Singular Loci of Varieties of Complexes, II

Journal of Algebra 235, 547᎐558 Ž2001. doi:10.1006rjabr.1999.8068, available online at http:rrwww.idealibrary.com on

Singular Loci of Varieties of Complexes, II Nicolae Gonciulea Department of Mathematics, Uni¨ ersity of Michigan, Ann Arbor, Michigan 48109 E-mail: [email protected] Communicated by Peter Littelmann Received November 20, 1998

In this paper we determine the singular loci of the irreducible components of the variety of complexes. 䊚 2001 Academic Press

INTRODUCTION Let W1 , . . . , Whq1 be finite-dimensional vector spaces over some field k such that dim Wi s n i for 1 F i F h q 1. Let Z be the affine space of all h-tuples of linear maps ⭈⭈⭈

f hy1

Wh

fh

Whq1 .

6

f2

6

W2

6

f1

6

Ž f 1 , . . . , f h . : W1

The subvariety C of Z defined by C s  Ž f 1 , f 2 , . . . , f h . g Z N f iq1 f i s 0 for 1 F i F h y 1 4 is called the ¨ ariety of complexes. For positive integers k 1 , . . . , k h such that k i q k iq1 F n iq1 for 0 F i F h Žwhere k 0 s k hq1 s 0., the varieties V Ž k 1 , . . . , k h . s  Ž f 1 , . . . , f h . g C N rank Ž f i . F k i for 1 F i F h4 are irreducible subvarieties of C . In this paper we determine Žthe irreducible components of. the singular locus of V Ž k 1 , . . . , k h .. Let V s V Ž k 1 , . . . , k h ., Vi s V Ž k 1 , . . . , k iy1 , k i y 1, k iq1 , . . . , k h .

for 1 F i F h,

Vj, jq1 s V Ž k 1 , . . . , k jy1 , k j y 1, k jq1 y 1, k jq2 , . . . , k h . for 1 F j F h y 1. 547 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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We show the following THEOREM 1. The irreducible components of Sing V are Vi , with i g ⍀, and Vj, jq1 , with j f ⍀, j q 1 f ⍀, where ⍀ is the set of all 1 F i F h such that k iy1 q k i - n i and k i q k iq1 - n iq1. As a consequence, we obtain THEOREM 2. If V s V Ž k 1 , . . . , k h . is an irreducible component of the ¨ ariety of complexes C , then the irreducible components of Sing V are V1, 2 , . . . , Vhy1, h . For h s 2, these results are proved in w3x. The varieties V Ž k 1 , . . . , k h . can be identified with opposite cells of certain Schubert varieties Žsee w4, 5x.; thus, using Theorem 1, one can determine the irreducible components of these Schubert varieties. We thank V. Lakshmibai for helpful discussions during the preparation of this paper.

1. PRELIMINARIES Let us fix the positive integers h G 2, n1 , . . . , n hq1 , and let n s Ž n1 , . . . , n hq1 .. Let W1 , . . . , Whq1 be finite-dimensional vector spaces over a field k with dim W1 s n1 , . . . , dim Whq1 s n hq1. Let Z be the affine space of all h-tuples of linear maps ⭈⭈⭈

f hy1

Wh

fh

Whq1 .

6

f2

6

W2

6

f1

6

Ž f 1 , . . . , f h . : W1

The group Gn s GLŽ n1 . = ⭈⭈⭈ = GLŽ n hq1 . acts on Z by y1 y1 Ž g 1 , g 2 , . . . , g hq1 . ⭈ Ž f 1 , f 2 , . . . , f h . s Ž g 2 f 1 gy1 1 , g 3 f 2 g 2 , . . . , g hq1 f h g h . .

The subvariety C of Z defined by C s  Ž f 1 , f 2 , . . . , f h . g Z N f iq1 f i s 0 for 1 F i F h y 1 4 is called the ¨ ariety of complexes. h Let Kn denote the set of all k s Ž k 1 , . . . , k h . g ⺪q , such that k i q k iq1 F n iq1 for 0 F i F h Žhere k 0 s k hq1 s 0.. For k s Ž k 1 , . . . , k h . g Kn , let V Ž k . s  Ž f 1 , . . . , f h . g C N rank Ž f i . F k i for 1 F i F h4 . Let X i be the n iq1 = n i matrix of indeterminates, for 1 F i F h, and let k w X 1 , . . . , X h x be the polynomial ring over the indeterminates in X 1 j ⭈⭈⭈ j X h . For k g Kn , let I Žk. be the ideal in k w X 1 , . . . , X h x generated by the entries of X iq1 X i , for 1 F i F h y 1, and the k i q 1 minors of X i , for 1 F i F h.

