arXiv:0704.0895v1  [math.AG]  6 Apr 2007
Gorenstein locus of minuscule Schubert varieties
Nicolas Perrin
Abstract
In this article, we describe explicitely the Gorenstein locus of all minuscule Schubert varieties.
This proves a special case of a conjecture of A. Woo and A. Yong [WY06b] on the Gorenstein
locus of Schubert varieties.
Introduction
The description of the singular locus and of the types of singularities appearing in Schubert
varieties is a hard problem. A first step in this direction was the proof by V. Lakshmibai and B.
Sandhya [LS90] of a pattern avoidance criterion for a Schubert variety in type A to be smooth. There
exist some other results in this direction, for a detailed account see [BL00]. Another important
result was a complete combinatorial description, still in type A, of the irreducible components of
the singular locus of a Schubert variety (this has been realised, almost in the same time, by L.
Manivel [Ma01a] and [Ma01b], S. Billey and G. Warrington [BW03], C. Kassel, A. Lascoux and C.
Reutenauer [KLR03] and A. Cortez [Co03]). The singularity at a generic point of such a component
is also given in [Ma01b] and [Co03]. However, as far as I know, this problem is still open for other
types. Another partial result in this direction is the description of the irreducible components of
the singular locus and of the generic singularity of minuscule and cominuscule Schubert varieties
(see Definition 1.2) by M. Brion and P. Polo [BP99].
In the same vein as [LS90], A. Woo and A. Yong gave in [WY06a] and [WY06b] a generalised
pattern avoidance criterion, in type A, to decide if a Schubert variety is Gorenstein. They do not
describe the irreducible components of the Gorenstein locus but give the following conjecture (see
Conjecture 6.7 in [WY06b]):
CONJECTURE 0.1. — Let X be a Schubert variety, a point x in X is in the Gorenstein locus of
X if and only if the generic point of any irreducible component of the singular locus of X containing
x is is the Gorenstein locus of X.
The interest of this conjecture relies on the fact that, at least in type A, the irreducible compo-
nents of the singular locus and the singularity of a generic point of that component are well known.
The conjecture would imply that one only needs to know the information on the irreducible com-
ponents of the singular locus to get all the information on the Gorenstein locus.
In this paper we prove this conjecture for all minuscule Schubert varieties thanks to a combi-
natorial description of the Gorenstein locus of minuscule Schubert varieties. To do this we use the
http://arxiv.org/abs/0704.0895v1
combinatorial tool introduced in [Pe07] associating to any minuscule Schubert variety a reduced
quiver generalising Young diagrams. First, we translate the results of M. Brion and P. Polo [BP99]
in terms of the quiver. We define the holes, the virtual holes and the essential holes in the quiver
(see Definitions 2.3 and 3.1) and prove the following:
THEOREM 0.2. — (ı) A minuscule schubert variety is smooth if and only if its associated quiver
has no nonvirtual hole.
(ıı) The irreducible components of the singular locus of a minuscule Schubert variety are indexed
by essential holes.
Furthermore we explicitely describe in terms of the quiver and the essential holes these irre-
ducible components and the singularity at a generic point of a component (for more details see
Theorem 3.2). In particular, with this description it is easy to say if the singularity at a generic
point of an irreducible component of the singular locus is Gorenstein or not. The essential holes
corresponding to irreducible components having a Gorenstein generic point are called Gorenstein
holes (see also Definition 3.8). We give the following complete description of the Gorenstein locus:
THEOREM 0.3. — The generic point of a Schubert subvariety X(w′) of a minuscule Schubert
variety X(w) is in the Gorenstein locus if and only if the quiver of X(w′) contains all the non
Gorenstein holes of the quiver of X(w).
COROLLARY 0.4. — Conjecture 0.1 is true for all minuscule Schubert varieties.
Example 0.5. — Let G(4, 7) be the Grassmannian variety of 4-dimensional subspaces in a 7-
dimensional vector space. Consider the Schubert variety
X(w) = {V4 ∈ G(4, 7) dim(V4 ∩W3) ≥ 2 and dim(V4 ∩W5) ≥ 3}
where W3 and W5 are fixed subspaces of dimension 3 and 5 respectively. The minimal length
representative w is the permutation (2357146). Its quiver is the following one (all the arrows are
going down):
We have circled the two holes on this quiver. The left hole is not a Gorenstein hole (this can be
easily seen because the two peaks above this hole do not have the same height, see Definition 2.3)
but the right one is Gorenstein (the two peaks have the same height). Let X(w′) be an irreducible
component of the singular locus of X(w). The possible quivers of such a variety X(w′) are the
following (for each hole we remove all the vertices above that hole):
These Schubert varieties correspond to the permutations: (1237456) and (2341567). Let X(w′) be
a Schubert subvariety in X(w) whose generic point is not in the Gorenstein locus. Then X(w′) has
to be contained in X(1237456).
Acknowledgements: I thank Frank Sottile and Jim Carrel for their invitation to the BIRS
workshop Comtemporary Schubert calculus during which the major part of this work has been
done.
1 Minuscule Schubert varieties
Let us fix some notations and recall the definitions of minuscule homogeneous spaces and minuscule
Schubert varieties. A basic reference is [LMS79].
In this paper G will be a semi-simple algebraic group, we fix B a Borel subgroup and T a
maximal torus in B. We denote by R the set of roots, by R+ and R− the set of positive and
negative roots. We denote by S the set of simple roots. We will denote by W the Weyl group of G.
We also fix P a parabolic subgroup containing B. We denote by WP the Weyl group of P
and by WP the set of minimal length representatives in W of the coset W/WP . Recall that the
Schubert varieties in G/P (that is to say the B-orbit closures in G/P ) are parametrised by WP .
DEFINITION 1.1. — A fundamental weight ̟ is said to be minuscule if, for all positive roots
α ∈ R+, we have 〈α∨,̟〉 ≤ 1.
With the notation of N. Bourbaki [Bo68], the minuscule weights are:
Type minuscule
An ̟1 · · ·̟n
Bn ̟n
Cn ̟1
Dn ̟1, ̟n−1 and ̟n
E6 ̟1 and ̟6
E7 ̟7
E8 none
F4 none
G2 none
DEFINITION 1.2. — Let ̟ be a minuscule weight and let P̟ be the associated parabolic subgroup.
The homogeneous space G/P̟ is then said to be minuscule. The Schubert varieties of a minuscule
homogeneous space are called minuscule Schubert varieties.
Remark 1.3. — It is a classical fact that to study minuscule homogeneous spaces and their
Schubert varieties, it is sufficient to restrict ourselves to simply-laced groups.
In the rest of the paper, the group G will be simply-laced, the subgroup P will be a maximal
parabolic subgroup associated to a minuscule fundamental weight ̟. The minuscule homogeneous
space G/P will be denoted by X and the Schubert variety associated to w ∈ WP will be denoted
by X(w) with the convention that the dimension of X(w) is the length of w.
2 Miniscule quivers
In [Pe07], we associated to any minuscule Schubert variety X(w) a unique quiver Qw. The definition
a priori depends on the choice of a reduced expression but does not depend on the commuting
relations. In the minuscule setting this implies that the following definitons do not depend on the
choosen reduced expression. Fix a reduced expression w = sβ1 · · · sβr of w (recall that w is in W
the set of minimal length representatives of W/WP ) where for all i ∈ [1, r], we have βi ∈ S.
