The results of this paper mainly concern the tangentially weakly defective join of subspace varieties

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VARIETIES

MING YANG AND WEN LIU

Abstract. Using Terracini’s Lemma, we study the tangential weak defectivity of join of subspace varieties.

1. Introduction

In [2], L. Chiantini and G. Ottaviani introduced the tangential weak defectivity of join of projective varieties, which was related to uniqueness conditions concerning tensor decompositions (see [1, 3, 4, 10]). In [12], the authors studied the non-tangentially weakly defective join of subspace varieties and used it to obtain some uniqueness results of block term tensor decompositions. The results of this paper mainly concern the tangentially weakly defective join of subspace varieties.

Throughout this paper, for basic definitions, notation and results, we follow [8].

Firstly we recall some basic concepts in algebraic geometry.

1.1. Notations. As in [8], for a finite dimensional complex vector space V,PV denotes the projective space associated to V, π denotes the projection of V\{0} onto PV; for a variety X ⊂PV, ˆX ⊂V denotes its inverse image under the projectionπ, which is the (affine) cone over X in V, and for x∈X, [x] denotes π(x).

Let S be a subset of PV, then the span hSi is by definition the range of π on the usual vector span of ˆS in V. The Zariski closure of S in PV will be denoted by ¯S.

When we need to specify the elements of S and its linear span, we use the notation {s₁, s₂,· · · } and hs₁, s₂,· · · i, respectively.

Forx∈X, ˆˆ T_[x]X := ˆT_xXˆ is the affine tangent space to X at [x].

1.2. Join of subspace varieties. LetA_j, 1≤j ≤n, be finite dimensional complex vector spaces.

Definition 1.1. (See Definition 1 in [9]) Let k_j ≤a_j := dimA_j, 1≤ j ≤ n be nonnegative integers. Subspace varieties, denotedSub_k₁_,...,k_n(A₁⊗. . .⊗A_n)∈P(A₁⊗. . .⊗A_n) are defined as

Sub_k₁_,...,k_n(A₁⊗. . .⊗A_n)

:={[T]∈P(A₁⊗. . .⊗A_n)| ∀j ∃A⁰_j ⊂A_j, dim A⁰_j =k_j, T ∈A⁰₁⊗. . .⊗A⁰_n}.

Definition 1.2. IfXi, i= 1, . . . , k,k ≤nare projective algebraic varieties ofPⁿ =PV, V = Cⁿ⁺¹, then the join of X₁, . . . , X_k is

J(X₁, . . . , X_k) :=[

{h[P₁], . . . ,[P_k]i|P_i ∈Xˆ_i, 1≤i≤k},

1991Mathematics Subject Classification. classical algebraic geometry.

Key words and phrases. subspace variety, joins of varieties, tangential weak defectivity.

1

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where P_i, i = 1, . . . , k, are linearly independent vectors in V. If X₁ = · · · =X_k = X, then we write J(X₁, ..., X_k) = σ_k(X) and we call this the k-th secant variety to X.

The following Terracini’s Lemma appears as Proposition 12.11 in [6] and we rephrased it in terms of join of projective varieties.

Theorem 1.3. Let Pi ∈ Xˆi be a general point of Xˆi for each i = 1, . . . , k, then for [P] :=

[P₁+. . .+P_k],

Tˆ_[P_]J(X₁, . . . , X_k) = ˆT_[P₁_]X₁+· · ·+ ˆT_[P_k_]X_k. (1.1)

Definition 1.4. (see Definition 2.6 in [2]) Let P_i ∈ Xˆ_i be a general point of ˆX_i, 1 ≤ i ≤ k. When for any j ∈ {1, . . . , k} and Q_j ∈ Xˆ_j, ˆT_[P₁_]X₁ +· · · + ˆT_[P_k_]X_k contains ˆT_[Q_j_]X_j only if [Q_j] ∈ {[P₁], . . . ,[P_k]}, we say J(X₁, . . . , X_k) is not tangentially weakly defective.