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Let G s GLŽ n.. Let T be the maximal torus consisting of diagonal matrices in G, let B be the Borel subgroup consisting of the upper triangular matrices in G, and let W s Sn be the Weyl group of G. Let Pd , 1 F d F n y 1, be the maximal parabolic subgroups in G, where Pd s

½

) AgGNAs 0 Ž nyd .=d

) )

ž

/5

.

Consider the parabolic subgroup Q s Pa1 l ⭈⭈⭈ l Pa h of G, i.e., the subgroup consisting of all elements of the form



A1 0 .. .

) A2 .. .

) ) .. .

⭈⭈⭈ ⭈⭈⭈

) ) .. .

) ) .. .

0 0

0 0

0 0

⭈⭈⭈ ⭈⭈⭈

Ah 0

) A hq 1

0

,

where A i is a matrix of size Ž a i y a iy1 . = Ž a i y a iy1 . for 1 F i F h q 1 Žhere a0 s 0 and a hq1 s n.. The Weyl group of Q is WQ s Sa = Sa ya 1 2 1 = ⭈⭈⭈ = Sny a h. The set of T-fixed points in GrQ for the action given by multiplication is precisely the set  e w, Q , w g W 4 , where for w g W, e w, Q is the point in GrQ corresponding to the coset wQ. For w g W, let X Q Ž w . be the Schubert ¨ ariety in GrQ associated to wWQ , i.e., the Zariski closure of Be w, Q in GrQ. It is well known that Schubert varieties are irreducible. Let Oy be the set of elements of G of the form



I1 ) .. .

0 I2 .. .

0 0 .. .

⭈⭈⭈ ⭈⭈⭈

0 0 .. .

0 0 .. .

) )

) )

) )

⭈⭈⭈ ⭈⭈⭈

Ih )

0 Ihq 1

0

,

where I j is the identity matrix of size Ž a j y a jy1 . = Ž a j y a jy1 . for 1 F j F h q 1 Žhere a0 s 0 and a hq1 s n.. Then the restriction of the canonical morphism G ª GrQ to Oy is an open immersion, and the image of Oy is isomorphic to the opposite big cell in GrQ, i.e., with Bye id, Q , where By is the Borel subgroup opposite to B, consisting of the lower triangular matrices in G. Thus the set Oy is identified with the opposite big cell in GrQ. Any Schubert variety X Q Ž w . ; GrQ has a non-empty intersection with the opposite big cell Oy, and X Q Ž w . l Oy is called the opposite cell in X Q Ž w ..

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The following result is proved in w5x: THEOREM 1.1. For k g Kn , the ¨ ariety V Žk. is isomorphic to the opposite cell in a certain Schubert ¨ ariety in GLŽ n.rQ, where n s n1 q ⭈⭈⭈ qn hq1 and Q s Pa1 l ⭈⭈⭈ l Pa h, a i s n1 q ⭈⭈⭈ qn i for 1 F i F h. Furthermore, I Žk. defines the reduced scheme structure on V Žk.. This isomorphism is the restriction of the map

␸ : M Ž n 2 = n1 . = ⭈⭈⭈ = M Ž n hq1 = n h . ª Oy ,

¡I

A1 0 .. .

0 I2 A2 .. .

0 0 I3 .. .

⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈

0 0 0

0 0 0

0 0 0

⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈

1

Ž A1 , . . . , A h . ¬

¢

0 0 0 .. .

0 0 0 .. .

0 0 0 .. .

Ihy 1 A hy 1 0

0 Ih Ah

0 0

¦ Ž mod Q .

