Can anyone confirm or refute this proof? Also, how do I formalize the second part?
Since ξ is nonsingular, it's contained in a unique irreducible component of X. So, there's no way for the subvarieties $C_1,C_2$ to both be contained in X, contain ξ and not both contain that component. If they did, it would imply that the irreducible subvariety containing ξ is not irreducible.
Therefore, $C_1=C_2$ in $Z$ if they touch at ξ. So, $C′_1=C′_2$ must have the same intersection with Z.
For the other part, I'm pretty sure $\sigma^{−1}$ only fails to be an isomorphism at ξ and instead maps to the tangent space of ξ, so If they coincide in this region, that means they have the same tangent line. I believe this is one of the key properties of the blowup, but I'm not sure how to exactly express this argument.
Edit:
The definition of the blowup is as follows:
Consider the subvariety $\prod\subset\mathbb{P}^n\times\mathbb{P}^{n-1}$ consisting of points $(x_0:...:x_n;y_1:...:y_n)$ satisfying $$x_iy_j=x_jy_i$$
for $i,j=1,...,n.$
The blowup (just "THE blowup") with center $\xi=(1:0:...:0)\in\mathbb{P}^n$ is the map $\prod\to\mathbb{P}^n$ defined by restricting the first projection to $\prod$.
The blowup of a quasiprojective variety $X\subset \mathbb{P}^n$ centered at nonsingular point $\xi\in X$ is the map $\sigma:Y\to X$ where $Y$ is defined as the component Y of $\sigma^{-1}(X)$ (which has two irreducible components, $(\xi \times \mathbb{P}^{N-1})$ and $Y$.)
The fact that $\sigma^{-1}(X)$ has the irreducible components which the definition assumes is proven in Shafarevich chapter 2 section 4 theorem 2.15. The rest of this definition is mostly copied from that section.