On a conjecture of Pappas and Rapoport about the standard local model for GL d

. In their study of local models of Shimura varieties for totally ramiﬁed extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n (cid:2) n matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The ﬁrst is our proof of a conjecture of Kreiman, Lakshmibai, Magyar


Introduction
Shimura varieties serve as a bridge between arithmetic geometry and automorphic forms, and as such they play a central role in the Langlands program. Rapoport and Zink ( [RZ96]) introduced the study of local models of Shimura varieties. These local models are intended to capture the behaviour that occurs when considering reduction mod p and often allow one to reduce arithmetic problems to questions about affine Grassmannians and flag varieties. These problems tend to be difficult but often tractible (e.g. [He13,Zhu14,HR19]).
Local models are defined over the ring of integers O E of a local field E (the reflex field). An essential desideratum is that the local model be flat over O E , which involves a set-theoretic condition called topological flatness and a scheme-theoretic condition about the location of embedded primes (see e.g. [GW10,§14.3]). For example, a topologically-flat family over O E with reduced special fiber is flat.
In [PR03], Pappas and Rapoport study the standard (local) model for GL d in the case of a totally ramified extension. As Pappas previously observed ( [Pap00]), this model is not topologically flat in general. Topological flatness does hold in the cases where, in their notation, the pr ϕ q ϕ differ by at most 1 ([PR03, Proposition 3.2]). In these cases, flatness of the standard model would follow from the reducedness of the following subscheme of nˆn matrices ([PR03, Theorem B (iii), Corollary 5.9]): N n,e " tA P Mat nˆn | A e " 0, detpλ´Aq " λ n u (1.1) Questions of flatness of local models have been studied by other authors. Görtz proved flatness of unramified local models for the general linear and symplectic groups [Gör01,Gör03]. We mention also results on topological flatness by Görtz and by Smithling ([Gör05,Smi11b,Smi11a,Smi14]).
1.0.1. Weyman's work. Soon after the Pappas-Rapoport conjecture was first announced in 2000, Weyman ([Wey02]) gave proofs of the conjecture (i) when the base field is characteristic zero, and (ii) when the base field has arbitrary characteristic and e " 2. Our main contribution is proving the conjecture in positive characteristic, which is the interesting case for arithmetic applications. We make no assumptions on the characteristic, so our argument also gives a new proof in characteristic zero.
1.1. How the space N n,e appears. We recall very briefly how N n,e appears in the work of Pappas and Rapoport. We refer the reader to [PR03,§2,3,4] for the details. This discussion is for motivation, and will not be needed for the rest of the paper.
Fix a complete discretely valued field F 0 with perfect residue field, and let F{F 0 be a totally ramified extension of degree e. Fix a uniformizer π 0 of F 0 , and let π be a uniformizer of F which is a root of the Eisenstein polynomial of the extension (see [PR03,Equation 2.1]). Fix a positive integer d and a rank-d free module Λ for the ring of integers O F of F. Finally, for each embedding ϕ : F Ñ F sep 0 of F into a fixed separable closure of F 0 , one fixes an integer r ϕ such that 0 ď r ϕ ď d. Associated to this data, one defines the reflex field E [PR03, Equation (2.3)] and its ring of integers O E . Pappas and Rapoport define a scheme M, which they call the standard model for GL d corresponding to r " pr ϕ q ϕ . This is a scheme over Spec O E . Additionally, they construct two auxiliary objects Ă M and N " Nprq, which are also schemes over O E . These three spaces are related by maps: One of Pappas and Rapoport's main theorems is the following. Let k be the residue field of O E . Let V " Λ b O F 0 k and let T " π b id k (these are denoted W and Π in [PR03,Equation 2.9]). Let n " ř r ϕ . Upon base changing to k, it is apparent from the definitions that (1.2) becomes: Here 1 S T is the subscheme of the Grassmannian Grpn, Vq consisting of T -invariant n-planes U such that detpλ´T | U q " λ n , Ą 1 S T is its preimage in the Stiefel variety, and the map ϕ comes from a canonical map from Ą 1 S T to Mat nˆn . See §2.1 and §2.3 for further explanation. By base change, we immediately have the following corollary.
In §2.3 we will explain briefly Pappas and Rapoport's proof of this corollary.