Otherwise, we say thatJ(X₁, . . . , X_k) istangentially weakly defective. J(X₁, . . . , X_k) isweakly defective if the general hyperplane which is tangent to X₁, . . . , X_k at some k general points [P₁], . . . ,[P_k], is also tangent at some other point [Q_j] 6= [P₁], . . . ,[P_k], Q_j ∈ Xˆ_j for some j ∈ {1, . . . , k}. Here general means in an open subset of the set of hyperplanes which are tangent toX₁, . . . , X_k at k general points [P₁],· · · ,[P_k] (see [5]).

Remark 1.5. By semicontinuity (see Theorem III.12.8 of [7]), if for one particular set of general points{P₁, . . . , P_k}, ˆT_[P₁_]X₁+· · ·+ ˆT_[P_k_]X_kcontains ˆT_[Q_j_]X_j only if [Q_j]∈ {[P₁], . . . ,[P_k]}, then J(X₁, . . . , X_k) is not tangentially weakly defective.

Remark 1.6. In [12], it is proved thatJ(Sub_1,L₁_,L₁(C^I⊗C^J⊗C^K), . . . ,Sub_1,L_R_,L_R(C^I⊗C^J⊗ C^K)) is not tangentially weakly defective, if

I ≥R, J, K ≥

R

X

r=1

L_r, L_i+L_j > L_k ∀1≤i, j, k ≤R.

The main results in this paper are the following.

Theorem 1.7. AssumeI ≥R, J(Sub_1,L₁_,L₁(C^I⊗C^J⊗C^K), . . . ,Sub_1,L_R_,L_R(C^I⊗C^J⊗C^K))is tangentially weakly defective, if for some[Xj1]∈σLj1(P^J−1×P^K−1), · · · , [Xjs]∈σL_js(P^J−1× P^K−1), j₁, . . . , j_s ∈ {1, . . . , R}, there exists [X_j⁰_t] ∈ h[X_j₁], · · · , [X_j_s]i ∩σ_L_jt(P^J−1×P^K−1), but [X_j⁰_t] is not in {[X_j₁], · · · , [X_j_s]}.

And we use the above theorem to obtain the following tangential weak defectivity results:

J(Sub_1,L₁_,L₁(C^I ⊗C^J ⊗ C^K),Sub_1,L₂_,L₂(C^I ⊗C^J ⊗C^K)) is tangentially weakly defective if I ≥ 2, J = K = ^2Lⁱ₂^+L^j, ∀1 ≤ i, j ≤ 2 (^2Lⁱ₂^+L^j is not independent from i, j) or I ≥ 2, min{J, K}= max{L₁, L₂}.

Theorem 1.8. J Sub_1,L₁_,L₁ C^I⊗C^J⊗C^K

, . . . , Sub_1,L_R_,L_R C^I⊗C^J ⊗C^K

is weakly defective if I ≥2, J, K ≥L₁+· · ·+L_R, L₁, . . . , L_R≥2.

Theorem 1.9. J Sub_2,L₁_,L₁ C^I⊗C^J ⊗C^K

, . . . , Sub_2,L_R_,L_R C^I ⊗C^J ⊗C^K

is tangentially weakly defective if I ≥2, J, K ≥L₁+· · ·+L_R, L₁, . . . , L_R≥2.

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2. Proof of Theorem 1.7 And Its Corollaries

Proof. Without loss of generality, we assume that X_j⁰_t =X₁+χ₂X₂+· · ·+χ_RX_R. Now a₁⊗X₁+· · ·+a_R⊗X_R

=a₁ ⊗X_j⁰_t −χ₂a₁⊗X₂− · · · −χ_Ra₁⊗X_R+a₂⊗X₂+· · ·+a_R⊗X_R

=a₁ ⊗X_j⁰_t + (a₂−χ₂a₁)⊗X₂+· · ·+ (a_R−χ_Ra₁)⊗X_R

=a₁ ⊗X_j⁰_t +a⁰₂⊗X₂+· · ·+a⁰_R⊗X_R.

