## 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 $$z^k$$:
$$V^bullet_{z^k}(M).$$
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
$$M^nu otimes partial_t^j ~~textrm{with nu leq k(alpha-j)},$$
where
$$M^nu:={min Mmid u(zpartial_z-nu)^p=0text{ for }pgg 0}$$
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 $$Delta$$,