Of central importance to Pappas and Rapoport is whether the space M is flat over O E . When the pr ϕ q ϕ above differ from each other by more than 1, they prove that M is not flat for dimension reasons. What remains are the cases where the pr ϕ q ϕ differ from each other by at most one. By [PR03,Theorem B (iii)], flatness in these cases would follow from the reducedness of N n,e . As a corollary of our work one obtains flatness of M in these cases, and therefore, in the terminology of [PR03], that the naïve local model coincides with the local model. 1.2. Our approach. Our solution follows naturally by building on the ideas of our previous work [MWY] and earlier work by the first two authors and Kamnitzer [KMW18]. One of our stated goals in [MWY,§1.3.2] is to use the framework developed there to understand the scheme structure of nilpotent orbit closures in positive characteristic. The present work is our first success in this program.
In [MWY], we proved a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman about the equations defining affine Grassmannians for SL n . Their conjecture is that the defining equations are given by certain explicit shuffle operators.
Our study led us to consider the general situation of a nilpotent operator T on a vector space V. We consider the Grassmannian Grpn, Vq, and define a subscheme S T by explicit T -shuffle operators generalizing the shuffle operators. Our first result (Theorem 2.3) is that 1 S T " S T , where 1 S T is the scheme appearing in the work of Pappas and Rapoport as explained above. This provides the link between our previous work and the work of Pappas and Rapoport. Theorem 2.3 should be of general interest because 1 S T is given as an explicit moduli functor, while S T is given by explicit homogeneous equations in the Plücker embedding of the Grassmannian.
In [KMW18], we developed some techniques for attacking a conjecture about the reduced scheme structure on Schubert varieties in the affine Grassmannian. We use this to prove that a certain family of S T are reduced (Proposition 3.1). In particular, we show that these schemes are Schubert varieties in the affine Grassmannian with their reduced scheme structure. Using the fact that Schubert varieties are Frobenius split compatibly with their Schubert subvarieties, we can take intersections and conclude that a larger family of S T are also reduced (Proposition 3.3). It follows that all corresponding Ă S T are reduced, since Ă S T Ñ S T is a GL n -torsor. We note that this strategy is similar to the proof of [Gör01, Theorem 4.5.1], where the special fiber M loc of a local model is expressed as an intersection of smooth Schubert varieties. One additional difficulty for us is that we consider intersections of singular Schubert varieties. Finally, we show that for each n and e that there is some S T (which we have shown is reduced) such that the corresponding map Ă S T Ñ N n,e is surjective and smooth. Surjectivity is a direct calculation (Proposition 3.7). Smoothness is due to Pappas and Rapoport (Theorem 2.7). They work in a more complicated situation than we consider, so for the benefit of the reader we explain in §2.3 how their proof works in our simpler setting. Therefore, we conclude that N n,e is reduced.
After analyzing our argument, we found a simplified but less conceptual version that is almost completely elementary. The only non-elementary ingredient is the Frobenius splitting of Schubert varieties. We present this simplified argument in §3.1.
1.3. Acknowledgements. We would like to thank an anonymous referee of [MWY] for suggesting the connection between our work and that of Pappas and Rapoport, and Geordie Williamson and Xinwen Zhu for their comments on an earlier version of this manuscript. We are also grateful to the Centre de Recherches Mathématiques and to Joel Kamnitzer and Hugh Thomas for organizing the program "Quiver Varieties and Representation Theory" where this project was started.

Preliminaries
2.1. Grassmannians and associated schemes. Let k be a field of arbitrary characteristic. Because our goal is to prove reducedness of a scheme, we may assume k is algebraically closed. Let V be a finite-dimensional vector space over k, and let n be a non-negative integer. Write Grpn, Vq for the Grassmannian of n-planes in V. Let V denote the trivial bundle on Grpn, Vq, and let T denote the tautological rank-n vector subbundle of V. Write Op´1q " Ź n T, which is a line subbundle of the trivial bundle Ź n V. This defines the Plücker embedding Grpn, Vq ãÑ Pp Ź n Vq.
Further, let us assume that T : V Ñ V is a nilpotent operator. We denote by G T pn, Vq the subscheme of Grpn, Vq consisting of T -invariant subspaces (see e.g. [MWY,§5.4]). When n and V are clear from context, we will simply write G T " G T pn, Vq.
For any T -invariant n-plane U, we can consider the characteristic polynomial detpλ´T | U q. This defines a regular function χ : G T Ñ krλs. Notice that set theoretically χ is equal to λ n , but this is not necessarily true scheme-theoretically. In particular, the non-leading coefficients of χ are all nilpotent elements of krG T s. We define 1 S T to be the closed subscheme of G T defined by the In [MWY,§6.1], we define for each d ě 1 shuffle operators, which are linear operators sh T d : For each sh T d we can consider the vanishing locus, denoted Vpsh T d q, which is the subvariety of Pp Ź n Vq given by the projectivization of the kernel of sh T d . By composing with all linear functionals on Ź n V, the shuffle operators define a subspace of linear forms on Pp Ź n Vq.
