d modules – V-filtration along a non-reduced hypersurface

Let $Delta$ be a small disk, and let $M$ be a regular holonomic right $mathcal{D}_Delta$-module. Let $z$ be the coordinate on $Delta$, and $kgeq 1$. Then we get the $V$-filtration on $M$ along $z^k$:
Recall that since for $k>1$ the function $z^k$ is not smooth, this is defined as
$$V^alpha_{z^k}(M):= (Motimes 1)cap V^{alpha}_t(i_{k+}M),$$
where $i_k:Deltato Delta^2$ is the map $cmapsto (c,c^k).$

I would guess that:
$$V^{alpha}_{z^k}(M) = V^{kalpha}_z(M),$$
but I cannot prove it.

Question 1: is this correct?

Besides not being able to prove it, the result in this question suggests that it might not be true. This result shows that $V^alpha_{t}(i_{k+}M)$ is generated over $mathcal{D}_Delta$ by
$$M^nu otimes partial_t^j ~~textrm{with $nu leq k(alpha-j)$},$$
$$M^nu:={min Mmid u(zpartial_z-nu)^p=0text{ for }pgg 0}$$
Now working this out means that we expect that
$$V^{alpha}_{t}(i_{k+}M)subsetneq V^{kalpha}_{t}(i_{1+}M).$$
However, I am not sure that this also automatically means that
$$V^{alpha}_{z^k}(M)=V^{alpha}_{t}(i_{k+}M)cap (Motimes 1)subsetneq V^{kalpha}_{t}(i_{1+}M)cap (Motimes 1)=V^{kalpha}_z(M)?$$

More generally I am not sure that
$$gr_{V_t}^alpha(i_+M)not=0 Leftrightarrow gr_{V_f}^alpha(M)not=0.$$
Where $i$ again denotes the graph embedding along $f$.

Question 2: is this equivalence true?

Notice that this does not follow from Kashiwara’s equivalence, since the filtered pieces in the $V$-filtration are not $D_X$ and $D_{Xtimes mathbb{C}}$-modules.

Ag.algebraic geometry – Non-reduced base location

I am looking for examples of abundant line bundles over (possibly smooth) projective varieties whose baseline is either unreduced or generically unreduced.

The only example I can find is a general type and general point curve $ p $ in order to $ h ^ 0 (2p) = 1 $,

Is there a criterion for the (generic) reduction of the base location? Or at least of small size? Perhaps large line bundles of Calabi Yau and / or Fano varieties have reduced the base location?

Representation Theory – Definition of the weight grid for a non-reduced root system

To let $ (V, Phi) $ be a root system with dual root system $ (V ^ { ast}, Phi ^ { vee}) $, To let $ Delta = { alpha_1, …, alpha_n } $ a series of simple roots for $ V $, and let $ Delta ^ { vee} = { alpha_1 ^ { vee}, …, alpha_n ^ { vee} } $ be the corals that correspond $ Delta $,

We have the basic weights $ has { Delta} = { omega_1, …, omega_n } $which by definition is the double base of $ Delta ^ { vee} $and we have the basic coweights $ has { delta} ^ { vee} = { omega_1 ^ { vee}, …, omega_n ^ { vee} } $which by definition is the double base of $ Delta $,

When $ Phi $ is reduced, the grid $$ {v in V: langle v, alpha ^ { vee} rangle in mathbb Z textrm {for all} alpha ^ { vee} in Phi ^ { vee} $$
and $$ mathbb Z omega_1 oplus cdots mathbb Z omega_n $$
are equal, and this is called a weight grid. But for $ Phi $ not necessarily reduced, the second grid is generally a proper subset of the first. This is so, though $ Delta $ is a base of $ Phi $, $ Delta ^ { vee} $ is generally not a base of $ Phi ^ { vee} $ (there if $ alpha in Delta $ is that $ 2 alpha $ is therefore a root $ alpha ^ { vee} $ is divisible by $ (2 alpha) ^ { vee} $).

How should the weight grid be defined in the unreduced case? Should we even bother to call that? $ omega_i $ "basic weights?" I know that $ has { Delta} $ and $ has { Delta} ^ { vee} $ will be treated in the unreduced case, for example in the proof of the Langlands classification here (section 2.2).

A possible alternative to this is to be defined $ Delta ^ { vee} $ not like the Koroots theorem $ alpha ^ { vee} $ to the $ alpha in Delta $but rather as a set of simple roots of $ Phi ^ { vee} $ according to the same chamber as $ Delta $,