§

Ihq1

to V Žk.. This result can also be obtained from the identification of qui¨ er ¨ arieties Žwhich are more general than the varieties V Žk. with k g Kn , discussed here. with opposite cells of Schubert varieties Žsee w4x.. It is easily seen that V Žk., with k s Ž k 1 , . . . , k h . g Kn , are all the closed irreducible Gn -stable subvarieties of C and that C s Dk g Kn V Žk.. Define the following partial order on the set Kn : for k s Ž k 1 , k 2 , . . . , k h ., k⬘ s Ž kX1 , kX2 , . . . , kXh . in Kn , k G k⬘ m k i G kXi , 1 F i F h. Clearly, V Žk⬘. : V Žk. if and only if k⬘ F k. Hence we have the following THEOREM 1.2. The irreducible components of C are its sub¨ arieties of the form V Žk., with k a maximal element of Kn Ž with respect to the partial order abo¨ e .. The following result is also proved in w5x: THEOREM 1.3.

Let k g Kn .

1. dim V Žk. s Ý1 F iF hq1Ž n i y k i .Ž k iy1 q k i ., where k 0 s k hq1 s 0. 1 2. codim Z V Žk. s Ý his1Ž n iq1 y k i .Ž n i y k i . q Ý hy is1 k i k iq1 . This theorem can also be deduced from w1x. Considering some fixed basis in each Wi , we have the identifications Wi ( k n i and Z ( M Ž n 2 = n1 . = ⭈⭈⭈ = M Ž n hq1 = n h ., where M Ž l = m. denotes the affine space of matrices over k with l rows and m columns. Then C can be identified with the set of all points Ž A1 , . . . , A h . in

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M Ž n 2 = n1 . = ⭈⭈⭈ = M Ž n hq1 = n h . such that A 2 A1 s 0, . . . , A h A hy1 s 0. For k g Kn , the variety V Žk. is identified with the set of all points Ž A1 , . . . , A h . in M Ž n 2 = n1 . = ⭈⭈⭈ = M Ž n hq1 = n h . such that A 2 A1 s 0, . . . , A h A hy1 s 0, rank A1 F k 1 , . . . , rank A h F k h . Note that V Žk. is the closure of the Gn -orbit in M Ž n 2 = n1 . = ⭈⭈⭈ = M Ž n hq1 = n h . through a point Ž A1 , . . . , A h . such that rank A1 s k 1 , . . . , rank A h s k h , and because of the action of Gn , we may assume, without loss of generality, that 0 Ai s M i

ž

0 0 ,

/

for 1 F i F h,

where Mi is the k i = k i identity matrix. 2. SINGULAR LOCUS OF V Ž k 1 , . . . , k h . In this section we determine the singular points of V Žk., using the Jacobian criterion. Recall that, by Theorem 1.1, I Žk. is the defining ideal of V Žk.. Let k s Ž k 1 , . . . , k h . g Kn and V s V Ž k 1 , k 2 , . . . , k h .. For 1 F i F h, let X i denote the n iq1 = n i matrix of variables, and let Mi be the set of all k i q 1 minors of X i . Then V is the set of zeros of the polynomials in M1 , . . . , Mh , X 2 X 1 , . . . , X h X hy1. Let x s Ž A1 , . . . , A h . be a point in V, A i being an n iq1 = n i matrix of rank at most k i , and let J x be the Jacobian matrix of V evaluated at x. The rows of J x are indexed by the polynomials in the sets M1 , . . . , Mh , X 2 X 1 , . . . , X h X hy1 , while the columns of J are indexed by the entries in X 1 , . . . , X h . The Ž M, ␣ .th entry of J x , where M is a minor in M1 j ⭈⭈⭈ j Mh and ␣ is an entry in X 1 j ⭈⭈⭈ j X h , is equal to " the minor obtained from M by deleting the row and the column of ␣ if ␣ appears in M, and 0 otherwise. Let f g X iq1 X i be the product of the r th row of X iq1 and the sth column of X i , for some 1 F i F h y 1, 1 F r F n iq2 , 1 F s F n i , and let ␣ be an entry in X 1 j ⭈⭈⭈ j X h ; then the Ž f, ␣ .th entry of J x is equal to the Ž j, s .th entry of X i if ␣ is the Ž r, j .th entry of X i for some 1 F j F n i , the Ž r, l .th entry of X iq1 if ␣ is the Ž l, s .th of X i for some 1 F l F n i , and 0 otherwise. The next lemma is obvious from the fact that the closure of the orbit through a point x s Ž A1 , . . . , A h . in V such that rank A i s k i for all 1 F i F h is V itself. We include the proof of this result to establish notations that will be used in the following, and for improved understanding of the Jacobian matrix of V. LEMMA 2.1. Let x s Ž A1 , . . . , A h . be a point in V such that rank A i s k i for all 1 F i F h. Then x is a smooth point of V.

