This is Exercise II.4.1 in Shafarevich's book Basic Algebraic Geometry, second edition.
Suppose that $\dim X = 2 $ and that $ \xi \in X $ is a nonsingular point. Let $ C_1, C_2 \in X $ be two curves passing through $ \xi $ and nonsingular there, $ \sigma: Y \to X $ the blowup centered at $ \xi $, and set $ C_i' = \overline{\sigma^{-1}(C_i \backslash \xi)} $ and $ Z = \sigma^{-1} (\xi) $. Prove that $ C_1'\cap Z=C_2'\cap Z $ if and only if $ C_1 $ and $ C_2 $ touch at $ \xi $.
I have been thinking about this exercise for many days now but I still don't even know where to start. I have of course read the relevant section, but I'm still lost. I believe the following is important as a background for the exercise:
Let $ X $ be a quasiprojective variety and $ \xi \in X $ a nonsingular point, and suppose that $ u_1, \cdots ,u_n $ are functions that are regular everywhere on $ X $ and such that (a) the equations $ u_1 = \cdots = u_n = 0 $ have the single solution $ \xi \in X $; and (b) $ u_1, \cdots, u_n $ form a local system of parameters on $ X $ at $ \xi $. $ Y \subseteq X \times \mathbb{P}^{n-1} $ consists of points $ (x; t_1 : \cdots : t_n ) $ with $ x \in X $ and $ (t_1 : \cdots : t_n ) \in \mathbb{P}^{n-1} $, such that $$ u_i(x)t_j = u_j(x)t_i $$ for $ i,j = 1, \cdots ,n $. The regular map $ \sigma: Y \to X $ obtained as the restriction to $ Y $ of the first projection $ X \times \mathbb{P}^{n-1} \to X $ is called the local blowup of $ X $ with center in $ \xi $.
Can anyone help me out?
I was also stuck on this problem for a while. I think I have a solution, but I find it a bit hand-wavey. Leaving it here for future visitors in hopes that somebody can improve it.
$X$ is 2-dimensional. We can choose $u_i$ so $X$ is locally given by $u_3=\ldots=u_N=0 ,$
$C_1$ given by $u_1=0$, $C_2$ by $F(u_1,u_2)=0$. On $X$ we have $\sigma^{-1}(u_1,u_2)=(u_1,u_2,0,\ldots,0,u_1:u_2:0:\ldots,0)$. We are interested in the image of $(0,0)$ under the restriction of this rational map to $C_i$. We are also really only interested in the first two homogeneous coordinates $(u_1:u_2)$ of the image, so from now on we will consider $\sigma^{-1}$ as a map to $\mathbb{P^1}$.
On a curve every rational map to a projective space is regular, so the restrictions of $\sigma^{-1}$ to $C_i$ must be regular. On $C_1$ the restriction takes the obvious form $\sigma^{-1}(0,u_2)=(0:1)$. On $C_2$ it must have some form $\sigma^{-1}(u_1,u_2)=(P(u_1,u_2):G(u_1:u_2))$. For this to be a restriction of $\sigma^{-1}$ from $X$ to $C_2$, we need it to satisfy $u_1G=u_2P\ \text{mod} F$, and for it to coincide with the restriction to $C_1$ we need it to satisfy $P(0,0)=0,\ G(0,0)=1$. Thus $F$ must have a nonzero term $ku_1$ and must have a zero coefficient before $u_2$, which happens exactly when $C_2$ touches $C_1$ at zero.