Let \(C\) and \(D\) be curves on a surface \(X\), meeting at a point \(P\). Let \(\pi: \tilde{X} \rightarrow X\) be the monoidal transformation with center \(P\).
Show that \[ \tilde{C} \cdot \tilde{D}=C . D-\mu_P(C) \cdot \mu_P(D) .\] Conclude that \(C . D=\sum \mu_P(C) \cdot \mu_P(D)\), where the sum is taken over all intersection points of \(C\) and \(D\), including infinitely near intersection points.
Solution
Write \[ r=\mu_P(C), \qquad s=\mu_P(D), \] and let \(E\) be the exceptional curve of the blowup \[ \pi:\widetilde X\longrightarrow X. \]
1. The total transforms satisfy\[ \pi^*C=\widetilde C+rE, \qquad \pi^*D=\widetilde D+sE. \]
For a curve through the centre of a blowup, the exceptional divisor occurs in the total transform with coefficient equal to the multiplicity at the centre. Thus the standard strict-transform formula FE-SRFBLOW gives \[ \pi^*C=\widetilde C+\mu_P(C)E \] and the analogous formula for \(D\).
2. The blowup intersection identities are\[ \pi^*C\cdot\pi^*D=C\cdot D, \qquad \pi^*C\cdot E=\pi^*D\cdot E=0, \qquad E^2=-1. \]
These are the standard intersection formulas for the blowup of a nonsingular surface at a point FE-SRFBLOW. Pullback preserves the intersection product of divisors from \(X\), every pulled-back divisor has intersection zero with the exceptional curve, and the exceptional curve has self-intersection \(-1\).
3. One has\[ \boxed{ \widetilde C\cdot\widetilde D = C\cdot D-rs.} \]
By step 1, \[ \widetilde C=\pi^*C-rE, \qquad \widetilde D=\pi^*D-sE. \] Therefore step 2 gives \[ \begin{aligned} \widetilde C\cdot\widetilde D &=(\pi^*C-rE)\cdot(\pi^*D-sE)\\ &=\pi^*C\cdot\pi^*D -s\,\pi^*C\cdot E -r\,\pi^*D\cdot E +rsE^2\\ &=C\cdot D-rs. \end{aligned} \] Substituting the definitions of \(r\) and \(s\) gives \[ \widetilde C\cdot\widetilde D = C\cdot D- \mu_P(C)\mu_P(D), \] which is the first assertion.
4. Assume \(C\) and \(D\) have no common irreducible component. There is afinite sequence of point blowups after which their strict transforms aredisjoint.
Resolve the reduced divisor \(C\cup D\) by finitely many point blowups on the nonsingular surface. After resolution, its irreducible strict transforms are nonsingular and meet, if at all, only transversely. If the strict transforms of \(C\) and \(D\) still meet at such a transverse point, blow up that point once more. Their two strict transforms then meet the exceptional curve at the two distinct points corresponding to their distinct tangent directions, so they no longer meet there.
There are only finitely many intersections after resolution. Blowing up each remaining common point therefore produces a surface on which the final strict transforms of \(C\) and \(D\) are disjoint.
5. Along the sequence in step 4, let\[ C_0=C, \qquad D_0=D, \]and let \(C_{i+1},D_{i+1}\) be the strict transforms after blowing up a point\(P_i\) on the \(i\)th surface. Then\[ C_i\cdot D_i - C_{i+1}\cdot D_{i+1} = \mu_{P_i}(C_i)\mu_{P_i}(D_i). \]
Apply step 3 on the \(i\)th surface. If the centre \(P_i\) lies on only one of the two strict transforms, the multiplicity of the other there is zero, so the same formula still applies and contributes zero. The nonzero terms are exactly the points at which the current strict transforms meet: the ordinary intersection points on \(X\) and their infinitely near successors.
6. The global intersection number is the sum of the products ofmultiplicities at all ordinary and infinitely near common points:\[ \boxed{ C\cdot D = \sum_P \mu_P(C)\mu_P(D).} \]
Sum the identities of step 5 over the finite blowup sequence. The left side telescopes: \[ \sum_i \bigl(C_i\cdot D_i-C_{i+1}\cdot D_{i+1}\bigr) = C_0\cdot D_0-C_N\cdot D_N. \] By step 4 the final strict transforms are disjoint, so \[ C_N\cdot D_N=0. \] Consequently \[ C\cdot D = \sum_i\mu_{P_i}(C_i)\mu_{P_i}(D_i). \] Omitting the zero terms leaves precisely the sum over all common points of \(C\) and \(D\), including infinitely near common points.
7. Q.E.D.
Step 3 proves the single-blowup formula, and steps <1>4–<1>6 give the iterated intersection-multiplicity formula.