DEFINITION 2.1. — (ı) The successor s(i) and the predecessor p(i) of an element i ∈ [1, r] are the
elements s(i) = min{j ∈ [1, r] / j > i and βj = βi} and p(i) = max{j ∈ [1, r] / j < i and βj = βi}.
(ıı) Denote by Qw the quiver whose set of vertices is the set [1, r] and whose arrows are given
in the following way: there is an arrow from i to j if and only if 〈β∨j , βi〉 6= 0 and i < j < s(i) (or
only i < j if s(i) does not exist).
Remark 2.2. — (ı) This quiver comes with a coloration of its vertices by simple roots via the
map β : [1, r] → S such that β(i) = βi.
(ıı) There is a natural order on the quiver Qw given by i4j if there is an oriented path from j
to i. Caution that this order is the reversed order of the one defined in [Pe07].
(ııı) Note that if we denote by Q̟ the quiver obtained from the longuest element in W
P , then
the quiver Qw is a subquiver of Q̟. The quivers of Schubert subvarieties are exactely the order
ideals in the quiver Q̟. We will call such a quiver reduced (meaning that it corresponds to a reduced
expression of an element in WP , see [Pe07] for more details on the shape of reduced quivers).
Recall also that we defined in [Pe07] some combinatorial objects associated to the quiver Qw.
DEFINITION 2.3. — (ı) We call peak any vertex of Qw maximal for the partial order 4. We
denote by Peaks(Qw) the set of peaks of Qw.
(ıı) We call hole of the quiver Qw any vertex i of Q̟ satisfying one of the following properties
• the vertex i is in Qw but p(i) 6∈ Qw and there are exactly two vertices j1<i and j2<i in Qw
with 〈β∨i , βjk〉 6= 0 for k = 1, 2.
• the vertex i is not in Qw, s(i) does not exist in Q̟ and there exist j ∈ Qw with 〈β
i , βj〉 6= 0.
Because the vertex of the second type of holes is not a vertex in Qw we call such a hole a virtual
hole of Qw. We denote by Holes(Qw) the set of holes of Qw.
(ııı) The height h(i) of a vertex i is the largest positive integer n such that there exists a sequence
(ik)k∈[1,n] of vertices with i1 = 1, in = r and such that there is an arrow from ik to ik+1 for all
k ∈ [1, n − 1].
Many geometric properties of the Schubert variety X(w) can be read on its quiver. In particular
we proved in [Pe07, Corollary 4.12]:
PROPOSITION 2.4. — A Schubert subvariety X(w′) in X(w) is stable under Stab(X(w)) if and
only if β(Holes(Qw′)) ⊂ β(Holes(Qw)).
An easy consequence of this fact and the result by M. Brion and P. Polo that the smooth locus
of X(w) is the dense Stab(X(w))-orbit is the following:
PROPOSITION 2.5. — A Schubert variety X(w) is smooth if and only if all the holes of its
quiver Qw are virtual.
We will be more precise in Theorem 3.2 and we will describe the irreducible components of
the singular locus and the generic singularity of this component in terms of the quiver. The
Gorensteiness of the variety is also easy to detect on the quiver as we proved in [Pe07, Corollary
4.19]:
PROPOSITION 2.6. — A Schubert variety X(w) is Gorenstein if and only if all the peaks of its
quiver Qw have the same height.
3 Generic singularities of minuscule Schubert varieties
In this section, we go one step further in the direction of reading on the quiver Qw the geometric
properties of X(w). We will translate the results of M. Brion and P. Polo [BP99] on the irreducible
components of the singular locus of X(w) and the singularity at a generic point of such a component
in terms of the quiver Qw. We will need the following notations:
DEFINITION 3.1. — (ı) Let i be a vertex of Qw, we define the subquiver Q
w of Qw as the full
subquiver containing the following set of vertices {j ∈ Qw / j < i}. We denote by Qw,i the full
subquiver of Qw containing the vertices of Qw \ Q
w. We denote by w
i (resp. wi) the elements in
WP associated to the quivers Qiw (resp. Qw,i).
(ıı) A hole i of the quiver Qw is said to be essential if it is not virtual and if there is no hole in
the subquiver Qiw.
(ııı) Following M. Brion and P. Polo, denote by J the set β(Holes(Qw))
We then prove the following:
THEOREM 3.2. — (ı) The set of irreducible components of the singular locus of X(w) is in one
to one correspondence with the set of essential holes of the quiver Qw. In particular, if i is an
essential hole of Qw, the corresponding irreducible component is the Schubert subvariety X(wi) of
X(w) whose quiver is Qw,i.
(ııı) Furthermore, the singularity of X(w) at a generic point of X(wi) is the same singularity
as the one of the B-fixed point in the Schubert variety X(wi) whose quiver is Qiw.
Remark 3.3. — The singularity of the B-fixed point in X(wi) is described in [BP99].
Proof — This result is a reformulation of the main results of M. Brion and P. Polo [BP99].
Proposition 2.4 shows that the essential holes are in one to one correspondence with maximal
Schubert subvarieties in X(w) stable under Stab(X(w)) and that if i is an essential hole, then the
corresponding Schubert subvariety X(wi) is associated to the quiver Qw,i. According to [BP99],
these are the irreducible components of the singular locus.
To describe the singularity of X(wi), M. Brion and P. Polo define two subsets I and I
′ of the
set of simple roots as follows:
• the set I is the union of the connected components of J ∩ wi(RP ) adjacent to β(i)
• the set I ′ is the union I ∪ {β(i)}.
We describe these sets thanks to the quiver.
PROPOSITION 3.4. — The set I ′ is β(Qiw).
Proof — The elements in J ∩ wi(RP ) are the simple roots γ ∈ J such that wi
−1(γ) ∈ RP .
Thanks to Lemma 3.5, these elements are the simple roots in J neither in β(Holes(Qw,i)) nor in
β(Peaks(Qw,i)).
An easy (but fastidious for types E6 and E7) look on the quivers shows that I
′ = β(Qiw). A
uniform proof of this statement is possible but needs an involved case analysis on the quivers. �
LEMMA 3.5. — Let β be a simple root, then we have
1. w−1(β) ∈ R− \R−
if β ∈ β(Peaks(Qw)),
2. w−1(β) ∈ R+ \R+
if β ∈ β(Holes(Qw)) = J
3. w−1(β) ∈ R+
otherwise.
Proof — Let w = sβ1 · sβr be a reduced expression for w, we want to compute w
−1(β) =
sβr · · · sβ1(β). We proceed by induction and deal with the three cases at the same time.
1. Take first β ∈ β(Peaks(Qw)), we may assume that β1 = β and w
−1(β) = sβr · · · sβ2(−β). Let
i ∈ Peaks(Qw) such that β(i) = β, the quiver obtained by removing i has s(i) for hole (possibly
virtual). We may apply induction and the result in case 2.
2.a. Let β ∈ Jc. Assume first that there is no k ∈ Qw with β(k) = β. Then there exist an
i ∈ Qw such that 〈β
∨, βi〉 6= 0. Let us prove that such a vertex i is unique. Indeed, the support
of w is contained in a subdiagram D of the Dynkin diagram not containing β. The diagram D
contains the simple root α corresponding to P (except if X(w) is a point in which case w = Id
and the lemma is easy). The quiver Qw is in particular contained in the quiver of the minuscule
homogeneous variety associated to α ∈ D. It is easy to check on these quivers (see in [Pe07] for
the shape of these quivers) that there is a unique such vertex i.