We will refer to the homogeneous ideal generated by these linear forms as the shuffle ideal.
We define S T " S T pn, Vq to be the closed subscheme of Grpn, Vq defined by the shuffle ideal, i.e. we define S T " Ş dě1 Vpsh T d q X Grpn, Vq. In our previous work, we proved the following.
Theorem 2.1. [MWY, Theorem 6.8] The subscheme S T is a closed subscheme of G T . This closed embedding induces an isomorphism of reduced schemes.
Remark 2.2. In fact we conjecture ([MWY, Conjecture 7.7]) that S T is reduced and is therefore equal to the reduced scheme of G T .
The Stiefel variety Č Grpn, Vq is the open subscheme of Hom k pk n , Vq consisting of rank-n linear transformations. We have a natural map Č Grpn, Vq Ñ Grpn, Vq, which is a GL n -torsor. For any closed subscheme X ãÑ Grpn, Vq, we define r X to be the preimage of X inside the Stiefel variety. For example, we define Ă S T this way.
2.2. Alternative description of S T .
Theorem 2.3. We have an equality S T " 1 S T as subschemes of Grpn, Vq.
Proof. Recall that S T is defined by intersecting the Grassmannian Grpn, Vq with the vanishing locus of shuffle operators sh T d : Consider these as operators on the trivial bundle sh T d : Ź n V Ñ Ź n V, which we can restrict to get morphisms: One can alternately define S T to be the intersection of the vanishing loci of (2.4). Observe by Theorem 2.1 that over G T , the maps sh T d : Op´1q Ñ Ź n V factor through maps sh T d : Op´1q Ñ Op´1q.
Consider the krzs-linear operator I`zT on V bkrzs, and consider its n-th wedge power Ź n p1`zT q (taken over krzs), which acts on Ź n pVq b krzs. By [MWY,Lemma 6.5], we have the following equality n ľ pI`zT q " I`n ÿ of operators on Ź n pVq b krzs.
Write A 1 " Spec krzs. We interpret (2.5) as an equality of endomorphisms of Ź n V b krzs, the trivial bundle on Grpn, VqˆA 1 with fiber Ź n V. If we restrict to G TˆA1 , we obtain a map: Op´1q is a line bundle, the map (2.6) is equivalent to the data of a regular function on G T valued in krzs. By (2.5), this map is given by the z n χp´z´1q, where χ : G T Ñ krzs is the characteristic polynomial map defined above.

2.3.
Smoothness of the map ϕ : Ă S T Ñ N n,e . We have a map ϕ : Ă S T Ñ Mat nˆn defined as follows. For a test ring R, we have Ă S T pRq " pU, ψq | U P S T pRq and ψ : R n " Ñ U ( . The map ϕ sends a pair pU, ψq to the nˆn matrix ψ´1˝T˝ψ. Because of the characteristic polynomial equation defining S T , the image of ϕ lies in the nilpotent cone N (even scheme theoretically). In general this map is poorly behaved, but in the special case we consider below it is smooth.
Let d, e ě 1 be integers, let V be a vector space of dimension de and suppose T is the nilpotent endomorphism of V that is given by a standard Jordan form corresponding to the partition pe d q. Let S T " S T pn, Vq for some n ď de. In this case, the morphism ϕ factors through a map ϕ : Ă S T Ñ N n,e .
The proof of [PR03, Theorem 4.1] holds in a more complicated arithmetic situation, and it simplifies greatly in the present one. For the benefit of the reader, we will briefly explain how their proof works in our situation.
Define a closed subscheme V of Mat n,nˆH om k pk n , Vq as follows. For any test ring R, consider the ring S " Rrts{t e . The operator T defines an S-module structure on V b R. Additionally, any point A P N n,e pRq defines an S-module structure on R n , which we denote by R n A . We define V by: VpRq " tpA, ϕq | A P N n,e pRq and ϕ P Hom S pR n A , V b Rqu (2.8) It is clear that Ă S T admits an open embedding into V with image equal to the set of pairs pA, ϕq where ϕ is of maximal rank.
