Blow-Up in p not singular in p

Question

I came upon a Lemma, which stated that the Blow-Up of an algebraic Curve $C \in \mathbb{C}[x,y]$ in a singular point $p$ of $C$ is non singular in $p$. For the proof the author referred to the Jacobian, after expressing the Blow-Up as a variety $\mathbb{V} = (f(x,y), xx_1-yx_0=0)$, where the $f$ denotes the defining equation of the curve, and $xx_1-yx_0$ the defining equation of the Blow-Up. Then the Jacobian is: \begin{pmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} &0&0\\ x_1 & -x_0 & x & -y \end{pmatrix}

I know that a point is non singular in a point $p$, if the rank of the the Jacobian in $p$ is $n-d$, where $n$ is the number of variables, and $d$ the dimension of the Variety. But the Jacobian in this example has rank 1, although I thought $n=4$, and $d =1$. Can somebody help with this proof, or am I missing something?

EDIT: I figured it out, since we are only looking at the Matrix in $p$, which is 2-dimensional, our Variety has two variables, so $n=2$. Now it all makes sense!

Answer 1

A single blow-up will not always resolve the singularity. The following two examples are exercises from chapter I of Hartshorne (undoubtedly covered in many other books as well).

• Blowing up the singularity at the origin of a tacnode $x^2=x^4+y^4$ amounts to substituting $x=yt$ and cancelling the common factor. This gives us the equation $t^2=y^2t^4+y^2$, and that curve still has an ordinary double point as solving for $y$ gives $y=\pm t/\sqrt{t^4+1}$ (implying two branches through the origin with tangents of slopes $\pm1$).
• Blowing up a higher order cusp of $y^3=x^5$ at the origin with the substitution $y=xt$ leads to the usual cusp $t^3=x^2$.

You see that in both cases above, another blowing up will resolve the singularity. IIRC a finite number of blowings up will always do. But, as we saw, a single one will not necessarily suffice.