LetP_i =a_i⊗X_i, 1≤i≤R and Q=a₁⊗X_j⁰_t. Using Terracini’s Lemma 1.3, we have Tˆ_[P₁+···+P_R]J(Sub_1,L₁_,L₁(C^I⊗C^J ⊗C^K), . . . ,Sub_1,L_R_,L_R(C^I⊗C^J⊗C^K))

= ˆT_[P₁_]Sub_1,L₁_,L₁(C^I⊗C^J⊗C^K) +· · ·+ ˆT_[P_R_]Sub_1,L_R_,L_R(C^I⊗C^J⊗C^K)

contains ˆT_[Q]Sub1,L_jt,L_jt(CÎ ⊗C^J ⊗C^K). By Remark 1.5, we have J(Sub1,L1,L1(CÎ ⊗C^J ⊗ C^K), . . . ,Sub_1,L_R_,L_R(CÎ⊗C^J ⊗C^K)) is tangentially weakly defective.

Corollary 2.1. J(Sub_1,L₁_,L₁(C^I⊗C^J⊗C^K),Sub_1,L₂_,L₂(C^I⊗C^J⊗C^K))is tangentially weakly defective if I ≥2, J =K = ^2Lⁱ₂^+L^j, ∀1≤i, j ≤2.

Proof. It is sufficient to prove the case L₁, L₂ < J =K <P2

r=1L_r. Let A, B and C denote complex vector spaces of dimensions I, J, K respectively. Split B = B₁ ⊕ B₀ ⊕B₂ and C =C₁⊕C₀ ⊕C₂, where B₁, B₀, B₂,C₁, C₀, and C₂ are of dimensions L₁ −l_b, l_b,L₂ −l_b, L₁−l_c, l_c,L₂−l_c, respectively, and l_b =l_c= ¹₂L_j.

Recall that there is a normal form for a general point [p] of σ_L(PB×PC) (L is smaller than dimB and dimC), which is of the form

p=b1⊗c1+· · ·+bL⊗cL.

We may assume that all the b_i,1 ≤ i ≤ L are linearly independent in B as well as all the c_i,1 ≤ i ≤ L (Otherwise one would have [p] ∈ σL−1(PB ×PC)). Then a general element [ϕ]∈Sub_1,L,L(PA⊗PB×PC) is of the form

ϕ=a⊗(b₁⊗c₁+· · ·+b_L⊗c_L), where a is a nonzero vector in A.

So we my consider

X₁ =b_1,1⊗c_1,1+· · ·+b_1,L₁−l_b⊗c_1,L₁−l_b +b_0,1 ⊗c_0,1+· · ·+b_0,l_b ⊗c_0,l_c

∈(B₁⊕B₀)⊗(C₁⊕C₀)∼=C^L¹ ⊗C^L¹ and

X₂ =b_2,1⊗c_2,1+· · ·+b_2,L₂_−l_c⊗c_2,L₂_−l_c +b_0,1⊗c_0,1+· · ·+b_0,l_b ⊗c_0,l_c

∈(B₂⊕B₀)⊗(C₂⊕C₀)∼=C^L² ⊗C^L². LetX_j⁰ be a general point of σLj(P^J−1×P^K−1) and set

X_j⁰ =X1−X2,

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then we have

X_j⁰ =b_1,1⊗c_1,1+· · ·+b_1,L₁−l_b ⊗c_1,L₁−l_b−b_2,1⊗c_2,1− · · · −b_2,L₂−lc ⊗c_2,L₂−lc

has rank equal to L_j, which implies that [X_j⁰] is a point in σ_L_j(P^J−1 ×P^K−1). But [X_j⁰] is not in{[X₁],[X₂]}. From Theorem 1.7, we knowJ(Sub_1,L₁_,L₁(C^I⊗C^J⊗C^K),Sub_1,L₂_,L₂(C^I⊗

C^J ⊗C^K)) is tangentially weakly defective.

Corollary 2.2. J(Sub1,L1,L1(C^I⊗C^J⊗C^K),Sub1,L2,L2(C^I⊗C^J⊗C^K))is tangentially weakly- defective if I ≥2, min{J, K}= max{L1, L2}.

