ALGEBRAIC GEOMETRY I - FINAL PROJECT

ADAM KAYE

Abstract. This paper begins with a description of the Schubert varieties of a Grassmannian variety Gr(k, n) over ℂ. Following the technique of Ryan [3] for determining the singularity of flag manifolds we will then describe the Jacobian of a Schubert variety X_{λ}. This will be used to determine which Schubert varieties are nonsingular.

Contents

1. Preliminaries 1 2. Schubert Varieties of Gr(2, 4) 2 3. The Jacobian of a Schubert Condition 4 4. The Jacobian of a Schubert Variety 6 5. (Non)singularity of X_{λ} 8 References 9

1. Preliminaries

In this paper we will be discussing Schubert varieties of the Grassmannian variety Gr(k, n) (the variety of k-dimensional planes in ℂ^{n}). Basic information on Schubert varieties can be found in [2] and proofs of the claims from this section can be found in Tengren’s paper. We fix a flag in ℂ^{n} which as a sequence of subspaces F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{n} = ℂ^{n} such that dim F_{i} = i. Since we are fixing a flag once and for all we can choose a basis v_{1}, …, v_{n} for ℂ^{n} such that F_{i} = ⟨v_{1}, …, v_{i}⟩ = ℂ^{i} (the subspace generated by v_{1}, …, v_{n}). A Young diagram λ is defined to be a nonincreasing finite sequence of non-negative integers (λ_{1}, λ_{2}, …, λ_{s}) and this can be represented by a grid where the ith row has λ_{i} blocks. For each λ that fits inside a rectangle of width n − k and height k we define a Schubert cell X_{λ}^{∘} in Gr(k, n) by

X_{λ}^{∘} := {V ∈ Gr(k, n) : dim(V ∩ ℂ^{j}) = i for n − k − λ_{i} + i ≤ j ≤ n − k − λ_{i+1} + i}

= {V ∈ Gr(k, n) : dim(V ∩ ℂ^{n−k−λ_{i}+i}) = i for all i}

(for the second equality we take the convention that λ_{0} = n − k). We also define the Schubert variety

X_{λ} := {V ∈ Gr(k, n) : dim(V ∩ ℂ^{n−k−λ_{i}+i}) ≥ i for all i}.

The Schubert variety X_{λ} is a variety and it is the closure of X_{λ}^{∘}. Furthermore, X_{λ}^{∘} is an open subset of X_{λ} isomorphic to affine space. It is also important to note that X_{λ}^{∘} ⊂ X_{μ}^{∘} if and only if μ ≤ λ (where μ ≤ λ means μ_{i} ≤ λ_{i} for all i) and X_{λ} = ⋃_{λ≤μ} X_{μ}^{∘}. We can represent some V ∈ Gr(k, n) by a k × n matrix M such that V is given by the span of the rows of M (in the v_{i} basis). Using row reduction M can be uniquely chosen so that the last nonzero term in each row is a 1, there are only 0’s below this 1 and the 1 furthest to the right in each row is to the right of all the 1’s in the higher rows. For example the plane spanned by the vectors (3, 1, 2, 2, 0), (4, 5, 1, 0, 1), (1, 1, 0, 0, 0) is Gr(3, 5) is represented by the matrix

Date: 12/20/10.

(\begin{matrix}3 & 1 & 2 & 2 & 0 \\ 4 & 5 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 \end{matrix})

which we can rewrite as

(\begin{matrix}1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 \\ −1 & 0 & 1 & 0 & 1 \end{matrix})_{.}

By writing elements of Gr(k, n) in this form, we can let GL_{n}(ℂ) act on Gr(k, n) by right multiplication. For each V ∈ Gr(k, n), the orbit of V under the subgroup of lower triangular matrices will be a Schubert cell: we know the orbit is contained in a Schubert cell because if V is spanned by the vectors w_{1}, …, w_{k} written in this special form, then w_{1} ∈ ℂ^{i} \ℂ^{i−1} implies w_{1}A ∈ ℂ^{i} \ℂ^{i−1} for any A in the subgroup of lower triangular matrices. The orbit will be the entire Schubert cell because every element of X_{λ}^{∘} can explicitly be written as V A for some lower triangular A where V is the plane given by the matrix with a 1 in the (n − k − λ_{i} + i)th spot of the ith row and 0’s everywhere else. For example the matrix above can be written as

