are $0$, then this product cannot be zero.

For nonlinear $Z\subseteq Y$, e.g. a union of linear subspaces, there might still be some cancelation giving $[Z]=0$. This can’t happen (as follows from Theorem D in [@KM]) if all the weights of $T$ acting on $Y$ live in an open half-space in $T^*$; in this very common case[^1] [*any*]{} closed $T$-invariant scheme $Z\subseteq Y$ has a nonzero multidegree $[Z]$.

\[thm:multidegree\] Let $\kappa : G\times^P Z\to Y$ be a Kempf collapsing, where $Lie(P)$ contains all the [*negative*]{} root spaces. Let $d$ be the number of components in a general fiber of $\kappa$. Assume that all the weights of $T$ acting on $Y$ live in an open half-space in $T^*$.

Let $m_0 = [Z]$, and construct a sequence of polynomials $m_1, m_2,\ldots$ by applying divided difference operators $\partial_\alpha := \frac{1}{\alpha} (1 - r_\alpha)$ to $m_0$, where $\alpha$ varies over the set of simple roots of $G$, and $r_\alpha$ acts on ${\mathrm{Sym}}^\bullet(T^*)$ from the reflection action on $T^*$. Don’t apply a divided difference operator if the result is $0$, and only stop when all $\partial_\alpha$ give the result $0$.

This process always terminates after the same number of steps (namely, $\dim G\times^P Z - 