Proof. It is sufficient to prove the case L1 ≤L2 =K. Let B and C denote vector spaces of dimensions J, K respectively. Split B =B1+B2 and C =C1 ⊕C2, where B1, B2, C1, and C₂ are of dimensions L₁,L₂,L₁, L₂−L₁, respectively.

Consider

X₁ =b_1,1⊗c_1,1+· · ·+b_1,L₁ ⊗c_1,L₁ ∈B₁⊗C₁ ∼=C^L¹ ⊗C^L¹ and

X₂ =b_2,1 ⊗c_1,1+· · ·+b_2,L₁ ⊗c_1,L₁ +b_2,L₁₊₁⊗c_1,L₁₊₁+· · ·+b_2,L₂ ⊗c_2,L₂

∈B2⊗(C1⊕C2)∼=C^L² ⊗C^L²,

where {b_1,1, . . . , b_1,L₁}, {b_2,1, . . . , b_2,L₂},{c_1,1, . . . , c_1,L₁}, and {c_2,L₁₊₁, . . . , c_2,L₂} are bases for B₁, B₂,C₁ and C₂, respectively.

Let [X_j⁰] be a general point of σ_L_j(P^J−1×P^K−1) and set X_j⁰ =X₁+X₂. Then we have

X_j⁰ = (b_1,1+b_2,1)⊗c_1,1+· · ·+ (b_1,L₁ +b_2,L₁)⊗c_1,L₁ +b_2,L₁₊₁⊗c_2,L₁₊₁+· · ·+b_2,L₂ ⊗c_2,L₂. X_j⁰ has rank equal to L2, which implies that [X_j⁰] is a point in σL2(P^J−1×P^K−1). But [X_j⁰] is not in{[X1],[X2]}. From Theorem 1.7, we knowJ(Sub1,L1,L1(C^I⊗C^J⊗C^K),Sub1,L2,L2(C^I⊗

C^J ⊗C^K)) is tangentially weakly defective.

3. Proof of Theorem 1.8

Lemma 3.1. Let A, B, C be complex vector spaces of dimensions I, J, K and ϕ = a_i ⊗ (b₁⊗c₁+· · ·+b_L⊗c_L), where a_i, b_i, c_i are basis for A, B, C, respectively. We have

Tb_ϕSub_1,L,L(A⊗B⊗C) =A⊗

* _L X

i=1

b_i⊗c_i +

+B⊗ ha_i⊗c_ii+C⊗ ha_i⊗b_ii Proof. Pick a curveϕ(t) = a_i(t)⊗

PL

i=1b_i(t)⊗c_i(t)

, whereϕ(0) = ϕ. Taking derivative with respect to t, we have

ϕ⁰(0) =a⁰_i(0)⊗

L

X

i=1

b_i⊗c_i

!

+a_i⊗

L

X

i=1

b⁰_i(0)⊗c_i

!

+a_i⊗

L

X

i=1

b_i⊗c⁰_i(0)

! .

Since a⁰(0), b⁰(0), c⁰(0) are arbitrary vectors in A, B, C, we obtained Lemma 3.1.

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Proof of Theorem 1.8. It is sufficient to prove the caseI = 2, J =K =PR r=1L_r.

Let A, B and C be complex vector spaces of dimensions I, J, K, respectively. Split B =L

1≤q≤RB_q and C =L

1≤r≤RC_r, where for 1≤q, r ≤R, B_q and C_r are of dimensions L_q, L_r, respectively.

Choose a general set {ϕ_p ∈ Subd_1,L_p_,L_p(C^I ⊗C^J ⊗C^K) : 1 ≤ p ≤ R}. Without loss of generality, we can assume

ϕp = (a1+λ^pa2)⊗(bp,1⊗cp,1+bp,2⊗cp,2+· · ·+bp,Lp⊗cp,Lp)∈Ap⊗Bp⊗Cp, for any 1≤p≤R, where {a₁+λ^pa₂},{b_p,1, . . . , b_p,L_p}and {c_p,1, . . . , c_p,L_p} are bases for A_p, B_p, C_p, respectively. Note that for a general set {ϕ_p ∈A_p⊗B_p⊗C_p : 1≤p≤R}.

