Let $X\subset\mathbb{P}^{n}$ be a smooth projective variety of pure dimension $d$. Let $Z\subset \mathbb{G}(n-d,n)\times\mathbb{P}^{n}$ be the space of pairs $(P,x)$ of a linear space $P\cong\mathbb{P}^{n-d}$ and a point $x\in P$. Let $Z_m\subset Z$ be the space of $(P,x)$ where $x\in X$ and $P\cap X$ has multiplicity at least $m$ at $x$. How does one compute the class of $[Z_m]\in A^{*}(Z)$?
If $d=1$ or $n-d=1$, the one can set up a (relative) jet bundle and express the contact order condition in terms of a degeneracy locus. However, already when $X\subset\mathbb{P}^{4}$ is a surface, I am not sure what to do, because the contact order condition does not seem to be "linear."