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Proof. Without loss of generality, we may assume that 0 Ai s M i

ž

0 0 ,

/

for 1 F i F h,

where Mi is the k i = k i identity matrix. Let us denote J x by just J. Next we describe all the nonzero rows of J. For 1 F i F h, a row of J indexed by a minor M g Mi of X i is nonzero if and only if M contains the identity block Mi . The only nonzero element in such a row is equal to "1, placed in the column indexed by the upper right corner entry of M. When computing the rank of J, we may assume that such a "1 entry of J x is actually equal to 1. Let Ri be the set of all minors in Mi containing Mi . Let Yi be the set of all right upper corners of minors in Ri , i.e., the set of all entries of X i not contained in any of the rows or columns of Mi . Note that Yi consists of the entries of the Ž n iq1 y k i . = Ž n i y k i . right upper block of X i , and hence its cardinality is Ž n iq1 y k i .Ž n i y k i .. Also, < Ri < s Ž n iq1 y k i .Ž n i y k i .. Now let f be an entry in X iq1 X i for some 1 F i F h y 1. In Figure 1, the blocks X i , X iq1 , X iq1 X i are displayed as shown below: Xi X iq1 X i X iq1

Let f be the product of the r th row of X iq1 and the sth column of X i , 1 F r F n iq2 , 1 F s F n i . Then the row of J indexed by f is nonzero if and only if either the r th row of X iq1 is nonzero Ži.e., it is one of the last k iq1 rows of X iq1 . or the sth column of X i is nonzero Ži.e., it is one of the first k i columns of X i .. The set of all such f ’s in X iq1 X i can be written as Bi, iq1 j L i, iq1 , where Bi, iq1 denotes the set of all entries in the last k iq1 rows of X iq1 X i , and L i, iq1 denotes the set of all entries in the first k i columns of X iq1 X i . Now we find a maximal set of linearly independent rows of J among the set of rows described above. The rows of J indexed by minors in R1 , . . . , Rh are linearly independent, since their nonzero entries are in different columns Žthese columns are indexed by the entries in Y1 , . . . , Yh , and each of them contains precisely one nonzero entry in the rows indexed by R1 , . . . , Rh .. For 1 F i F h y 1, let K i, iq1 s Bi, iq1 l L i, iq1 Žnote that K i, iq1 is the k iq1 = k i left lower block of X iq1 X i .. A row of J indexed by f g K i, iq1 contains precisely two nonzero entries Žequal to 1., neither of which is in

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553

FIGURE 1

the same column with a nonzero entry of another nonzero row of J. The set of nonzero entries in the rows indexed by all f g K i, iq1 is in the columns indexed by the entries in the left upper block of X i of size k iq1 = k i , which we denote by Qq i, iq1, and the right lower block of X iq1 of size k iq1 = k i , which we denote by Qy i, iq1. A row of J indexed by f g Ž Bi, iq1 j L i, iq1 . _ K i,iq1 contains precisely one nonzero entry Žequal to 1. in a column indexed either by an element in the first k iq1 rows of Yi or by an element in the last k i columns of Yiq1. Therefore these rows are identical to some of the rows indexed by minors in R1 , . . . , Rh . Consequently, the rows of J indexed by minors in R1 , . . . , Rh and entries of K 1, 2 , . . . , K hy1, h give a maximal set of linearly independent rows of J. Therefore, rank J s < R1 < q ⭈⭈⭈ q< Rh < q < K 1, 2 < q ⭈⭈⭈ q< K hy1, h < s Ž n1 y k 1 . Ž n 2 y k 2 . q ⭈⭈⭈ q Ž n h y k h . Ž n hq1 y k h . q k 1 k 2 q ⭈⭈⭈ qk hy1 k h s codim V . This shows that x is a smooth point of V. Notation. For the rest of the paper, y q y J , Y1 , . . . , Yh , Qq 1, 2 , Q1, 2 , . . . , Q hy1, h, Q hy1, h