(\begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) · (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 \\ −1 & 0 & 1 & 0 & 1 \end{matrix})_{.}

The rest of this paper will examine the singularities of Schubert varieties. Beyond the fact that these give lots of examples of nontrivial varieties, the singularities of Schubert varieties are important because of applications to representation theory and combinatorics. They are also important for the study of the Grassmannian because they can tell us about the cohomology of Gr(k, n) (see [2]). However, before we go on to look at Schubert varieties in general, we will examine all the Schubert varieties in Gr(2, 4) as an example. This is the first nontrivial example of a Grassmannian and it will show how singular Schubert varieties can arise, as well as help to solidify the basic ideas just presented about Schubert cells and varieties.

2. Schubert Varieties of Gr(2, 4)

Let’s look at the Schubert varieties of Gr(2, 4) and see where they have singularities. Each Schubert variety will correspond to a Young diagram that fits in the 2 × 2 square, so there are six Schubert varieties: X_{(0)}, X_{(1)}, X_{(2)}, X_{(1,1)}, X_{(2,1)}, and X_{(2,2)}. The Schubert cell X_{λ}^{∘} consists of the planes V such that dim(V ∩ ℂ^{2−λ_{i}+i}) = i, so each V ∈ X_{λ}^{∘} is spanned by vectors v, w such that v ∈ ℂ^{2−λ_{1}+1} \ ℂ^{2−λ_{1}} and w ∈ ℂ^{2−λ_{2}+2} \ ℂ^{2−λ_{2}+1}. Using row reduction we can then assume that v has the form (∗, …, ∗, 1, 0, …, 0) where the 1 is in the 2−λ_{1} +1 = (3−λ_{1})th spot and w has the same form, but with a 1 in the (4−λ_{2})th spot and a 0 in the (3 − λ_{1})th spot. Carrying out this procedure for each of the Young diagrams we can describe the Schubert cells:

X_{(0)}^{∘} = \{(\begin{matrix}∗ & ∗ & 1 & 0 \\ ∗ & ∗ & 0 & 1 \end{matrix})\} , X_{(1)}^{∘} = \{(\begin{matrix}∗ & 1 & 0 & 0 \\ ∗ & 0 & ∗ & 1 \end{matrix})\} ,

X_{(2)}^{∘} = \{(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & ∗ & ∗ & 1 \end{matrix})\} , X_{(1,1)}^{∘} = \{(\begin{matrix}∗ & 1 & 0 & 0 \\ ∗ & 0 & 1 & 0 \end{matrix})\} ,

X_{(2,1)}^{∘} = \{(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & ∗ & 1 & 0 \end{matrix})\} , X_{(2,2)}^{∘} = \{(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix})\} .

Using the Plücker embedding Gr(2, 4) ↪ ℙ^{5} whose image is defined by x_{12}x_{34} − x_{13}x_{24} +x_{14}x_{23} (where x_{ij}, the Plücker coordinate, is the determinant of the minor given by taking the ith and j column) we can use the Jacobian criterion for nonsingularity to determine which of these Schubert varieties is nonsingular: it was shown in class that an affine variety X = V (f_{1}, …, f_{l}) ⊂ 𝔸^{n} is nonsingular if the Jacobian (∂f_{i}/∂x_{j}) has rank n − dim_{x} X for all x ∈ X. We have seen that X_{λ} = ∪_{μ≥λ}X_{μ}^{∘}. In particular this shows that X_{(0)} = Gr(2, 4). If we look at the affine open subset where x_{12} ≠ 0, the Grassmannian is defined by z_{34} − z_{13}z_{24} + z_{14}z_{23} where z_{ij} = x_{ij}/x_{12}. Ordering the coordinates by z_{13}, z_{14}, z_{23}, z_{24}, z_{34}, the Jacobian is then given by the 1 × 5 matrix

(−z_{24}, z_{23}, z_{14}, −z_{13}, 1)

which always has rank 1 because of the 1 in the last coordinate (which comes from the derivative with respect to z_{34}. Since Gr(2, 4) has codimension 1 in ℙ^{5} this shows that it is nonsingular on the open subset of x_{12} ≠ 0. The same argument shows that it is nonsingular on the other open affine charts, so this shows that the Grassmanian is nonsingular.