Now consider the quivers Qiw and Qw,i. Recall that we denote by w
i and wi the associated
elements in W . We have w = wiwi. We compute w
i−1(β) and because all simple roots β(x)
for x ∈ Qiw with x 6= i are orthogonal to β we have w
i−1(β) = sβi(β) = β + βi. We then
have w−1(β) = w−1i (β + βi). Because i was the only vertex such that 〈β
∨, βi〉 6= 0, we have
w−1i (β) = β ∈ R
and by induction (note that i is now a hole of Qw,i) we have w
i (βi) ∈ R
+ \R+
and we have the result.
2.b. Now assume that there exist k ∈ Holes(Qw) with β(k) = β and let i a vertex maximal for
the property 〈β∨, βi〉 6= 0. Remark that we have k < i. Consider one more time the quivers Q
and Qw,i and the elements w
i and wi. We have w
−1(β) = w−1i (βi + β). But as before we have by
induction w−1
(βi) ∈ R
+ \ R+
so that we can conclude by induction as soon as k is not a peak of
Qw,i. But because k is an hole, there exist a vertex j ∈ Qw with j 6= i and such that there is an
arrow j → k in Qw. Because i was taken maximal j is a vertex of Qw,i and k is not a peak of this
quiver.
3. If β is not in the support of w but is not in β(Holes) then w−1(β) = β ∈ R+
Let β in β(Qw) but not in β(Holess(Qw)) or β(Peaks(Qw)) and let k the highest vertex such
that β(k) = β. There exists a unique vertex i ∈ Qw such that i ≻ k and 〈β
∨, β(i)〉 6= 0. We
have w−1(β) = w−1i (βi + β) and the vertex k is a peak of Qw,i so that wi = sβ(k)wk = sβwk and
w−1(β) = w−1
(βi). Now it is easy to see that either s(i) does not exists and in this case it is not a
virtual hole or it exists but is neither a peak nor a hole of Qw,k. We conclude by induction on the
third case. �
The Theorem is now a corollary of the description of the singularities thanks to I and I ′ done
by M. Brion and P. Polo. �
Remark 3.6. — In their article M. Brion and P. Polo also deal with the cominucule Schubert
varieties. We believe that, in that case, Theorem 0.3 should hold true as well as Corollary 0.4.
It is now easy to decide which generic singularity is Gorenstein:
COROLLARY 3.7. — Let i be an essential hole of the quiver Qw. The generic point of the
irreducible component X(wi) of the singular locus is Gorenstein if and only if all the peaks of Q
are of the same height.
We describe the Schubert subvarieties X(w′) in X(w) that are expected to be Gorenstein at
their generic point by the conjecture of A. Woo and A. Yong. Let us give the following
DEFINITION 3.8. — (ı) An essential hole is said to be Gorenstein if the generic point of the
associated irreducible component of the singular locus is in the Gorenstein locus.
(ıı) A Schubert subvariety X(w′) in X(w) is said to have the property (WY) if the generic point
of any irreductible component of the singular locus of X(w) containing X(w′) is in the Gorenstein
locus of X(w).
We have the following:
PROPOSITION 3.9. — Let X(w′) be a Schubert subvariety of the Schubert variety X(w). If the
generic point of X(w′) is Gorentein in X(w), then X(w′) has the property (WY).
Proof — Let X(v) be an irreducible component of the singular locus of X(w) containing X(w′).
Because the property of beeing non Gorenstein is stable under closure, this implies that the generic
point of X(v) is Gorenstein in X(w). �
Remark that, because all the irreducible components of the singular locus of X(w) are stable
under Stab(X(w)), the property (WY) need only to be checked on Stab(X(w))-stable Schubert
subvarieties.
PROPOSITION 3.10. — (ı) The Schubert subvarieties X(w′) in X(w) stable under Stab(X(w))
are exactely those such that the associated quiver Qw′ satisfies
Qw′ =
i∈Holes(Qw)
Qw,ski(i)
where the (ki)i∈Holes(Qw) are integers greater or equal to −1 (if ki = −1, the quiver Qw,ski(i) is Qw
by definition).
(ıı) A Stab(X(w))-stable Schubert subvariety X(w′) of X(w) has the property (WY) if and only
if the only essential holes in the difference Qw \Qw′ are Gorenstein. Equivalentely, writing
Qw′ =
i∈Holes(Qw)
Qw,ski(i),
if and only if the only holes in of the quivers (Q
ski (i)
w )i∈Holes(Qw) are Gorenstein holes. Another
equivalent formulation is that Qw′ contains all the non Gorenstein essential holes of Qw.
Proof — (ı) Consider the subquiver Qw′ in Qw and for each hole i of Qw define the integer
ki = min{k ≥ 0 / s
k(i) ∈ Qw′} − 1. Because of the fact (see for example [LMS79]) that the
strong and weak Bruhat orders coincide for minuscule Schubert varieties, the quiver Qw′ has to be
contained in the intersection
i∈Holes(Qw)
Qw,ski(i).
We therefore need to remove some vertices to Q′ to get Qw′. But removing a vertex j of the quiver
Q′ (it has to be a peak of Q′) creates a hole in s(j) (or a virtual hole in j if s(j) does not exist).
Because X(w′) is Stab(X(w))-stable, the last removed vertex j is such that β(j) ∈ β(Holes(Qw)).
This implies that no more vertex can be removed from Q′ to get Qw′ and in particular Qw′ = Q
(ıı) The Schubert subvariety has the property (WY) if and only if all the irreducible components
X(wi) of the singular locus of X(w) containing X(w
′) are such that i is a Gorenstein hole. But
X(w′) is contained in X(wi) if and only if Qw′ is contained in Qw,i. This is equivalent to the fact
that Qiw is contained in Qw \Qw′ and the proof follows. �
4 Relative canonical model and Gorenstein locus
In this section, we recall the explicit construction given in [Pe07] of the relative canonical model
of X(w). Recall that we described in [Pe07] the Bott-Samelson resolution π : X̃(w) → X(w) as a
configuration variety à la Magyar [Ma98]:
X̃(w) ⊂
G/Pβi
where Pβi is the maximal parabolic associated to the simple root βi. The map π : X̃(w) → X(w)
is given by the projection
G/Pβi → G/Pβm(w) where m(w) is the smallest element in Qw.
We define a partition on the peaks of the quiver Qw and a partition of the quiver itself:
DEFINITION 4.1. — (ı) Define a partition (Ai)i∈[1,n] of Peaks(Qw) by induction: A1 is the set of
peaks with minimal height and Ai+1 is the set of peaks in Peaks(Qw)\
k=1Ak with minimal height
(the integer n is the number of different values the height function takes on the set Peaks(Qw)).
(ıı) Define a partition (Qw(i))i∈[1,n] of Qw by induction:
Qw(i) = {x ∈ Qw / ∃j ∈ Ai : x 4 j and x 64 k ∀k ∈ ∪j>iAj}.
We proved in [Pe07] that these quivers Qw(i) are quivers of minuscule Schubert varieties and
in particular have a minimal element mw(i). We defined the variety X̂(w) as the image of the
Bott-Samelson resolution X̃(w) (seen as a configuration variety) in the product
i=1 G/Pβmw(i) .
Because mw(n) = m(w) we have a map π̂ : X̂(w) → X(w) and a factorisation
X̃(w)
// X̂(w)
X(w).
We proved the following result in [Pe07]:
THEOREM 4.2. — (ı) The variety X̂(w) together with the map π̂ realise X̂(w) as the relative
canonical model of X(w).