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will denote the Jacobian matrix, respectively the sets, associated, as in Lemma 2.1, to the point Ž A1 , . . . , A h . in V with 0 Ai s M i

ž

0 0 ,

/

for 1 F i F h,

where Mi is the k i = k i identity matrix. Remark 2.2. Note that the nonzero columns of J are the columns y q y indexed by the entires in Y1 , . . . , Yh , Qq 1, 2, Q 1, 2, . . . , Q hy1, h, Q hy1, h. For q 1 F j F h y 1, each column indexed by an entry in Q j, jq1 is identical to a column indexed by the corresponding entry in Qy j, jq1. A maximal set of linearly independent columns of J consists of all distinct nonzero columns " < Ž< Q j,"jq1 < of J. We have codim V s < Y1 < q ⭈⭈⭈ q< Yh < q < Q1,"2 < q ⭈⭈⭈ q< Q hy1, h q y denotes the cardinality of each of Q j, jq1 and Q j, jq1 .. A nonzero column whose index is not included in the set of indices of the columns in a maximal set ⌺ of linearly independent columns of J is indexed by an entry y in Qq j, jq1 or Q j, jq1, for some j, and it is identical to a column whose index y belongs to Q j, jq1, respectively Qq j, jq1, and is among the indices of columns in ⌺. Remark 2.3. For 1 F j F h, a maximal set of linearly independent columns of J indexed by entries in X 1 , . . . , X jy1 , X jq1 , . . . , X h consists of all such distinct nonzero columns. The cardinality of such a maximal set is codim V y < Yj <. The set of indices of the columns in such a maximal set y must include Qq jy1, j and Q j, jq1. Remark 2.4. For 1 F j F h y 1, a maximal set of linearly independent columns of J indexed by X 1 , . . . , X jy1 , X jq2 , . . . , X h consists of all such distinct nonzero. The cardinality of such a maximal set is codim V y Ž< Yj < q < Yjq1 < q < Q j,"jq1 <.. The set of indices of the columns in such a maximal y set must include Qq jy1, j and Q jq1, jq2. LEMMA 2.5. Let x s Ž A1 , . . . , A h . be a point of V such that rank A j - k j for some j, 1 F j F h. If x is smooth, then either k jy1 q k j s n j or k j q k jq1 s n jq1 Ž here k 0 s k hq1 s 0.. Proof. Let t i s rank A i , for 1 F i F h, with t j - k j . Without loss of generality, we may assume that 0 Ai s N i

ž

0 0 ,

/

for 1 F i F h,

where Ni is the t i = t i identity matrix. As noted in Remark 2.3, there are precisely codim V y < Yj < linearly independent columns of J indexed by entries in X 1 , . . . , X jy1 , X jq1 , . . . , X h . Since the nonzero columns of J x are

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555

among the nonzero columns of J, the set T of linearly independent columns of J x indexed by entries in X 1 , . . . , X jy1 , X jq1 , . . . , X h has cardinality at most codim V y < Yj < Žnote that T consists of all distinct nonzero columns of J x .. Note that a column of J indexed by an entry in X 1 , . . . , X jy1 , X jq1 , . . . , X h is either identical to some column indexed by an entry in X j or it is linearly independent of the columns indexed by entries in X j . Since x is smooth, rank J x s codim V, so there exists a set S of cardinality at least < Yj < of linearly independent columns of J x indexed by entries in X j such that S j T is a set of linearly independent columns of J x . The only nonzero columns indexed by entries in X j are actually indexed by entries in the first k jq1 rows of X j and the last k jy1 columns of X j q Žsince rank A j - k j .. The columns indexed by entries in Qy jy1, j j Q j, jq1 cannot be in S, since they are in T Žsee Remark 2.3., and S j T is a set of linearly independent columns of J x . Thus S consists of columns indexed by entries in the first k jq1 rows of Yj and the last k jy1 columns of Yj . Since the cardinality of S must be at least < Yj <, we deduce that the set of entries in the first k jq1 rows and the last k jy1 columns of Yj must actually contain the set of entries in Yj . This implies that either k jy1 q k j s n j or k j q k jq1 s n jq1. LEMMA 2.6. Let x s Ž A1 , . . . , A h . be a point of V such that rank A j - k j for some j, 1 F j F h, and rank A i s k i for 1 F i F h, i / j. Then x is smooth if and only if either k jy1 q k j s n j or k j q k jq1 s n jq1 Ž here k 0 s k hq1 s 0.. Proof. In view of Lemma 2.5, we only have to show that if k jy1 q k j s n j or k j q k jq1 s n jq1 then x is smooth. It is easily seen that the columns of J x that are indexed by the elements in the disjoint sets y q q Y1 , . . . , Yjy1 , Yjq1 , . . . , Yh , Qy 1 , 2 , . . . , Q jy2, jy1, Q jq1, jq2 , . . . , Q hy1, h, ⌫j ,