We now carry out a similar procedure for the other Schubert varieties. X_{(1)} = Gr(2, 4) \ X_{(0)}^{∘} and this is equal to Gr(2, 4) ∩ V (x_{34}) because by looking at the Schubert cells we can see that a point V ∈ Gr(2, 4) has nonzero x_{34} coordinate if and only if it is in X_{(0)}^{∘}. Thus X_{(1)}^{∘} = V (x_{34}, x_{12}x_{34} − x_{13}x_{24} + x_{14}x_{23}) = V (x_{34}, x_{13}x_{24} − x_{14}x_{23}). Localizing to the affine open subset x_{12} ≠ 0 again, we get the Jacobian

(\begin{matrix}0 & 0 & 0 & 0 & 1 \\ z_{24} & −z_{23} & −z_{14} & z_{13} & 0 \end{matrix}) .

The Schubert cell X_{(1)}^{∘} is an open subset of X_{(1)} that is isomorphic to ℂ^{3} so X_{(1)} has codimension 2 in ℙ^{5}. Therefore X_{(1)} will be nonsingular if this Jacobian has rank 2 everywhere. However the Jacobian only has rank 1 at the point (x_{12}, x_{13}, x_{14}, x_{23}, x_{24}, x_{34}) = (1, 0, 0, 0, 0, 0) ∈ X_{(1)}. Thus X_{(1)} is singular (in fact this shows that X_{(1)} is the closure of V (xy − zw) in ℙ^{4}).

Now we’ll look at an example of a proper Schubert variety that is nonsingular. We know that X_{(1,1)} is the union X_{(1,1)}^{∘} ∪ X_{(2,1)}^{∘} ∪ X_{(2,2)}^{∘} and by looking at our description of these cells it is clear that their union is defined by x_{34} = x_{24} = x_{14} = 0. This is because each of these polynomials vanishes on the union and x_{34} ≠ 0 on X_{(0)}^{∘}, x_{24} ≠ 0 on X_{(1)}^{∘}, and x_{14} ≠ 0 on X_{(2)}^{∘}, so V (x_{34}, x_{24}, x_{14}) = X_{(1,1)} in Gr(2, 4). Thus X_{(1,1)} is equal to V (x_{34}, x_{24}, x_{14}, x_{12}x_{34} − x_{13}x_{24} + x_{14}x_{23}) = V (x_{34}, x_{24}, x_{14}) in ℙ^{5}. But this means that X_{(1,1)} is isomorphic to ℙ^{2} which is a nonsingular variety. You can use the same method to look at each of the other Schubert varieties X_{(2)}, X_{(2,1)}, X_{(2,2)} and determine which ones are nonsingular. Do you have any conjectures about which Schubert varieties will be nonsingular in a general Gr(k, n)?

In the rest of this paper we will answer this question, however we will not be able to explicitly describe each Schubert variety of Gr(k, n), or write out the Jacobian exactly like we did for Gr(2, 4). Instead we will focus on the unique point in X_{(n−k,n−k)}^{∘} and use polynomials that locally define the Schubert variety. We will use the Jacobian condition to determine if the Schubert variety is nonsingular at this point and that will allow us to determine when the variety is nonsingular. This method follows the same steps that Ryan uses in [3] to look at singularities of general flag varieties; we have just adjusted it to only take care of the Grassmannian case.

3. The Jacobian of a Schubert Condition

In this section we consider the Schubert variety X in Gr(k, n) defined by the Schubert condition dim(V ∩ ℂ^{m}) ≥ d (note that we must have d ≤ min(k, m) for X to be nonempty). This will correspond to the Young diagram with λ_{i} = n−k−m+d for i ≤ d and λ_{i} = 0 for i ≥ d. Let {f_{l}} be polynomials in the variables x_{ij} (1 ≤ i ≤ n, 1 ≤ j ≤ k) defining X. Let E ∈ Gr(k, n) be the element (a_{ij}) given by the k × k identity matrix with 0s to the right:

( Id_{k×k} 0_{(n−k)×k} )

and U the neighborhood of E in Gr(k, n) consisting of matrices (x_{ij}) such that x_{ij} = 1 if i = j and x_{ij} = 0 if i ≠ j, j ≤ k. Then U is isomorphic ℂ^{(n−k)k} (see Tengren’s paper for a proof), so we can describe the tangent space of E in Gr(k, n) as the span of the vectors e_{ij} with a 0 in every coordinate except the one corresponding to x_{ij}. Let J(X) be the Jacobian (∂f_{l}/∂x_{ij}). Using the basis we will give an explicit description of J(X)|_{E}.