(ıı) The variety X̂(w) is a tower of locally trivial fibrations with fibers the Schubert varieties
associated to the quivers Qw(i). In particular X̂(w) is Gorenstein.
We will use this resolution to prove our main result. Indeed, we will prove that the generic fibre
of the map π̂ : X̂(w) → X(w) above a (WY) Schubert subvariety X(w′) is a point. In other words,
the map π̂ is an isomorphism on an open subset of X(w′). As a consequence, the generic point of
X(w′) will be in the Gorenstein locus.
Let us recall some facts on X̃(w) and X̂(w) (see [Pe07]):
FACT 4.3. — (ı) To each vertex i of Qw one can associated a divisor Di on X̃(w) and all these
divisors intersect transversally.
(ıı) For K a subset of the vertices of Qw, we denote by ZK the transverse intersection of the
Di for i ∈ K.
(ııı) The image of the closed subset ZK by the map π is the Schubert variety X(wK) whose
quiver QwK is the biggest reduced subquiver of Qw not containing the vertices in K.
The quiver Qw(i) defines a element w(i) in W and the fact that these quivers realise a partition
of Qw implies that we have an expression w = w(1) · · ·w(n) with l(w) =
l(w(i)). We prove the
following generalisation of this fact:
PROPOSITION 4.4. — Let K be a subset of the vertices of Qw. The image of the closed subset
ZK by the map π̃ is a tower of locally trivial fibrations with fibers the Schubert varieties X(wK(i))
whose quiver QwK(i) is the biggest reduced subquiver of Qw(i) not containing the vertices of K∩Qw(i).
This variety is the image by π̃ of Z∪n
QK(i).
Proof — As we explained in [Pe07, Proposition 5.9], the Bott-Samelson resolution is the quotient
of the product
Ri where theRi are certain minimal parabolic subgroups by a product of Borel
subgroups
i=1 Bi. The variety X̂(w) is the quotient of a product
i=1 Ni of parabolic subgroups
such that the multiplication in G maps
k∈Qw(i)
Rk to Ni by a product
i=1Mi of parabolic
subgroups. The map π̃ is induced by the product from
Ri to
i=1Ni. In particular, this
means that for i ∈ [1, n] fixed, the map
k∈Qw(i)
→ Ni induces the map from the Bott-Samelson
resolution X̃(w(i)) to X(w(i)). We may now apply part (ııı) of the preceding fact because the
quiver Qw(i) is minuscule. �
We now remark that the quivers Qw′ associated to Schubert subvarieties X(w
′) in the Schu-
bert variety X(w) having the property (WY) have a nice behaviour with repect to the partition
(Qw(i))i∈[1,n] of Qw.
PROPOSITION 4.5. — Let X(w′) be a Stab(X(w))-stable Schubert subvariety of X(w) having
the property (WY). Let us denote by (Cj)j∈[1,k] the connected components of the subquiver Qw \Qw′
of Qw. Then for each j, there exist an unique ij ∈ [1, n] such that Cj ⊂ Qw(ij).
Proof — Recall from Proposition 3.10 that, denoting by GorHol(Qw) the set of Gorenstein holes
in Qw, we may write
Qw \Qw′ =
i∈GorHol(Qw)
with ki an integer greater or equal to −1 and with the additional condition that Q
ski(i)
w contains
only Gorenstein holes. Because the quivers Q
ski(i)
w are connected, any connected component of
Qw \Qw′ is an union of such quivers. But we have the following:
LEMMA 4.6. — Let i ∈ Holes(Qw) and assume that Q
sk(i)
w meets at least two subquivers of the
partition (Qw(i))i∈[1,n], then Q
sk(i)
w contains a non Gorenstein hole.
Proof — The quiver Q
sk(i)
w meets two subquivers of the partition (Qw(i))i∈[1,n], in particular it
contains two peaks of Qw of different heights. By connexity of Q
sk(i)
w , we may assume that these
two peaks are adjacent. In particular there is a hole between these two peaks and this hole is not
Gorenstein and is contained in Q
sk(i)
w . �
The proposition follows. �
We describe the inverse image by π̂ of a Stab(X(w))-stable Schubert subvariety of X(w) having
the property (WY). To do this, first remark that the map π is B-equivariant and that the inverse
image π−1(X(w′)) has to be a union of closed subsets ZK for some subsets K of Qw. Let ZK ⊂
π−1(X(w′)) be such that π : ZK → X(w
′) is dominant. We will denote by Qw
w (i) the intersection
Qw′ ∩Qw(i) and by w
′(i) the associated element in W .
PROPOSITION 4.7. — The image of ZK in X̂(w) by π̃ is the same as the image of ZQw\Qw′ .
Proof — Thanks to Proposition 4.4 we only need to compute the quivers QwK(i). Consider the
decomposition into connected components Qw \Qw′ = ∪
j=1Cj . We may decompose K accordingly
as K = ∪kj=1Kj where Kj = K ∩ Cj. But because each connected component of Qw \ Qw′ is
contained in one of the quivers (Qw(i))i∈[1,n] this implies that QwK(i) is exactely QwK ∩ Qw(i)
where QwK is the biggest reduced quiver in Qw Qw not containing the vertices in K (see Fact 4.3).
We get QwK = Qw′ (because ZK is sent onto X(w
′)) and the result follows. �
THEOREM 4.8. — Let X(w′) be a Schubert subvariety in X(w). Then X(w′) has the property
(WY) if and only if its generic point is in the Gorenstein locus of X(w).
Proof — We have already seen in Proposition 3.9 that if the generic point of X(w′) is in the
Gorenstein locus of X(w) then X(w′) has the property (WY).
Conversely let X(w′) be a Schubert subvariety having the property (WY). The previous propo-
sition implies that its inverse image π̂−1(X(w′)) is the variety π̃(ZQw\Qw′ ). But this implies that
the map π̂ : π̃(ZQw\Qw′ ) = π̂
−1(X(w′)) → X(w′) is birational (because the varieties have the
same dimension given by the number of vertices in the quiver). In particular, the map π̂ is an
isomorphism on an open subset of X(w) meeting X(w′) non trivially. Therefore, because X̂(w) is
Gorenstein, it is the case of the generic point in X(w′) as a point in X(w). �
References
[BL00] Sara Billey and Venkatramani Lakshmibai : Singular loci of Schubert varieties. Progress in Math-
ematics, 182. Birkhäuser Boston, Inc., Boston, MA, 2000.
[BW03] Sara Billey and Gregory Warrington: Maximal singular loci of Schubert varieties in SL(n)/B.
Trans. Amer. Math. Soc. 355 (2003), no. 10, 3915–3945.
[Bo68] Nicolas Bourbaki : Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie.
Chapitre IV : Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des
réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337
Hermann, Paris 1968.
[BP99] Michel Brion and Patrick Polo: Generic singularities of certain Schubert varieties. Math. Z. 231
(1999), no. 2, 301–324.
[Co03] Aurélie Cortez : Singularités génériques et quasi-résolutions des variétés de Schubert pour le
groupe linéaire. Adv. Math. 178 (2003), no. 2, 396–445.
[KLR03] Christian Kassel, Alain Lascoux and Chritophe Reutenauer : The singular locus of a Schubert
variety. J. Algebra 269 (2003), no. 1, 74–108.
[LMS79] Venkatramani Lakshmibai, Chitikila Musili and Conjeerveram S. Seshadri : Geometry of G/P .
