ALGEBRAIC GEOMETRY I - FINAL PROJECT

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1 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 Xλ 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 λi+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 λi+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 Date: 12/20/10 1

2 2 ADAM KAYE 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 which we can rewrite as 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 Xλ 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 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 Xλ consists of the planes V such that dim(v C 2 λi+i ) = i, so each V Xλ is spanned by vectors v, w such that v C 2 λ1+1 \ C 2 λ1 and w C 2 λ2+2 \ C 2 λ2+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

3 ALGEBRAIC GEOMETRY I - FINAL PROJECT 3 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) 1 0 =, X (1) =, 0 1 {( )} {( )} X(2) =, X (1,1) =, {( )} {( )} X(2,1) =, X (2,2) = 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 ( ) z 24 z 23 z 14 z 13 0 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

4 4 ADAM KAYE 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: ( Idk 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 v 1 c k M = v m c 1 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

5 ALGEBRAIC GEOMETRY I - FINAL PROJECT 5 divided up into four blocks ( ) Idm m 0 M = A B with the m m identity matrix in the top left corner and (A B) = c 1 the matrix representing V c k Lemma 31 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 ( ) Idm m 0 0 D 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 i1,, v ir, c j1,, c js 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 i1,, v ir whose projection onto the v l1,, v lm+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 li : 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 i1,, v ir Therefore the projection onto the plane v li : 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 j1,, c js onto v li : 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 j1,, c js 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 = x 1,m+1 x 1,n x k,m+1 x k,n We can use this description of the polynomials defining X to describe the image of J(X) E :

6 6 ADAM KAYE Proposition 32 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 e 1,k+1 e 1,n e m,k+1 e m,n If d = k then the image of J(X) E is spanned by the e ij appearing in the matrix e 1,m+1 e 1,n e k,m+1 e k,n 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 e 1,m+1 e 1,n e k,m+1 e k,n 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 This completes the proof e 1,k+1 e 1,n e m,k+1 e m,n 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

7 ALGEBRAIC GEOMETRY I - FINAL PROJECT 7 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 λi+i ) i For each 1 i k let X i be the variety given by the single Schubert condition dim(v C n k λi+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 32 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 32 tells us that J(X k ) E is the span of the e ij occurring in the matrix e 1,n a+1 e 1,n e k,n a+1 e k,n 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 32 tells us that im(j(x s ) E ) is the span of the e ij appearing in the matrix e 1,k+1 e 1,n e s,k+1 e s,n 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 41 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 e 1,n a+1 e 1,n e k,n a+1 e k,n and J(X λ ) E has rank ka + ns ks as Proof We have shown that, e 1,k+1 e 1,n e s,k+1 e s,n 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 e 1,n a+1 e 1,n e 1,k+1 e 1,n A 1 := A 2 := e k,n a+1 e k,n e s,k+1 e s,n,

8 8 ADAM KAYE Note that n a k (because n k a) and k s Therefore A 1 and A 2 overlap in the matrix e 1,n a+1 e 1,n A 3 := e s,n a+1 e s,n 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 Xλ 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 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 k k n k λ j = k(n k) λ j = k(n k) λ j=1 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 51 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 41 states that both J(X λ ) E and J(X µ ) E have rank ka + ns ks as j=1

9 ALGEBRAIC GEOMETRY I - FINAL PROJECT 9 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, ie 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 52 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 Xµ (for Xµ X λ ) if and only if it is singular at every point in Xµ 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, [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,