where ⌫j is the set of entries in the first k jq1 rows and the last k jy1 columns of X j , are linearly independent. Note that Yj is contained in ⌫j Žsince k jy1 q k j s n j or k j q k jq1 s n jq1 ., and the cardinality of ⌫j is < < q < precisely < Yj < q < Qy jy1, j q Q j, jq1 . Since the number of these columns is codim V, we conclude that rank J x s codim V, and hence x is smooth. Remark 2.7. An alternate proof of Lemma 2.6 can be obtained as follows. First note that if k jy1 q k j s n j , then < L jy1, j < s < Rj <, and if k j q k jq1 s n jq1 , then < Bj, jq1 < s < Rj <. Let

¡L ⌬ s~ B ¢L j

jy1 , j

if k jy1 q k j s n j ,

j, jq1

if k j q k jq1 s n jq1 ,

jy1 , j

or Bj, jq1

if k jy1 q k j s n j and k j q k jq1 s n jq1 .

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We have < ⌬ j < s < Rj <, and the rows of J indexed by elements in Rj are identical to the rows of J indexed by ⌬ j . It is easily seen that the rows of J x indexed by the codim V elements in the sets R1 , . . . , Rjy1 , ⌬ j , Rjq1 , . . . , Rh , K 1, 2 , . . . , K hy1, h are linearly independent, and hence x is smooth. LEMMA 2.8. Let x s Ž A1 , . . . , A h . be a point of V such that if rank A j - k j for some 1 F j F h, then 1. either 1 F j y 1 F h and rank A jy1 s k jy1 , or 1 F j q 1 F h and rank A jq1 s k jq1; 2. either k jy1 q k j s n j , or k j q k jq1 s n jq1 Ž here k 0 s k hq1 s 0.. Then x is a smooth point of V. Proof. Let ⌬ j be as in Remark 2.7. Then the rows of J x indexed by the codim V elements in the sets Ri , with i such that rank A i s k i , ⌬ j , with j such that rank A j - k j , K 1, 2 , . . . , K hy1, h are linearly independent. Therefore x is a smooth point of V. Remark 2.9. An alternative proof of Lemma 2.8 can be obtained as follows. For 1 F j F h with rank A j - k j , let ⌫j be as in the proof of Lemma 2.6. It is easily seen that the columns of J x that are indexed by the elements in the disjoint sets Yi , with i such that rank A i s k i , Qq i, iq1, with i such that rank A i s k i , rank A iq1 s t iq1 , ⌫j , with j such that rank A j - k j are linearly independent. Since the number of these columns is codim V, we conclude that rank J x s codim V, and hence x is smooth. LEMMA 2.10. Let x s Ž A1 , . . . , A h . be a point of V such that rank A j - k j and rank A jq1 - k jq1 for some j, 1 F j F h y 1. Then x is a singular point of V. Proof. Let rank A i s t i , with t j - k j , t jq1 - k jq1. Without loss of generality, we may assume that 0 Ai s N i

ž

0 0 ,

/

where Ni is the t i = t i identity matrix.