III. Standard monomial theory for a quasi-minuscule P . Proc. Indian Acad. Sci. Sect. A Math.
Sci. 88 (1979), no. 3, 93–177.
[LS90] Venkatramani Lakshmibai and B. Sandhya: Criterion for smoothness of Schubert varieties in
Sl(n)/B. Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 1, 45–52.
[Ma98] Peter Magyar : Borel-Weil theorem for configuration varieties and Schur modules. Adv. Math.
134 (1998), no. 2, 328–366.
[Ma01a] Laurent Manivel : Le lieu singulier des variétés de Schubert. Internat. Math. Res. Notices 2001,
no. 16, 849–871.
[Ma01b] Laurent Manivel : Generic singularities of Schubert Varieties. math.AG/0105239 (2001).
[Pe07] Nicolas Perrin: Small-resolutions of minuscule Schubert varieties. Preprint math.AG/0601117 to
appear in Compositio Mathematica (2007).
[WY06a] Alexander Woo and Alexander Yong: When is a Schubert variety Gorenstein? Adv. Math. 207
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math.AG/0603273 (2006).
Université Pierre et Marie Curie - Paris 6
UMR 7586 — Institut de Mathématiques de Jussieu
175 rue du Chevaleret
75013 Paris, France.
email : nperrin@math.jussieu.fr

  In this article, we describe explicitely the Gorenstein locus of all
minuscule Schubert varieties. This proves a special case of a conjecture of A.
Woo and A. Yong (see math.AG/0603273) on the Gorenstein locus of Schubert
varieties.

Introduction
The description of the singular locus and of the types of singularities appearing in Schubert
varieties is a hard problem. A first step in this direction was the proof by V. Lakshmibai and B.
Sandhya [LS90] of a pattern avoidance criterion for a Schubert variety in type A to be smooth. There
exist some other results in this direction, for a detailed account see [BL00]. Another important
result was a complete combinatorial description, still in type A, of the irreducible components of
the singular locus of a Schubert variety (this has been realised, almost in the same time, by L.
Manivel [Ma01a] and [Ma01b], S. Billey and G. Warrington [BW03], C. Kassel, A. Lascoux and C.
Reutenauer [KLR03] and A. Cortez [Co03]). The singularity at a generic point of such a component
is also given in [Ma01b] and [Co03]. However, as far as I know, this problem is still open for other
types. Another partial result in this direction is the description of the irreducible components of
the singular locus and of the generic singularity of minuscule and cominuscule Schubert varieties
(see Definition 1.2) by M. Brion and P. Polo [BP99].
In the same vein as [LS90], A. Woo and A. Yong gave in [WY06a] and [WY06b] a generalised
pattern avoidance criterion, in type A, to decide if a Schubert variety is Gorenstein. They do not
describe the irreducible components of the Gorenstein locus but give the following conjecture (see
Conjecture 6.7 in [WY06b]):
CONJECTURE 0.1. — Let X be a Schubert variety, a point x in X is in the Gorenstein locus of
X if and only if the generic point of any irreducible component of the singular locus of X containing
x is is the Gorenstein locus of X.
The interest of this conjecture relies on the fact that, at least in type A, the irreducible compo-
nents of the singular locus and the singularity of a generic point of that component are well known.
The conjecture would imply that one only needs to know the information on the irreducible com-
ponents of the singular locus to get all the information on the Gorenstein locus.
In this paper we prove this conjecture for all minuscule Schubert varieties thanks to a combi-
natorial description of the Gorenstein locus of minuscule Schubert varieties. To do this we use the
http://arxiv.org/abs/0704.0895v1
combinatorial tool introduced in [Pe07] associating to any minuscule Schubert variety a reduced
quiver generalising Young diagrams. First, we translate the results of M. Brion and P. Polo [BP99]
in terms of the quiver. We define the holes, the virtual holes and the essential holes in the quiver
(see Definitions 2.3 and 3.1) and prove the following:
THEOREM 0.2. — (ı) A minuscule schubert variety is smooth if and only if its associated quiver
has no nonvirtual hole.
(ıı) The irreducible components of the singular locus of a minuscule Schubert variety are indexed
by essential holes.
Furthermore we explicitely describe in terms of the quiver and the essential holes these irre-
ducible components and the singularity at a generic point of a component (for more details see
Theorem 3.2). In particular, with this description it is easy to say if the singularity at a generic
point of an irreducible component of the singular locus is Gorenstein or not. The essential holes
corresponding to irreducible components having a Gorenstein generic point are called Gorenstein
holes (see also Definition 3.8). We give the following complete description of the Gorenstein locus:
THEOREM 0.3. — The generic point of a Schubert subvariety X(w′) of a minuscule Schubert
variety X(w) is in the Gorenstein locus if and only if the quiver of X(w′) contains all the non
Gorenstein holes of the quiver of X(w).
COROLLARY 0.4. — Conjecture 0.1 is true for all minuscule Schubert varieties.
Example 0.5. — Let G(4, 7) be the Grassmannian variety of 4-dimensional subspaces in a 7-
dimensional vector space. Consider the Schubert variety
X(w) = {V4 ∈ G(4, 7) dim(V4 ∩W3) ≥ 2 and dim(V4 ∩W5) ≥ 3}
where W3 and W5 are fixed subspaces of dimension 3 and 5 respectively. The minimal length
representative w is the permutation (2357146). Its quiver is the following one (all the arrows are
going down):
We have circled the two holes on this quiver. The left hole is not a Gorenstein hole (this can be
easily seen because the two peaks above this hole do not have the same height, see Definition 2.3)
but the right one is Gorenstein (the two peaks have the same height). Let X(w′) be an irreducible
component of the singular locus of X(w). The possible quivers of such a variety X(w′) are the
following (for each hole we remove all the vertices above that hole):
These Schubert varieties correspond to the permutations: (1237456) and (2341567). Let X(w′) be
a Schubert subvariety in X(w) whose generic point is not in the Gorenstein locus. Then X(w′) has
to be contained in X(1237456).
Acknowledgements: I thank Frank Sottile and Jim Carrel for their invitation to the BIRS
workshop Comtemporary Schubert calculus during which the major part of this work has been
done.
1 Minuscule Schubert varieties
Let us fix some notations and recall the definitions of minuscule homogeneous spaces and minuscule
Schubert varieties. A basic reference is [LMS79].
In this paper G will be a semi-simple algebraic group, we fix B a Borel subgroup and T a
maximal torus in B. We denote by R the set of roots, by R+ and R− the set of positive and
negative roots. We denote by S the set of simple roots. We will denote by W the Weyl group of G.
We also fix P a parabolic subgroup containing B. We denote by WP the Weyl group of P
and by WP the set of minimal length representatives in W of the coset W/WP . Recall that the
Schubert varieties in G/P (that is to say the B-orbit closures in G/P ) are parametrised by WP .
DEFINITION 1.1. — A fundamental weight ̟ is said to be minuscule if, for all positive roots
α ∈ R+, we have 〈α∨,̟〉 ≤ 1.
With the notation of N. Bourbaki [Bo68], the minuscule weights are:
Type minuscule
An ̟1 · · ·̟n
Bn ̟n
Cn ̟1
Dn ̟1, ̟n−1 and ̟n
E6 ̟1 and ̟6
E7 ̟7
E8 none
F4 none
G2 none
DEFINITION 1.2. — Let ̟ be a minuscule weight and let P̟ be the associated parabolic subgroup.
The homogeneous space G/P̟ is then said to be minuscule. The Schubert varieties of a minuscule
homogeneous space are called minuscule Schubert varieties.