Using Terracini’s Lemma 1.3 and 3.1, a general hyperplane tangent to J Sub_1,L₁_,L₁ C^I⊗C^J ⊗C^K

, . . . , Sub_1,L_R_,L_R C^I⊗C^J⊗C^K at [ϕ1+· · ·+ϕR] is of the form

H = X

1≤p≤R

(λ_pa^∗₁−a^∗₂)⊗



 X

1≤j6=k≤L_p

µ_p,j,kb^∗_p,j⊗c^∗_p,k+µ⁰_p,j,k b^∗_p,j⊗c^∗_p,j−b^∗_p,k⊗c^∗_p,k





It is straightforward to see that H is tangent to Sub1,Lp,Lp C^I⊗C^J ⊗C^K at ς_p = (a₁+λ_pa₂)⊗X

j6=k

j −k

|j −k|µ_p,j,kb_p,j ⊗c_p,k+µ⁰_p,j,k(b_p,j⊗c_p,j+b_p,k⊗c_p,k), which is different from ϕ₁,· · · , ϕ_R. So

J Sub_1,L₁_,L₁ C^I⊗C^J ⊗C^K

, . . . , Sub_1,L_R_,L_R C^I⊗C^J⊗C^K

is weakly defective.

4. Proof of Theorem 1.9 Proof. It is sufficient to prove the caseI = 2, J =K =PR

r=1L_r.

Let A, B and C be complex vector spaces of dimensions I, J, K, respectively. Split B =L

1≤q≤RB_q and C =L

1≤r≤RC_r, where for 1≤q, r ≤R, B_q and C_r are of dimensions Lq, Lr, respectively.

Choose a general set {ϕ_p ∈ Subd_2,L_p_,L_p(C^I⊗C^J ⊗C^K) : 1 ≤ p ≤ R}. Using Weierstrass normal forms (see Chapter 10 in [8] and Chapter IX in [11]) of tensors of multilinear rank (2, L_p, L_p), we can assume

ϕ₁ =

L1

X

i=1

(a₁+λ_ia₂)⊗b_i ⊗c_i,

ϕ₂ =

L1+L2

X

i=L1+1

(a₁+λ_ia₂)⊗b_i⊗c_i, ...

ϕ_R=

L1+···+L_R

X

i=L1+···+LR−1+1

(a₁ +λ_ia₂)⊗b_i⊗c_i,

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where {a₁, a₂},{b₁, . . . , b_L₁_+···+L_R},{c₁, . . . , c_L₁_+···+L_R} are bases for A, B, C.

Consider

ψ₁ =

L1−1

X

i=1

(a₁+λ_ia₂)⊗b_i⊗c_i+ (a₁+λ_L₁₊₁a₂)⊗b_L₁₊₁⊗c_L₁₊₁

ψ₂ =

L1+L2

X

i=L1+2

(a₁+λ_ia₂)⊗b_i⊗c_i+ (a₁+λ_L₁a₂)⊗b_L₁ ⊗c_L₁.

It is obviously thatϕ₁+· · ·+ϕ_R=ψ₁+ψ₂+ϕ₃· · ·+ϕ_R. So from Terracini’s Lemma 1.3, we knowJ Sub_2,L₁_,L₁ C^I⊗C^J ⊗C^K

, . . . , Sub_2,L_R_,L_R C^I⊗C^J ⊗C^K

is tangentially weakly

defective.

5. Acknowledgement

I am very grateful to Prof. Landsberg for many inspiring discussions, and unfailingly useful suggestions. I am also very grateful to Prof. Ottaviani to teach us the concept of tangential weak defectivity.

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Department of Computer Science/Mathematics, Southern Illinois University Carbon- dale, Carbondale, IL 62901, USA

E-mail address: yangmingmath@gmail.com

(Wen Liu)Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

E-mail address: wenliu79@tamu.edu