The first step is to give a more detailed description of the polynomials f_{l} that define X in the neighborhood U. If V is an element of U then let c_{1}, …, c_{k} be the row vectors defining V, so V corresponds to the matrix

(x_{ij}) = ( c_{1} ; ⋮ ; c_{m})_{.}

v_{i} denotes the row vector with all 0’s except a 1 in the ith spot and ℂ^{m} is spanned by v_{1}, …, v_{m}. Saying that dim(V ∩ℂ^{m}) ≥ d is equivalent to saying that dim(V +ℂ^{m}) ≤ m + k − d. Thus V will satisfy this condition if and only if every m + k − d + 1 × m + k − d + 1 minor of the matrix

M = ( v_{1} ; ⋮ ; v_{m} ; c_{1} ; ⋮ ; c_{k})

has determinant 0. The determinants of the minors of this matrix are polynomials in the x_{ij} coordinates, so these polynomials define X in U. This matrix can be divided up into four blocks

M = ( \begin{matrix}Id_{m×m} & 0 \\ A & B \end{matrix} )

with the m × m identity matrix in the top left corner and (A|B) = ( c_{1} ; ⋮ ; c_{k} ) the

matrix representing V.

Lemma 3.1. With M, B as above, every m + k − d + 1 × m + k − d + 1 minor of M has determinant 0 if and only if every k − d + 1 × k − d + 1 minor of B has determinant 0.

Proof. If D is a k − d + 1 × k − d + 1 minor of B with nonzero determinant then

(\begin{matrix}Id_{m×m} & 0 \\ 0 & D \end{matrix})

is a m + k − d + 1 × m + k − d + 1 minor of M with nonzero determinant.

If D is a m+k −d+1×m+k −d+1 minor of M with nonzero determinant then suppose D is given by taking the rows corresponding to v_{i_{1}}, …, v_{i_{r}}, c_{j_{1}}, …, c_{j_{s}} and the columns numbered by l_{1}, …, l_{m+k−d+1}. For each of the v rows chosen, there will be a row of all zeros in D unless the column where v has a 1 is also chosen. Subtracting the v_{i} rows chosen from the c_{j} rows chosen gives s vectors in V +⟨v_{i_{1}}, …, v_{i_{r}}⟩ whose projection onto the ⟨v_{l_{1}}, …, v_{l_{m+k−d+1}}⟩ plane has dimension at least m + k − d + 1 − r. We can choose to subtract the v rows such that the projection onto the ⟨v_{l_{i}} : 1 ≤ l_{i} ≤ m⟩ plane has dimension at most t − r where t is the number of l_{i} such that l_{i} ≤ m. This is done by subtracting the v rows such that there are only 0’s in the columns corresponding to v_{i_{1}}, …, v_{i_{r}}. Therefore the projection onto the plane ⟨v_{l_{i}} : m+1 ≤ i ≤ n⟩ has dimension at least m + k − d + 1 − r − (t − r) = m + k − d + 1 − t. But this projection is the same as the projection of ⟨c_{j_{1}}, …, c_{j_{s}}⟩ onto ⟨v_{l_{i}} : m + 1 ≤ i ≤ n⟩ because subtracting the v rows doesn’t affect the coordinates m + 1 through n. This means that the s vectors c_{j_{1}}, …, c_{j_{s}} span a space of dimension at least m + k − d + 1 − t after projecting onto the ⟨v_{m+1}, …, v_{n}⟩ plane. Since t ≤ m this implies that there is a subset of the c vectors that span a space of dimension at least k − d + 1 after projecting onto the ⟨v_{m+1}, …, v_{n}⟩. But this tells us that there is a k − d + 1 × k − d + 1 minor of B with nonzero determinant. □

Thus the polynomials f_{l} defining our Schubert condition dim(V ∩ ℂ^{m}) ≥ d are precisely the polynomials that define the determinants of all k − d + 1 × k − d + 1 minors of the matrix

B = (\begin{matrix}x_{1,m+1} & … & x_{1,n} \\ ⋮ & ... & ⋮ \\ x_{k,m+1} & … & x_{k,n} \end{matrix})_{.}

We can use this description of the polynomials defining X to describe the image of J(X)|_{E}:

Proposition 3.2. The Jacobian J(X)|_{E} has rank 0 unless d = min(k, m). If d = m then the image of J(X)|_{E} is spanned by the e_{ij} appearing in the matrix