Remark 1.3. — It is a classical fact that to study minuscule homogeneous spaces and their
Schubert varieties, it is sufficient to restrict ourselves to simply-laced groups.
In the rest of the paper, the group G will be simply-laced, the subgroup P will be a maximal
parabolic subgroup associated to a minuscule fundamental weight ̟. The minuscule homogeneous
space G/P will be denoted by X and the Schubert variety associated to w ∈ WP will be denoted
by X(w) with the convention that the dimension of X(w) is the length of w.
2 Miniscule quivers
In [Pe07], we associated to any minuscule Schubert variety X(w) a unique quiver Qw. The definition
a priori depends on the choice of a reduced expression but does not depend on the commuting
relations. In the minuscule setting this implies that the following definitons do not depend on the
choosen reduced expression. Fix a reduced expression w = sβ1 · · · sβr of w (recall that w is in W
the set of minimal length representatives of W/WP ) where for all i ∈ [1, r], we have βi ∈ S.
DEFINITION 2.1. — (ı) The successor s(i) and the predecessor p(i) of an element i ∈ [1, r] are the
elements s(i) = min{j ∈ [1, r] / j > i and βj = βi} and p(i) = max{j ∈ [1, r] / j < i and βj = βi}.
(ıı) Denote by Qw the quiver whose set of vertices is the set [1, r] and whose arrows are given
in the following way: there is an arrow from i to j if and only if 〈β∨j , βi〉 6= 0 and i < j < s(i) (or
only i < j if s(i) does not exist).
Remark 2.2. — (ı) This quiver comes with a coloration of its vertices by simple roots via the
map β : [1, r] → S such that β(i) = βi.
(ıı) There is a natural order on the quiver Qw given by i4j if there is an oriented path from j
to i. Caution that this order is the reversed order of the one defined in [Pe07].
(ııı) Note that if we denote by Q̟ the quiver obtained from the longuest element in W
P , then
the quiver Qw is a subquiver of Q̟. The quivers of Schubert subvarieties are exactely the order
ideals in the quiver Q̟. We will call such a quiver reduced (meaning that it corresponds to a reduced
expression of an element in WP , see [Pe07] for more details on the shape of reduced quivers).
Recall also that we defined in [Pe07] some combinatorial objects associated to the quiver Qw.
DEFINITION 2.3. — (ı) We call peak any vertex of Qw maximal for the partial order 4. We
denote by Peaks(Qw) the set of peaks of Qw.
(ıı) We call hole of the quiver Qw any vertex i of Q̟ satisfying one of the following properties
• the vertex i is in Qw but p(i) 6∈ Qw and there are exactly two vertices j1<i and j2<i in Qw
with 〈β∨i , βjk〉 6= 0 for k = 1, 2.
• the vertex i is not in Qw, s(i) does not exist in Q̟ and there exist j ∈ Qw with 〈β
i , βj〉 6= 0.
Because the vertex of the second type of holes is not a vertex in Qw we call such a hole a virtual
hole of Qw. We denote by Holes(Qw) the set of holes of Qw.
(ııı) The height h(i) of a vertex i is the largest positive integer n such that there exists a sequence
(ik)k∈[1,n] of vertices with i1 = 1, in = r and such that there is an arrow from ik to ik+1 for all
k ∈ [1, n − 1].
Many geometric properties of the Schubert variety X(w) can be read on its quiver. In particular
we proved in [Pe07, Corollary 4.12]:
PROPOSITION 2.4. — A Schubert subvariety X(w′) in X(w) is stable under Stab(X(w)) if and
only if β(Holes(Qw′)) ⊂ β(Holes(Qw)).
An easy consequence of this fact and the result by M. Brion and P. Polo that the smooth locus
of X(w) is the dense Stab(X(w))-orbit is the following:
PROPOSITION 2.5. — A Schubert variety X(w) is smooth if and only if all the holes of its
quiver Qw are virtual.
We will be more precise in Theorem 3.2 and we will describe the irreducible components of
the singular locus and the generic singularity of this component in terms of the quiver. The
Gorensteiness of the variety is also easy to detect on the quiver as we proved in [Pe07, Corollary
4.19]:
PROPOSITION 2.6. — A Schubert variety X(w) is Gorenstein if and only if all the peaks of its
quiver Qw have the same height.
3 Generic singularities of minuscule Schubert varieties
In this section, we go one step further in the direction of reading on the quiver Qw the geometric
properties of X(w). We will translate the results of M. Brion and P. Polo [BP99] on the irreducible
components of the singular locus of X(w) and the singularity at a generic point of such a component
in terms of the quiver Qw. We will need the following notations:
DEFINITION 3.1. — (ı) Let i be a vertex of Qw, we define the subquiver Q
w of Qw as the full
subquiver containing the following set of vertices {j ∈ Qw / j < i}. We denote by Qw,i the full
subquiver of Qw containing the vertices of Qw \ Q
w. We denote by w
i (resp. wi) the elements in
WP associated to the quivers Qiw (resp. Qw,i).
(ıı) A hole i of the quiver Qw is said to be essential if it is not virtual and if there is no hole in
the subquiver Qiw.
(ııı) Following M. Brion and P. Polo, denote by J the set β(Holes(Qw))
We then prove the following:
THEOREM 3.2. — (ı) The set of irreducible components of the singular locus of X(w) is in one
to one correspondence with the set of essential holes of the quiver Qw. In particular, if i is an
essential hole of Qw, the corresponding irreducible component is the Schubert subvariety X(wi) of
X(w) whose quiver is Qw,i.
(ııı) Furthermore, the singularity of X(w) at a generic point of X(wi) is the same singularity
as the one of the B-fixed point in the Schubert variety X(wi) whose quiver is Qiw.
Remark 3.3. — The singularity of the B-fixed point in X(wi) is described in [BP99].
Proof — This result is a reformulation of the main results of M. Brion and P. Polo [BP99].
Proposition 2.4 shows that the essential holes are in one to one correspondence with maximal
Schubert subvarieties in X(w) stable under Stab(X(w)) and that if i is an essential hole, then the
corresponding Schubert subvariety X(wi) is associated to the quiver Qw,i. According to [BP99],
these are the irreducible components of the singular locus.
To describe the singularity of X(wi), M. Brion and P. Polo define two subsets I and I
′ of the
set of simple roots as follows:
• the set I is the union of the connected components of J ∩ wi(RP ) adjacent to β(i)
• the set I ′ is the union I ∪ {β(i)}.
We describe these sets thanks to the quiver.
PROPOSITION 3.4. — The set I ′ is β(Qiw).
Proof — The elements in J ∩ wi(RP ) are the simple roots γ ∈ J such that wi
−1(γ) ∈ RP .
Thanks to Lemma 3.5, these elements are the simple roots in J neither in β(Holes(Qw,i)) nor in
β(Peaks(Qw,i)).
An easy (but fastidious for types E6 and E7) look on the quivers shows that I
′ = β(Qiw). A
uniform proof of this statement is possible but needs an involved case analysis on the quivers. �
LEMMA 3.5. — Let β be a simple root, then we have
1. w−1(β) ∈ R− \R−
if β ∈ β(Peaks(Qw)),
2. w−1(β) ∈ R+ \R+
if β ∈ β(Holes(Qw)) = J
3. w−1(β) ∈ R+
otherwise.
Proof — Let w = sβ1 · sβr be a reduced expression for w, we want to compute w
−1(β) =
sβr · · · sβ1(β). We proceed by induction and deal with the three cases at the same time.
