique point where $C$ meets $x=1$, so invert $x-1$. There's an integral relation $s^3 = x/(x-1)$ over $C[1/(x-1)]$. – BCnrd Aug 29 '10 at 2:27