We also have a map V Ñ N n,e , but for general T this map will not be well behaved. However, our assumptions on the Jordan type of T imply that V b R is a free S-module of rank d. Using this, one concludes that the map V Ñ N n,e is a (trivial) vector bundle of rank nd. Therefore, the map Ă S T Ñ N n,e is smooth because it factors as an open embedding into a vector bundle followed by projection from the vector bundle to its base.
Remark 2.9. For a general T with T e " 0, the fibers of V Ñ N n,e are closely related to the work of Shayman [Sha82, Section 5], who studied certain locally closed subvarieties of the varieties S T .
2.4. Type A affine Grassmannians. Let n ě 2. Recall the GL n affine Grassmannian Gr GL n parameterizing lattices L Ă kpptqq n such t a L 0 Ď L Ď t´bL 0 for some integers a, b ě 0 where L 0 " krrtss n (see e.g. [Zhu17, Section 1.1]). We further require that dim k pL X L 0 q{L 0´d im k pL X L 0 q{L " 0. This condition means that we are only considering the connected component of the full affine Grassmannian of GL n containing the point L 0 . We will not use the other connected components, so our notation should cause no confusion.
For each pair of integers a, b ě 0, we can consider the linear operator T a,b on t´bL 0 {t a L 0 given by multiplication by t. We therefore obtain a closed embedding G T a,b " G T a,b pna, t´bL 0 {t a L 0 q ãÑ Gr GL n . Explicitly, we identify G T a,b with the subscheme of Gr GL n parameterizing lattices L Ă kpptqq n such that t a L 0 Ă L Ă t´bL 0 and dim`L{t a L 0˘" na. Taking direct limits we have (2.10) This realizes Gr GL n as an ind-scheme of ind-finite type. We define the affine Grassmannian Gr SL n for SL n to be the induced reduced structure of Gr GL n . Explicitly: pG T a,b q red (2.11) We may also consider S T a,b " S T a,b pna, t´bL 0 {t a L 0 q. A priori, S T a,b is a subscheme of Gr GL n , but below (Corollary 3.9) we will show that S T a,b is in fact equal to the reduced scheme pG T a,b q red and is therefore a subscheme of Gr SL n .
2.4.1. Big cells. Consider the decomposition kpptqq n " L 0 't´1krt´1s n and the induced projection map kpptqq n Ñ L 0 . The big cell of Gr GL n is the open locus consisting of lattices L such that the restricted projection map L Ñ L 0 is an isomorphism. As is well known (see e.g. [Fal03, Lemma 2]), the big cell is isomorphic to the ind-scheme GL p1q n rt´1s whose R-points are given by tApt´1q P GL n pRrt´1sq | Apt´1q " 1 mod t´1u. The open immersion GL p1q n rt´1s ãÑ Gr GL n is given by sending a matrix polynomial Apt´1q to the lattice spanned by its columns. We will use this identification and simply write GL p1q n rt´1s for the big cell. We can intersect the big cell of Gr GL n with Gr SL n to obtain the big cell of the latter. As above, we can identify the big cell of Gr SL n with SL p1q n rt´1s, which is analogously defined. Observe that the group SL n rrtss acts on Gr GL n by left multiplication. We note that every orbit meets the big cell as we consider only the L 0 connected component of the full affine Grassmannian of GL n .

Schubert varieties.
Recall that the SL n rrtss orbits on Gr SL n are indexed by dominant cocharacters. For a dominant cocharacter µ, let Gr µ be the corresponding orbit closure in Gr SL n . By definition Gr µ has reduced scheme structure. Additionally, Gr SL n is isomorphic to a partial flag variety for the Kac-Moody group y SL n , and the subvarieties Gr µ are Schubert subvarieties. In particular, each Gr µ admits a Frobenius splitting compatible with all of its Schubert subvarieties ([Mat88, Ch. VIII], see also [Fal03,§4] and [BK05, Chapter 2]). Therefore, the scheme theoretic intersection of two Schubert varieties is Frobenius split and hence reduced. This result is the only consequence of Frobenius splitting we will use.