1. Take first β ∈ β(Peaks(Qw)), we may assume that β1 = β and w
−1(β) = sβr · · · sβ2(−β). Let
i ∈ Peaks(Qw) such that β(i) = β, the quiver obtained by removing i has s(i) for hole (possibly
virtual). We may apply induction and the result in case 2.
2.a. Let β ∈ Jc. Assume first that there is no k ∈ Qw with β(k) = β. Then there exist an
i ∈ Qw such that 〈β
∨, βi〉 6= 0. Let us prove that such a vertex i is unique. Indeed, the support
of w is contained in a subdiagram D of the Dynkin diagram not containing β. The diagram D
contains the simple root α corresponding to P (except if X(w) is a point in which case w = Id
and the lemma is easy). The quiver Qw is in particular contained in the quiver of the minuscule
homogeneous variety associated to α ∈ D. It is easy to check on these quivers (see in [Pe07] for
the shape of these quivers) that there is a unique such vertex i.
Now consider the quivers Qiw and Qw,i. Recall that we denote by w
i and wi the associated
elements in W . We have w = wiwi. We compute w
i−1(β) and because all simple roots β(x)
for x ∈ Qiw with x 6= i are orthogonal to β we have w
i−1(β) = sβi(β) = β + βi. We then
have w−1(β) = w−1i (β + βi). Because i was the only vertex such that 〈β
∨, βi〉 6= 0, we have
w−1i (β) = β ∈ R
and by induction (note that i is now a hole of Qw,i) we have w
i (βi) ∈ R
+ \R+
and we have the result.
2.b. Now assume that there exist k ∈ Holes(Qw) with β(k) = β and let i a vertex maximal for
the property 〈β∨, βi〉 6= 0. Remark that we have k < i. Consider one more time the quivers Q
and Qw,i and the elements w
i and wi. We have w
−1(β) = w−1i (βi + β). But as before we have by
induction w−1
(βi) ∈ R
+ \ R+
so that we can conclude by induction as soon as k is not a peak of
Qw,i. But because k is an hole, there exist a vertex j ∈ Qw with j 6= i and such that there is an
arrow j → k in Qw. Because i was taken maximal j is a vertex of Qw,i and k is not a peak of this
quiver.
3. If β is not in the support of w but is not in β(Holes) then w−1(β) = β ∈ R+
Let β in β(Qw) but not in β(Holess(Qw)) or β(Peaks(Qw)) and let k the highest vertex such
that β(k) = β. There exists a unique vertex i ∈ Qw such that i ≻ k and 〈β
∨, β(i)〉 6= 0. We
have w−1(β) = w−1i (βi + β) and the vertex k is a peak of Qw,i so that wi = sβ(k)wk = sβwk and
w−1(β) = w−1
(βi). Now it is easy to see that either s(i) does not exists and in this case it is not a
virtual hole or it exists but is neither a peak nor a hole of Qw,k. We conclude by induction on the
third case. �
The Theorem is now a corollary of the description of the singularities thanks to I and I ′ done
by M. Brion and P. Polo. �
Remark 3.6. — In their article M. Brion and P. Polo also deal with the cominucule Schubert
varieties. We believe that, in that case, Theorem 0.3 should hold true as well as Corollary 0.4.
It is now easy to decide which generic singularity is Gorenstein:
COROLLARY 3.7. — Let i be an essential hole of the quiver Qw. The generic point of the
irreducible component X(wi) of the singular locus is Gorenstein if and only if all the peaks of Q
are of the same height.
We describe the Schubert subvarieties X(w′) in X(w) that are expected to be Gorenstein at
their generic point by the conjecture of A. Woo and A. Yong. Let us give the following
DEFINITION 3.8. — (ı) An essential hole is said to be Gorenstein if the generic point of the
associated irreducible component of the singular locus is in the Gorenstein locus.
(ıı) A Schubert subvariety X(w′) in X(w) is said to have the property (WY) if the generic point
of any irreductible component of the singular locus of X(w) containing X(w′) is in the Gorenstein
locus of X(w).
We have the following:
PROPOSITION 3.9. — Let X(w′) be a Schubert subvariety of the Schubert variety X(w). If the
generic point of X(w′) is Gorentein in X(w), then X(w′) has the property (WY).
Proof — Let X(v) be an irreducible component of the singular locus of X(w) containing X(w′).
Because the property of beeing non Gorenstein is stable under closure, this implies that the generic
point of X(v) is Gorenstein in X(w). �
Remark that, because all the irreducible components of the singular locus of X(w) are stable
under Stab(X(w)), the property (WY) need only to be checked on Stab(X(w))-stable Schubert
subvarieties.
PROPOSITION 3.10. — (ı) The Schubert subvarieties X(w′) in X(w) stable under Stab(X(w))
are exactely those such that the associated quiver Qw′ satisfies
Qw′ =
i∈Holes(Qw)
Qw,ski(i)
where the (ki)i∈Holes(Qw) are integers greater or equal to −1 (if ki = −1, the quiver Qw,ski(i) is Qw
by definition).
(ıı) A Stab(X(w))-stable Schubert subvariety X(w′) of X(w) has the property (WY) if and only
if the only essential holes in the difference Qw \Qw′ are Gorenstein. Equivalentely, writing
Qw′ =
i∈Holes(Qw)
Qw,ski(i),
if and only if the only holes in of the quivers (Q
ski (i)
w )i∈Holes(Qw) are Gorenstein holes. Another
equivalent formulation is that Qw′ contains all the non Gorenstein essential holes of Qw.
Proof — (ı) Consider the subquiver Qw′ in Qw and for each hole i of Qw define the integer
ki = min{k ≥ 0 / s
k(i) ∈ Qw′} − 1. Because of the fact (see for example [LMS79]) that the
strong and weak Bruhat orders coincide for minuscule Schubert varieties, the quiver Qw′ has to be
contained in the intersection
i∈Holes(Qw)
Qw,ski(i).
We therefore need to remove some vertices to Q′ to get Qw′. But removing a vertex j of the quiver
Q′ (it has to be a peak of Q′) creates a hole in s(j) (or a virtual hole in j if s(j) does not exist).
Because X(w′) is Stab(X(w))-stable, the last removed vertex j is such that β(j) ∈ β(Holes(Qw)).
This implies that no more vertex can be removed from Q′ to get Qw′ and in particular Qw′ = Q
(ıı) The Schubert subvariety has the property (WY) if and only if all the irreducible components
X(wi) of the singular locus of X(w) containing X(w
′) are such that i is a Gorenstein hole. But
X(w′) is contained in X(wi) if and only if Qw′ is contained in Qw,i. This is equivalent to the fact
that Qiw is contained in Qw \Qw′ and the proof follows. �
4 Relative canonical model and Gorenstein locus
In this section, we recall the explicit construction given in [Pe07] of the relative canonical model
of X(w). Recall that we described in [Pe07] the Bott-Samelson resolution π : X̃(w) → X(w) as a
configuration variety à la Magyar [Ma98]:
X̃(w) ⊂
G/Pβi
where Pβi is the maximal parabolic associated to the simple root βi. The map π : X̃(w) → X(w)
is given by the projection
G/Pβi → G/Pβm(w) where m(w) is the smallest element in Qw.
