Hartshorne Exercise V.3.2.
Let $C$ and $D$ be curves on a surface $X$, meeting at a point $P$. Let $\pi:\tilde X\ra X$ be the monoidal transformation with center $P$. Show that $\tilde C.\tilde D=C.D-\mu_P(C)\mu_P(D)$. Conclude that $C.D=\sum\mu_P(C)\mu_P(D)$, where the sum is taken over all intersection points of $C$ and $D$, including infinitely near intersection points.
We have,
$$\tilde C.\tilde D=(\pi ^* C-\mu_P(C)E).(\pi ^*D-\mu_P(D)E)=C.D-\mu_P(C)\mu_P(D).$$
Assume that $C$ and $D$ are linearly equivalent to divisors that share no irreducible components, otherwise the question makes no sense. It suffices to show that for each point $P\in C\cap D$, $\op{mult}_{P}(C\cap D)=\sum_i\mu_{P_i}(C)\mu_{P_i}(D)$, where the sum is taken over all the points infinitely near to $P$. We may thus assume that $C$ and $D$ are irreducible curves and intersect only at $P\in X$. After blowing up once at $P$, every point of intersection in $\tilde C\cap\tilde D$ is in the exceptional locus, infinitely near the point $P$. We obtain a sequence of blowups $\varepsilon_n:X_n\ra X_{n-1}$, with $X_0=X$, exceptional divisors $E_n$, and blowup point $P_{n}$ chosen as any point in $ C_{n}\cap D_{n}\subset E_n\subset X_{n-1}$. Here $C_0:=C,D_0:=D$ and $C_n,D_n$ are the strict transforms all the way up to $X_n$. Observe that $C_n.D_n=C.D-\sum_{j=1}^{n-1}\mu_{P_{j-1}}(C_{j-1})\mu_{P_{j-1}}(D_{j-1})$. We stop when $C_{n+1}\cap D_{n+1}=\varnothing\implies C_{n+1}.D_{n+1}=0$, at which point we get the desired equality,
$$C.D=\sum_{j=1}^{n}\mu_{P_{j}}(C_{j})\mu_{P_{j}}(D_{j})+\mu_{P}(C)\mu_{P}(D).$$
By construction, each point $P_j$ is infinitely near $P$, and when we stop, we’ve covered all of them, since $C_j\cap D_j$ lives in the preimage of $E\subset X_1$. The process does indeed terminate since $C_n.D_n < C_{n-1}.D_{n-1}$, the strict inequality being due to the choice of $P_{n-1}\subset C_{n-1}\cap D_{n-1}$.
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