(\begin{matrix}e_{1,k+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{m,k+1} & … & e_{m,n} \end{matrix})_{.}

If d = k then the image of J(X)|_{E} is spanned by the e_{ij} appearing in the matrix

(\begin{matrix}e_{1,m+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{k,m+1} & … & e_{k,n} \end{matrix})_{.}

Proof. Fix some f_{l} corresponding to a specific minor N. Then for a given x_{ij}, ∂f_{l}/∂x_{ij} will certainly be 0 if x_{ij} is not contained in N. If x_{ij} is contained in N, then expanding the determinant along cofactors shows us that ∂f_{l}/∂x_{ij} = det(N_{ij}) where N_{ij} is the cofactor of N given by removing the row and column containing x_{ij} (unless N is a 1 × 1 matrix in which case ∂f_{l}/∂x_{ij} = 1). Evaluating ∂f_{l}/∂x_{ij} at E will then give the determinant of the minor N_{ij} in the matrix E. Since E = (a_{ij}) is defined by a_{ij} = 1 if i = j and 0 otherwise, the only ways ∂f_{l}/∂x_{ij}|_{E} can be nonzero is if the N_{ij} minor of E is a k − d × k − d identity matrix or d − k = 0 and in both these cases we will have ∂f_{l}/∂x_{ij}|_{E} = 1.

The nonzero entries of E that appear in the B matrix are c_{m+1,m+1} = ⋯ = c_{k,k} = 1 (where c_{i} = (c_{i1}, …, c_{in})), and these will form a k − d × k − d identity matrix contained in B if and only if k − d ≤ k − m. But this implies that d ≥ m, so we must have d = m. The only other way that ∂f_{l}/∂x_{ij}|_{E} can be nonzero is if N is a 1 × 1 matrix which implies that k − d + 1 = 1, so k = d. Thus ∂f_{l}/∂x_{ij}|_{E} = 0 for all x_{ij} unless d = k or d = m. Therefore J(X)|_{E} has rank 0 unless d = k or d = m.

If d = k then the matrix N corresponding to f_{l} will be a 1 × 1 block for all f_{l}, so the collection of f_{l} defining X will just be the x_{ij} appearing in the B matrix. Thus the image of J(X)|_{E} will be the span of the e_{ij} appearing in the matrix

(\begin{matrix}e_{1,m+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{k,m+1} & … & e_{k,n} \end{matrix})_{.}

If d = m then the x_{ij} such that ∂f_{l}/∂x_{ij}|_{E} ≠ 0 are exactly the the x_{ij} that are contained in some minor N with cofactor N_{ij} the k − d × k − d identity matrix in the lower left (and for each such N there is a unique i, j such that ∂f_{l}/∂x_{ij}|_{E} ≠ 0). Thus x_{ij} must satisfy 1 ≤ i ≤ d and m − d + k + 1 ≤ j ≤ n. But d = m so the image of J(X)|_{E} will be the span of the e_{ij} appearing in the matrix

(\begin{matrix}e_{1,k+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{m,k+1} & … & e_{m,n} \end{matrix})_{.}

This completes the proof. □

4. The Jacobian of a Schubert Variety

Now that we know what J(X)|_{E} looks like for a fixed Schubert condition we can calculate the rank of the Jacobian of any Schubert variety at E. Let λ be a Young diagram contained in the rectangle of width n − k and height k and consider the corresponding Schubert variety X_{λ}. Then X_{λ} is defined by the Schubert conditions dim(V ∩ ℂ^{n−k−λ_{i}+i}) ≥ i. For each 1 ≤ i ≤ k let X_{i} be the variety given by the single Schubert condition dim(V ∩ ℂ^{n−k−λ_{i}+i}) ≥ i. Then im(J(X_{λ})|_{E}) will be spanned by the im(J(X_{i})|_{E}) because the polynomials defining X can be taken to be the union of the polynomials defining each X_{i}. Proposition 3.2 implies that J(X_{i})|_{E} = 0 unless i = k or i = n − k − λ_{i} + i. The second case occurs if and only if λ_{i} = n − k.

