pt\to G/P$ is a birational isomorphism. Writing this as a composite $$BS_{\vec\alpha}\cdot pt\to G/B {\mathop{\twoheadrightarrow}}G/P,$$ the first map is birational, by the assumption of reducedness, to the (opposite) Schubert variety $\overline{B w_0 w_0^P B}/B$. The fiber over $gP \in G/P$ of the second map is $gP/B = g\overline{B w_0^P B}/B$. Hence the fiber over $gP$ of the composite is the intersection $$g\overline{B w_0^P B}/B \cap \overline{B w_0 w_0^P B}/B$$ which for generic $g$ is a point, since the $w_0$ makes these opposed Schubert varieties.

Multidegrees, $K$-polynomials, and the proofs of Theorems \[thm:multidegree\] and \[thm:Kpoly\] {#sec:mdegs}
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Multidegrees and the proof of Theorem \[thm:multidegree\].
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Let a torus $T$ act on a vector space $Y$. To each $T$-invariant subscheme $Z\subseteq Y$, we can associate a [[****]{}multidegree]{} $[Z]_Y$ living in the symmetric algebra on the weight lattice $T^*$ of $T$, satisfying the following properties:

1.  If $Z=Y=\{0\}$, then $[Z]_Y=1$.

2.  If as a cycle $Z = \sum_i m_i Z_i$, where the $\{Z_i\}$ are varieties occurring with multiplicities $\{m_i\}$, then $[Z]_Y = \sum_i m_i [Z_i]_Y$.

3.  If $H \leq Y$ is a $T$-invariant hyperplane, and $Z$ is a variety, then

    1.  if $Z \not\subseteq H$, then $[Z]_Y = [Z\cap H]_H$, but

    2.  if 