Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z\_T:=T\cap Z$. Is there a formula relating $mult\_p(Z)$ and $mult\_p(Z\_T)$? 



Multiplicity of a singular point
The answer is yes (auniket's comment meant probably $T$ is generic and not $p$). This is a classical result : if $A$ is a noetherian local ring of dimension $d>0$, with **infinite** residue field, then there exists a system of parameters $(f\_1,...,f\_d)$ such that $\mathrm{mult}(A)=\mathrm{mult}(A/(f\_1,...,f\_{d-1}))$, see Zariski-Samuel vol. II, Chap. VIII, §10, Theorem 22 and Remark page 296. 


Note that the result is false if the residue field is finite. But it is true that $\mathrm{mult}\_p(Z)$ is a finite integral combination (with possibly negative coefficients) of multiplicities at $p$ of $Z\cap T\_i$'s. See Proposition 5.9 in [here](http://www.math.u-bordeaux1.fr/~liu/articles/index-var.pdf). 


