$Z \subseteq H$, then $[Z]_Y = [Z]_H \cdot wt(Y/H)$, where $wt(Y/H) \in T^*$ is the $T$-weight on the line $Y/H$.

The multidegree generalizes the notion of degree of a projective variety ${\mathbb P}Z \subseteq {\mathbb P}Y$. If $T$ is just a circle acting on $Y$ by rescaling, and $Z$ is the affine cone (hence $T$-invariant) over a projective variety ${\mathbb P}Z$, then $[Z]_Y = (\deg {\mathbb P}Z) a^{{\mathrm{codim}\,}_Y Z}$ where $a$ is the generator of $T^*$. Multidegrees (in ${\mathrm{Sym}}(T^*)$) are a special case of equivariant Chow classes (in $A_T(Y)$); since $Y$ is equivariantly contractible we have $A_T(Y) {\mathop{\cong}}A_T(pt) {\mathop{\cong}}{\mathrm{Sym}}(T^*)$.

It is easy to see that properties (1)-(3) characterize multidegrees. One can show existence in several ways, one being through multigraded Hilbert series, as in the next section. Multidegrees were introduced by [@Jo]. Our reference for them is [@MS].

We only use three results about them. One that follows immediately from the properties above is that for $Z\leq Y$ a linear subspace, $[Z]_Y$ is the product of the weights in $Y/Z$. The second is that if all the $T$-weights in $Y$ lie in an open half-space, then $[Z]_Y
\neq 0$ for $Z$ nonempty. (This follows from Theorem D in [@KM], and 