The Schubert varieties relevant to us are Gr pn̟ 1 and Gr qn̟ n´1 , where p, q ě 1 are integers. Here n̟ 1 and n̟ n´1 are n times the first and last fundamental coweight, which are the minimal multiples that lie in the cocharacter (equivalently, coroot) lattice for SL n . We have Gr pn̟ 1 " pG T ppn´1q,p q red , Gr qn̟ n´1 " pG T q,qpn´1q q red (2.12) which is an explicit set-theoretic description of these Schubert varieties in terms of lattices. Because of Theorem 2.1, one immediately obtains closed embeddings Gr pn̟ 1 ãÑ S T ppn´1q,p , Gr qn̟ n´1 ãÑ S T q,qpn´1q (2.13) that are bijective on points. Below (Proposition 3.1) we will show that these are actually isomorphisms of schemes. In particular, the naïve lattice description of these Schubert varieties is scheme-theoretically correct once we add a further condition about the characteristic polynomial of t (as in the defintion of 1 S T ). . This induces an involution ω of SGr that preserves Gr GL n and Gr SL n . On the level of Schubert varieties, ω maps Gr µ isomorphically onto Gr µ˚w here˚is induced by the unique non-trivial diagram automorphism of SL n . In particular, ω maps Gr pn̟ 1 isomorphically onto Gr pn̟ n´1 . On the level of big cells GL p1q n rt´1s and SL p1q n rt´1s, ω is given by the following map on matrix polynomials: where J is the nˆn antidiagonal matrix with entry p´1q j in column j, and where the superscript T denotes the inverse transpose.

The main proof
Proposition 3.1. For all p, q ě 1, the embeddings Gr pn̟ 1 ãÑ S T ppn´1q,p and Gr qn̟ n´1 ãÑ S T q,qpn´1q from (2.13) are isomorphisms. In particular, S T ppn´1q,p and S T q,qpn´1q are reduced.
Proof. First consider the embedding Gr pn̟ 1 ãÑ S T ppn´1q,p . Both spaces are invariant under the action of SL n rrtss, so it suffices to check that this map is an isomorphism on the big cell GL p1q n rt´1s Ď Gr GL n . Observe that S T ppn´1q,p XGL p1q n rt´1s maps into the closed subfunctor of GL p1q n rt´1s consisting of matrix polynomials with degree less than or equal to p. In [KMW18, Corollary 2] (see also the proof of [BL94, Proposition 6.1]), we proved that Gr pn̟ 1 X GL p1q n rt´1s is equal as a scheme to the subscheme of matrix polynomials Apt´1q " 1`A 1 t´1`¨¨¨`A p t p such that detpApt´1qq " 1. In [MWY,Theorem 4.27] we showed that the coefficients of detpApt´1qq lie in the shuffle ideal defining S T ppn´1q,p . Therefore the embedding is an isomorphism.
The second isomorphism follows by applying the diagram automorphism ( §2.4.3) and observing that the shuffle ideal considered in [MWY] is invariant under it (see [MWY,Corollary 4.13, Theorem 4.27]).
Remark 3.2. Let L be an R-point of S T ppn´1q,p X GL p1q n rt´1s. Then L may be uniquely represented by a matrix polynomial Apt´1q P GL p1q n rt´1s. It is not difficult to show that there is an R-basis of t´pL 0 {L under which the matrix of t is the companion matrix of the matrix polynomial Apt´1q " 1`A 1 t´1`¨¨¨`A p t´p. In particular, the characteristic polynomial of this matrix is λ pn¨A pλ´1q. We see that the characteristic polynomial equation defining S T ppn´1q,p (via Theorem 2.3) corresponds exactly to the equation detpApt´1qq " 1 defining Gr pn̟ 1 X GL p1q n rt´1s inside matrix polynomials with degree less than or equal to p. This in particular gives another proof of [MWY,Theorem 4.27].
Proof. It is clear that S T M,N is equal set theoretically to the intersection Gr pn̟ _ 1 X Gr qn̟ _ n´1 . Furthermore, it is known that the scheme-theoretic intersection Gr pn̟ _ 1 X Gr qn̟ _ n´1 is reduced by Frobenius splitting (see §2.4.2).
Denote K " maxtp, qpn´1qu. Then we have a Cartesian diagram of closed embeddings and a diagram of closed embeddings which is not a priori Cartesian. By Proposition 3.1, the objects in these two diagrams agree in all but the northwest corner. By the universal property of the fiber product, we obtain a closed embedding S T M,N ãÑ Gr pn̟ _ 1 X Gr qn̟ _ n´1 . Because Gr pn̟ _ 1 X Gr qn̟ _ n´1 is reduced and this map is a bijection on points, it is an isomorphism.
Remark 3.6. The intersection Gr pn̟ 1 X Gr qn̟ n´1 is irreducible and is therefore also a Schubert variety for Gr SL n (see [KMW18,Proposition 5.4]). In fact it is also not hard to see that S T q,p is isomorphic to Gr pn̟ 1 X Gr qn̟ n´1 and is in particular reduced.