Let a = λ_{k}. Then X_{k} is given by the Schubert condition with d = i = k and m = n − k − a + k = n − a. Therefore proposition 3.2 tells us that J(X_{k})|_{E} is the span of the e_{ij} occurring in the matrix

(\begin{matrix}e_{1,n−a+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{k,n−a+1} & … & e_{k,n} \end{matrix})_{.}

If λ_{i} = n−k then the Schubert condition is given by dim(V ∩ℂ^{i}) ≥ i. If λ_{1} = λ_{2} = ⋯ = λ_{s} = i then the single condition dim(V ∩ ℂ^{s}) ≥ s implies all the conditions corresponding to i ≤ s. Therefore im(J(X_{s})|_{E}) = im(J(X_{1})|_{E})+⋯+im(J(X_{s})|_{E}). The condition corresponding to X_{s} is given by m = d = s so proposition 3.2 tells us that im(J(X_{s})|_{E}) is the span of the e_{ij} appearing in the matrix

(\begin{matrix}e_{1,k+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{s,k+1} & … & e_{s,n} \end{matrix})_{.}

Since all the other J(X_{i})|_{E} don’t contribute anything to the rank of J(X)|_{E} we can just take the span of all vectors appearing in either of these matrices to describe the image of J(X)|_{E}. Since the the e_{ij} are linearly independent and we have an easy description of which e_{ij} are in the image we can actually count the dimension of im J(X)|_{E}:

Proposition 4.1. Let X_{λ} be the Schubert variety in Gr(k, n) corresponding to the Young diagram (λ_{1}, …, λ_{k}) where s is the largest integer such that λ_{i} = n − k for all i ≤ s and a = λ_{k}. Then the image of J(X_{λ})|_{E} is given by the span of the e_{ij} appearing in the matrices

(\begin{matrix}e_{1,n−a+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{k,n−a+1} & … & e_{k,n} \end{matrix})_{,} (\begin{matrix}e_{1,k+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{s,k+1} & … & e_{s,n} \end{matrix})

and J(X_{λ})|_{E} has rank ka + ns − ks − as.

Proof. We have shown that

im(J(X_{λ})|_{E}) = im(J(X_{1})|_{E}) + ⋯ + im(J(X_{k})|_{E}) = im(J(X_{s})|_{E}) + im(J(X_{k})|_{E})

which has image given by the span of the e_{ij} appearing in the matrices

A_{1} := (\begin{matrix}e_{1,n−a+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{k,n−a+1} & … & e_{k,n} \end{matrix})_{,} A_{2} := (\begin{matrix}e_{1,k+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{s,k+1} & … & e_{s,n} \end{matrix})_{.}

Note that n−a ≥ k (because n−k ≥ a) and k ≥ s. Therefore A_{1} and A_{2} overlap in the matrix

A_{3} := (\begin{matrix}e_{1,n−a+1} & … & e_{1,n} \\ ⋮ & ... & ⋮ \\ e_{s,n−a+1} & … & e_{s,n} \end{matrix})_{.}

Thus the rank of the Jacobian is equal to |A_{1}| + |A_{2}| − |A_{3}| where |A_{i}| denotes the number of entries in the matrix A_{i}. |A_{1}| = ak, |A_{2}| = (n − k)s, and |A_{3}| = as, so

rk J(X_{λ})|_{E} = ka + (n − k)s − as = ka + ns − ks − as.
□

5. (Non)singularity of X_{λ}

In this section we will determine when X_{λ} is nonsingular at E. We already know the rank of the Jacobian at E, so all we need to do is compare it to the codimension of X_{λ}. So we begin by calculating the dimension of X_{λ}. X_{λ}^{∘} is given by all matrices having a 1 in the (n − k − λ_{j} + j)th column of the jth row, all entries 0 below these 1’s and nonzero entries appearing only in the entries to the left of these 1’s. For example the Schubert cell in Gr(3, 6) corresponding to (3, 1, 0) is the set of planes that can be represented by a matrix of the form

(\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & ∗ & ∗ & 1 & 0 & 0 \\ 0 & ∗ & ∗ & 0 & ∗ & 1 \end{matrix})_{.}

Thus the dimension of this Schubert cell is the number of stars that appear in this matrix. The jth column will have n − k − λ_{j} stars because there will be n − k − λ_{j} + (j − 1) spots to the left of the 1 in the jth row, but j − 1 of them will be required to be 0 because of the 1’s in the rows above. Thus the Schubert cell has dimension

∑_{j=1}^{k} n − k − λ_{j} = k(n − k) − ∑_{j=1}^{k} λ_{j} = k(n − k) − |λ|.