We define a partition on the peaks of the quiver Qw and a partition of the quiver itself:
DEFINITION 4.1. — (ı) Define a partition (Ai)i∈[1,n] of Peaks(Qw) by induction: A1 is the set of
peaks with minimal height and Ai+1 is the set of peaks in Peaks(Qw)\
k=1Ak with minimal height
(the integer n is the number of different values the height function takes on the set Peaks(Qw)).
(ıı) Define a partition (Qw(i))i∈[1,n] of Qw by induction:
Qw(i) = {x ∈ Qw / ∃j ∈ Ai : x 4 j and x 64 k ∀k ∈ ∪j>iAj}.
We proved in [Pe07] that these quivers Qw(i) are quivers of minuscule Schubert varieties and
in particular have a minimal element mw(i). We defined the variety X̂(w) as the image of the
Bott-Samelson resolution X̃(w) (seen as a configuration variety) in the product
i=1 G/Pβmw(i) .
Because mw(n) = m(w) we have a map π̂ : X̂(w) → X(w) and a factorisation
X̃(w)
// X̂(w)
X(w).
We proved the following result in [Pe07]:
THEOREM 4.2. — (ı) The variety X̂(w) together with the map π̂ realise X̂(w) as the relative
canonical model of X(w).
(ıı) The variety X̂(w) is a tower of locally trivial fibrations with fibers the Schubert varieties
associated to the quivers Qw(i). In particular X̂(w) is Gorenstein.
We will use this resolution to prove our main result. Indeed, we will prove that the generic fibre
of the map π̂ : X̂(w) → X(w) above a (WY) Schubert subvariety X(w′) is a point. In other words,
the map π̂ is an isomorphism on an open subset of X(w′). As a consequence, the generic point of
X(w′) will be in the Gorenstein locus.
Let us recall some facts on X̃(w) and X̂(w) (see [Pe07]):
FACT 4.3. — (ı) To each vertex i of Qw one can associated a divisor Di on X̃(w) and all these
divisors intersect transversally.
(ıı) For K a subset of the vertices of Qw, we denote by ZK the transverse intersection of the
Di for i ∈ K.
(ııı) The image of the closed subset ZK by the map π is the Schubert variety X(wK) whose
quiver QwK is the biggest reduced subquiver of Qw not containing the vertices in K.
The quiver Qw(i) defines a element w(i) in W and the fact that these quivers realise a partition
of Qw implies that we have an expression w = w(1) · · ·w(n) with l(w) =
l(w(i)). We prove the
following generalisation of this fact:
PROPOSITION 4.4. — Let K be a subset of the vertices of Qw. The image of the closed subset
ZK by the map π̃ is a tower of locally trivial fibrations with fibers the Schubert varieties X(wK(i))
whose quiver QwK(i) is the biggest reduced subquiver of Qw(i) not containing the vertices of K∩Qw(i).
This variety is the image by π̃ of Z∪n
QK(i).
Proof — As we explained in [Pe07, Proposition 5.9], the Bott-Samelson resolution is the quotient
of the product
Ri where theRi are certain minimal parabolic subgroups by a product of Borel
subgroups
i=1 Bi. The variety X̂(w) is the quotient of a product
i=1 Ni of parabolic subgroups
such that the multiplication in G maps
k∈Qw(i)
Rk to Ni by a product
i=1Mi of parabolic
subgroups. The map π̃ is induced by the product from
Ri to
i=1Ni. In particular, this
means that for i ∈ [1, n] fixed, the map
k∈Qw(i)
→ Ni induces the map from the Bott-Samelson
resolution X̃(w(i)) to X(w(i)). We may now apply part (ııı) of the preceding fact because the
quiver Qw(i) is minuscule. �
We now remark that the quivers Qw′ associated to Schubert subvarieties X(w
′) in the Schu-
bert variety X(w) having the property (WY) have a nice behaviour with repect to the partition
(Qw(i))i∈[1,n] of Qw.
PROPOSITION 4.5. — Let X(w′) be a Stab(X(w))-stable Schubert subvariety of X(w) having
the property (WY). Let us denote by (Cj)j∈[1,k] the connected components of the subquiver Qw \Qw′
of Qw. Then for each j, there exist an unique ij ∈ [1, n] such that Cj ⊂ Qw(ij).
Proof — Recall from Proposition 3.10 that, denoting by GorHol(Qw) the set of Gorenstein holes
in Qw, we may write
Qw \Qw′ =
i∈GorHol(Qw)
with ki an integer greater or equal to −1 and with the additional condition that Q
ski(i)
w contains
only Gorenstein holes. Because the quivers Q
ski(i)
w are connected, any connected component of
Qw \Qw′ is an union of such quivers. But we have the following:
LEMMA 4.6. — Let i ∈ Holes(Qw) and assume that Q
sk(i)
w meets at least two subquivers of the
partition (Qw(i))i∈[1,n], then Q
sk(i)
w contains a non Gorenstein hole.
Proof — The quiver Q
sk(i)
w meets two subquivers of the partition (Qw(i))i∈[1,n], in particular it
contains two peaks of Qw of different heights. By connexity of Q
sk(i)
w , we may assume that these
two peaks are adjacent. In particular there is a hole between these two peaks and this hole is not
Gorenstein and is contained in Q
sk(i)
w . �
The proposition follows. �
We describe the inverse image by π̂ of a Stab(X(w))-stable Schubert subvariety of X(w) having
the property (WY). To do this, first remark that the map π is B-equivariant and that the inverse
image π−1(X(w′)) has to be a union of closed subsets ZK for some subsets K of Qw. Let ZK ⊂
π−1(X(w′)) be such that π : ZK → X(w
′) is dominant. We will denote by Qw
w (i) the intersection
Qw′ ∩Qw(i) and by w
′(i) the associated element in W .
PROPOSITION 4.7. — The image of ZK in X̂(w) by π̃ is the same as the image of ZQw\Qw′ .
Proof — Thanks to Proposition 4.4 we only need to compute the quivers QwK(i). Consider the
decomposition into connected components Qw \Qw′ = ∪
j=1Cj . We may decompose K accordingly
as K = ∪kj=1Kj where Kj = K ∩ Cj. But because each connected component of Qw \ Qw′ is
contained in one of the quivers (Qw(i))i∈[1,n] this implies that QwK(i) is exactely QwK ∩ Qw(i)
where QwK is the biggest reduced quiver in Qw Qw not containing the vertices in K (see Fact 4.3).
We get QwK = Qw′ (because ZK is sent onto X(w
′)) and the result follows. �
THEOREM 4.8. — Let X(w′) be a Schubert subvariety in X(w). Then X(w′) has the property
(WY) if and only if its generic point is in the Gorenstein locus of X(w).
Proof — We have already seen in Proposition 3.9 that if the generic point of X(w′) is in the
Gorenstein locus of X(w) then X(w′) has the property (WY).
Conversely let X(w′) be a Schubert subvariety having the property (WY). The previous propo-
sition implies that its inverse image π̂−1(X(w′)) is the variety π̃(ZQw\Qw′ ). But this implies that
the map π̂ : π̃(ZQw\Qw′ ) = π̂
−1(X(w′)) → X(w′) is birational (because the varieties have the
same dimension given by the number of vertices in the quiver). In particular, the map π̂ is an
isomorphism on an open subset of X(w) meeting X(w′) non trivially. Therefore, because X̂(w) is
Gorenstein, it is the case of the generic point in X(w′) as a point in X(w). �
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Université Pierre et Marie Curie - Paris 6
UMR 7586 — Institut de Mathématiques de Jussieu
175 rue du Chevaleret
75013 Paris, France.
email : nperrin@math.jussieu.fr