Proposition 3.7. Let e be an integer with n ě e ě 1. The map ϕ : Č S T 1,e´1 Ñ N n,e is surjective.
Proof. As surjectivity is a set-theoretic statement, it suffices to work with reduced schemes. Dividing with remainder, write n " ce`f where 0 ď f ă e, and let τ " pe c , fq be the partition with c parts of size e and one part of size f. Then the reduced scheme of N n,e is exactly the nilpotent orbit closure O τ Ď Mat nˆn . The map ϕ is GL n -equivariant, so it suffices to check that every nilpotent orbit O σ Ă O τ has non-empty intersection with the image. This is a straightforward calculation.
Theorem 3.8. For every n, e with 1 ď e ď n, the scheme N n,e is reduced.
Proof. The scheme S T 1,e´1 is reduced by Proposition 3.3 (choosing p " e´1 and q " 1). The map Č S T 1,e´1 Ñ S T 1,e´1 is a torsor for GL n , so Č S T 1,e´1 is also reduced. By Theorem 2.7 and Proposition 3.7, the map ϕ : Č S T 1,e´1 Ñ N n,e is smooth and surjective. In particular, it is faithfully flat, and the property of being reduced descends along faithfully flat morphisms.
3.1. Another proof. After closely inspecting the above argument, we found the following direct proof of Theorem 3.8. The idea is essentially the same as the more conceptual proof above, but gets to the punchline quickly and by more elementary means. The only non-elementary ingredient is the Frobenius splitting of Schubert varieties.
Another proof of Theorem 3.8. For any p ě 0, define Z p to be the closed subscheme of Mat nˆn rt´1s that represents the functor sending any test ring R to: ! 1`B 1 t´1`¨¨¨`B p t´p P Mat nˆn pRrt´1sq | detp1`B 1 t´1¨¨¨`B p t pq q " 1 ) We see that Z p is a subscheme of SL p1q n rt´1s, and it is equal as a scheme to the intersection Gr pn̟ 1 X SL p1q n rt´1s by [KMW18, Corollary 2] (see also the proof of [BL94, Proposition 6.1]). In particular, Z p is reduced.
Let X be the closed subscheme Mat nˆn rt´1s that represents the functor: XpRq " ! 1`C 1 t´1`¨¨¨`C n´1 t´p n´1q P Mat nˆn pRrt´1sq | detp1´C 1 t´1q " 1 and C i " C i 1 ) Observe that the condition detp1´C 1 t´1q " 1 is equivalent to requiring that the characteristic polynomial of C 1 is λ n . By the Cayley-Hamilton theorem, we have C n 1 " 0. Therefore, 1`C 1 t´1`¨¨¨`C n´1 t´p n´1q " p1´C 1 t´1q´1 and detp1`C 1 t´1`¨¨¨`C n´1 t´p n´1q q " 1. So X is equal to a closed subscheme of Gr pn´1qn̟ 1 X SL p1q n rt´1s. The involution (2.14) of SL p1q n rt´1s gives an isomorphism between X and Z 1 . We therefore conclude that X is equal as a scheme to the intersection Gr n̟ n´1 X SL p1q n rt´1s. In particular, X is reduced. More directly, the map X Ñ Mat nˆn sending 1`C 1 t´1`¨¨¨`C n´1 t´p n´1q to C 1 induces an isomorphism between X and the nilpotent cone N. Now consider the scheme theoretic intersection XXZ e´1 , which we compute inside Z n´1 . Because Z n´1 admits a Frobenius splitting compatible with X and Z e´1 , the intersection XXZ e´1 is reduced.
On the other hand, we directly compute pX X Z e´1 qpRq to be the set of 1`C 1 t´1`¨¨¨C n´1 t´p n´1q P Mat nˆn pRrt´1sq such that detp1´C 1 t´1q " 1 , C i " C i 1 , and C i " 0 for i ě e. Therefore, under the isomorphism X Ñ N, we see that the X X Z e´1 maps isomorphically to N n,e .
Recall that all schemes of the form S T a,b are reduced (Proposition 3.3, Remark 3.6). We can also argue this directly using the proof above. For all a, b, the map ϕ : Ć S T a,b Ñ N na,a`b is smooth (but not necessarily surjective). Because reducedness ascends along smooth maps, and Ć S T a,b Ñ S T a,b is a torsor for GL na , we obtain another proof of the following: Corollary 3.9. For all a, b ě 1, the scheme S T a,b is reduced.