If I take two curves $C,D$ on a surface $M$ with isolated intersection point $p$, then Noether gives a formula equating the intersection multiplicity $i_p(C,D)$ of $C$ and $D$ at $p$ in terms of their proper transforms $\tilde{C},\tilde{D}$ on the blow-up $\tilde{M}$ of $M$ at $p$. In particular, we have $i_p(C,D)=m_p(C)\cdot m_p(D)+\sum_{q\in E}i_q(\tilde{C},\tilde{D})$, where $m_p(C)$ is the multiplicity of $C$ at $p$ and $E$ is the exceptional divisor in $\tilde{M}$.
In "Positivity and Excess Intersection," Fulton and Lazarsfeld mention that this blown-up definition of intersection multiplicity does not work in general for varieties of dimension greater than 2, since the proper transforms of two subvarieties with proper (isolated) intersection need not intersect properly. Do such examples exist when $C\cap D$ is a zero dimensional local complete intersection? Or is the proper transform of a local complete intersection again lci?
For context, I'm interested in studying non-transverse intersection points of (zero dimensional) complete intersections (in $\mathbb{P}^n$) by reducing to the case of transverse intersections. If Noether's formula holds for complete intersections, it would give me a way to do so by taking repeated blow-ups.