for 1 F i F h,

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The nonzero columns of J x are among the nonzero columns of J. Let T be the indices of a maximal set of linearly independent columns of J x indexed by entries in X 1 , . . . , X jy1 , X jq2 , . . . , X h . Note that a column of J indexed by an entry in X 1 , . . . , X jy1 , X jq2 , . . . , X h is either identical to some column indexed by an entry in X j j X jq1 or linearly independent of the columns indexed by entries in X j j X jq1. Thus there exists a set S ; X j j X jq1 such that the columns of J x indexed by elements of S j T give a maximal set of linearly independent columns of J x , and we have rank J x s < S < q < T <. As noted in Remark 2.4, there are precisely codim V y Ž< Yj < q < Yjq1 < q < Q j,"jq1 <. linearly independent columns of J indexed by entries in X 1 , . . . , X jy1 , X jq2 , . . . , X h . Therefore < T < F codim V y Ž< Yj < q < Yjq1 < q < Q j,"jq1 <.. Now we show that < S < - < Yj < q < Yjq1 < q < Q j,"jq1 <. Each column indexed by an element of S contains precisely one nonzero entry in a row indexed by an entry of X j X jy1 , X jq1 X j , or X jq2 X jq1 Žsince rank A j - k j and rank q A jq1 - k jq1 .. The columns indexed by entries in Qy jy1, j and Q jq1, jq2 are identical to columns in T Žsee Remark 2.4.. Therefore S l Qy jy1, j s ⭋, q y S l Qq s ⭋, and S ; Y j Y j Q j Q . The set of jq1, jq2 j jq1 j, jq1 j, jq1 columns indexed by entries in Qq is the same as the set of columns j, jq1 y indexed by entries in Qy j, jq1, so we may assume that S l Q j, jq1 s ⭋. Thus q we have < S < s < S l Yj < q < S l Yjq1 < q < S l Q j, jq1 <. Obviously, < S l Yj < F < Yj < and < S l Yjq1 < F < Yjq1 < Žnote that one can have equalities here.. On the < < " < other hand, we have < S l Qq j, jq1 F t j t jq1 - k j k jq1 s Q j, jq1 . Consequently, < S < - < Yj < q < Yjq1 < q < Q j,"jq1 <. Thus rank J x - codim V, which shows that x is a singular point of V. Next we describe the singular locus of V. For 1 F i F h, let Vi denote the set of all points Ž A1 , . . . , A h . in V such that rank A i - k i , i.e., Vi s V Ž k 1 , . . . , k iy1 , k i y 1, k iq1 , . . . , k h . . For 1 F j F h y 1, let Vj, jq1 denote the set of all points Ž A1 , . . . , A h . in V such that rank A j - k j and rank A jq1 - k jq1 , i.e., Vj, jq1 s V Ž k 1 , . . . , k jy1 , k j y 1, k jq1 y 1, k jq2 , . . . , k h . . Note that Vj, jq1 s Vj l Vjq1. THEOREM 2.11. The irreducible components of Sing V are the sub¨ arieties of V of the form Vi , with i g ⍀, and Vj, jq1 , with j f ⍀, j q 1 f ⍀, where ⍀ is the set of all 1 F i F h such that k iy1 q k i - n i and k i q k iq1 - n iq1.

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Proof. Note that Sing V is a union of Gn -orbit closures, i.e., subvarieties of V of the form V Žr., with r g Kn . Next we determine for which r s Ž r 1 , . . . , r h . g Kn the subvariety V Žr. is singular. To check that V Ž r 1 , . . . , r h . is singular, it is enough to check that the point x s Ž A1 , . . . , A h . with rank A i s ri is singular. If there is no i with ri - k i , by Lemma 2.1 V Žr. is not singular. If there is precisely one index i with ri - k i , by Lemma 2.6 the subvariety V Žr. s Vi is singular if and only if i g ⍀. Next, we analyze the subvarieties V Žr. with ri - k i for more than one index i. If there is some j with r j - k j and r jq1 - k jq1 , then V Žr. ; Vj, jq1. By Lemma 2.10 Vj, jq1 is singular, and therefore V Žr. is singular. If for each j we have either r j s k j or r jq1 s k jq1 , then by Lemma 2.8 V Žr. is singular only if i g ⍀ for some i with ri - k i . But in this case V Žr. ; Vi , and since Vi is singular, V Žr. is also singular. Thus we have determined all the subvarieties of V of the form V Žr. with r g Kn , r F k, that are contained in Sing V. By taking the maximal such subvarieties Žwith respect to set inclusion., we obtain the required description for the irreducible components of Sing V. As a consequence, we obtain the following result that describes the singular loci of the irreducible components of the variety of complexes: THEOREM 2.12. If V is an irreducible component of the ¨ ariety of complexes C , then the irreducible components of Sing V are V1, 2 , . . . , Vhy1, h . Proof. In this case the set ⍀ is empty Žby Theorem 1.2., and the result follows immediately from Theorem 2.11.

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