Multiplicity on strict tranform and exceptional divisor : $m_x(\hat{C}\cap E)\geq m_x(\hat{C})$ for $x\in \hat{C}\cap E$

Question

Let $S$ be a surface, $C$ an irreducible curve on $S$ and $p\in C$ a point. Then consider the blow-up at $p$ and write $E$ the exceptional divisor. I want to show that if $x\in \hat{C}\cap E$, then $m_x(\hat{C}\cap E)\geq m_x(\hat{C})$.

I have tried using definitions that is $m_x(\hat{C}\cap E)=\dim \mathcal O_{S,p}/(f,g)$ and $m_x(\hat{C})$ being the maximal integer such that the local equation of $\hat{C}$ at $x$ is still in the power of the maximal ideal of $\mathcal O_{\hat{S},x}$ and creating a sequence of subspaces in $\mathcal O_{S,p}/(f,g)$ of size at least $m_x(\hat{C})$ but I can't get it to work.

Do you have ideas or other results ?

Answer 1

Let's remind ourselves of some facts about surfaces.

Fact 1. If $C,D$ are curves (i.e. effective divisors) on a smooth projective surface sharing no common irreducible component, then $C.D= \sum_{p\in C\cap D} i(C,D;p)$ where $i(C,D;p)$ is the intersection multiplicity at $p$.

Fact 2. Intersection theory on a blowup $\pi:\widetilde{X}\to X$ of a smooth projective surface $X$ at a point $p$ with exceptional divisor $E$ follows the following rules:

• $C.D=(\pi^*C).(\pi^*D)$
• $E^2=-1$
• $\pi^*C=\widetilde{C}+\mu_P(C)E$
• $(\pi^*C).E=0$
• $\widetilde{C}.E=\mu_P(C)$

Combining the statements in fact 2 and applying a little light algebra, we can show that if $C,D$ are curves on a smooth projective surface $X$ not sharing any common component, then after blowing up $X$ at a point $P$ we have $C.D = \widetilde{C}.\widetilde{D} + \mu_P(C)\mu_P(D)$.

Now we can show the result you want. Writing $C.D = \sum_{P\in C\cap D} i(C,D;P)$ and $\widetilde{C}.\widetilde{D} = \sum_{Q\in \widetilde{C}\cap \widetilde{D}} i(\widetilde{C},\widetilde{D};Q)$ and noting that the blowup map is an isomorphism on $X\setminus \{P\}$, we can subtract from both sides the contributions from the intersection multiplicity at points located in $X\setminus\{P\}\cong Bl_P X \setminus E$ to arrive at the equation $$ i(C,D;P) = \mu_P(C)\mu_P(D) + \sum_{Q\mapsto P, Q\in\widetilde{C}\cap\widetilde{D}} i(\widetilde{C},\widetilde{D},Q).$$

Applying this in your situation with the curves $\widehat{C}$ and $E$ on the surface $Bl_p S$ and noting that as $E$ is smooth, $\mu_x(E)=1$ for all $x\in E$, one has the result.