Since the Schubert cell X_{λ}^{∘} is dense in X_{λ} and it is isomorphic to ℂ^{k(n−k)−|λ|} we may conclude that X_{λ} is irreducible and has dimension k(n − k) − |λ| (so the codimension of X_{λ} is is k(n − k) − |λ| at every point). Therefore dim_{E} X_{λ} = k(n−k)−|λ|. The Jacobian condition for nonsingularity says that X_{λ} is nonsingular at E if and only if the rank of the Jacobian is equal to the codimension of X_{λ} at E. Thus X_{λ} will be nonsingular if and only if

rk J(X_{λ})|_{E} = dim Gr(k, n) − dim_{E} X_{λ} = k(n − k) − (k(n − k) − |λ|) = |λ|.

Theorem 5.1. X_{λ} is nonsingular at E if and only if λ has the form (n − k, n − k, …, n − k, a, a, …, a) (where the last a is λ_{k}).

Proof. Take some Young diagram λ and as earlier let s be the largest integer such that λ_{i} = n − k for all i ≤ s and a = λ_{k}. Let μ be the Young diagram given by μ_{i} = n − k for 1 ≤ i ≤ s and μ_{i} = a for s + 1 ≤ i ≤ k. Then μ is contained in λ, so |λ| ≥ |μ| with equality if and only if μ = λ. We have just shown that X_{λ} will be nonsingular at E if and only if |λ| = rk J(X_{λ})|_{E} and proposition 4.1 states that both J(X_{λ})|_{E} and J(X_{μ})|_{E} have rank ka + ns − ks − as.

Since μ contains s rows of length n−k and k−s rows of length a, |μ| = s(n−k)+ (k − s)a = ka + ns − ks − as. Therefore X_{μ} is nonsingular at E because the rank of the Jacobian and the codimension of X_{μ} at E are both ka + ns − ks − as. If λ ≠ μ then |λ| > |μ| so X_{λ} will have codimension strictly greater than ka + ns − ks − as, but it will still have a Jacobian of rank ka + ns − ks − as at E. Therefore X_{λ} is singular at E if λ ≠ μ. Therefore we can conclude that X_{λ} will be singular at E unless it has the form (n − k, …, n − k, a, …, a) with a = λ_{k}. □

We now use an argument which can be found in [1] page 1382 to say which Schubert varieties are nonsingular. The idea is that the singular points form a closed subset and the action of lower triangular matrices sends singular points to singular points, so E is the “worst” point in the Schubert variety, i.e. if there is a singular point anywhere then E must be a singular point. Thus a Schubert variety will be nonsingular if and only if E is a nonsingular point.

Corollary 5.2. The Schubert variety X_{λ} in Gr(k, n) is nonsingular if and only if λ has the form (n − k, …, n − k, a, …, a) with a = λ_{k}.

Proof. Given X_{λ} and x ∈ X_{λ}, rk J(X_{λ})|_{x} = |λ| is equivalent to the nonvanishing of a collection of polynomials which define the determinants of the |λ| × |λ| minors of J(X_{λ})|_{x}. Since the entries of J(X_{λ})|_{x} are given by polynomials in the coordinates of x this means that the nonsingular points of X_{λ} form an open subset. In addition, the lower triangular matrices in Gl(n, ℂ) act transitively by isomorphisms on each Schubert cell contained in X_{λ}. Therefore X_{λ} is singular at some x ∈ X_{μ}^{∘} (for X_{μ}^{∘} ⊂ X_{λ}) if and only if it is singular at every point in X_{μ}^{∘}. Since the singular points form a closed subset this would then imply that every x ∈ X_{μ} is a singular point of X_{λ}. But X_{λ} = ⋃_{λ⊂μ} X_{μ}^{∘} and E ∈ X_{μ} for all μ so if X_{λ} contains any nonsingular point, every point in X_{μ} will be singular for some μ and E will be contained in this X_{μ} so E will be a nonsingular point. Thus X_{λ} is nonsingular if and only if E is a nonsingular point of X_{λ}. Boom Shakalaka. □

References

[1] Doedhar V. Poincaré duality and nonsingularity of Schubert varieties. Comm. Algebra. 13, 1379-1388. 1985.

[2] Fulton, William. Young Tableaux. Cambridge University Press. 1997.

[3] Ryan, Kevin M. On Schubert varieties in the flag manifold of Sl(n, ℂ). Math. Ann. 276, 205-224. 1987.