Proposition 19.9.27.987D754E15F6 For this flag,
\[ \{ \nu (s):0\neq s\in W_ m\} =\{ (a_1,\ldots ,a_{n-1},0):a_ i\in \mathbb N, \ \textstyle \sum _ i a_ i\leq km\} . \]
Consequently
\[ \Delta (W_\bullet )= \{ (x_1,\ldots ,x_{n-1},0):x_ i\geq 0, \ \textstyle \sum _ i x_ i\leq k\} . \]
Its $n$-dimensional volume is zero. When $k>0$, its volume in the coordinate hyperplane $x_ n=0$, with the Euclidean measure for which the coordinate lattice has covolume one, is
\[ \frac{k^{n-1}}{(n-1)!} =\lim _{m\to \infty }\frac{\dim W_ m}{m^{n-1}}. \]
For $k=0$ the body is a point and $\dim W_ m=1$.
Proof.
By proposition 19.8.13, sections of $L_ k^ m$ are homogeneous polynomials of degree $km$. Dividing successively by the highest powers of $z_1,\ldots ,z_{n-1}$ that divide the section, and restricting at each step, leaves a nonzero multiple of $z_ n^{km-a_1-\cdots -a_{n-1}}$, where $a_ i$ is the $i$th valuation. This residual section has no zeros on $Y_{n-1}$, so the last valuation is zero, and the preceding ones are nonnegative with sum at most $km$.
Conversely, every such tuple is the valuation of the monomial
\[ z_1^{a_1}\cdots z_{n-1}^{a_{n-1}} z_ n^{km-a_1-\cdots -a_{n-1}}. \]
The normalized values lie in the displayed simplex and, for $m=1$, include all its vertices, so their closed convex hull is the simplex. For $k>0$, its coordinate-lattice volume is $k^{n-1}/(n-1)!$, the leading coefficient of the binomial count in proposition 19.8.13. For $k=0$, only the constant polynomials remain.
$\square$
Pome
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