A TRIPLE INTERSECTION THEOREM FOR THE VARIETIES SO(n)/P_{d}

Sinan Sertöz Bilkent University, Ankara Appeared in: Fundamenta Mathematicae 142 (1993), 201–220

Abstract We study the Schubert calculus on the space of d-dimensional linear subspaces of a smooth n-dimensional quadric lying in the projective space. Following Hodge and Pedoe we develop the intersection theory of this space in a purely combinatorial manner. We prove in particular that if a triple intersection of Schubert cells on this space is nonempty then a certain combinatorial relation holds among the Schubert symbols involved, similar to the classical one. We also show when these necessary conditions are also sufficient to obtain a nontrivial intersection. Several examples are calculated to illustrate the main results.

INTRODUCTION

The aim of the present paper is to establish Schubert calculus on a certain class of homogeneous spaces. To be more precise, let Q_{n} be a non-singular quadric hypersurface in ℙ^{n+1} and let G(d, Q_{n}) be the set of d-dimensional linear subspaces which lie on Q_{n}. The orthogonal group O(n + 2) acts transitively on G(d, Q_{n}) in a natural way so that G(d, Q_{n})≃O(n + 2)/P_{d+1}, where P_{d+1} is the stabilizer of an arbitrary element in G(d, Q_{n}). If d < [\frac{n}{2}], then SO(n+2), the special orthogonal group, operates on G(d, Q_{n}) transitively, and hence G(d, Q_{n})≃SO(n + 2)/SP_{d+1}, where SP_{d+1} = SO(n + 2) ∩ P_{d+1}. These spaces G(d, Q_{n}) are the objects we study in this paper.

These spaces, which are also described as A_{s}^{(m)}, the space of normalized complex s-substructures of ℝ^{m}, were studied by Dibağ, [3], where they appeared as fibers in certain global obstruction problems. He defined some Schubert cells on them which form bases of the cohomology rings of the space in question, and found that these Schubert cells have beautiful duality properties. This discovery was our motivation to establish Schubert symbolism on G(d, Q_{n}).

G(d, Q_{n}) is, by definition, a subvariety of G(d, ℙ^{n+1}), the Grassmann variety of ddimensional linear subspaces in a complex projective space of dimension n + 1. Our method here is to follow and generalize the classical treatment of Hodge and Pedoe in [7], where they develop the intersection theory on Grassmannians in a purely combinatorial manner. Thus in this paper we prove that if triple intersection of Schubert cells on G(d, Q_{n}) is non-empty, then there follows a combinatorial relation, similar to the classical one [7].

In the classical case, the combinatorial relation mentioned above implies the nonempty triple intersection, which amounts to the Pieri formula and the Giambelli formula. In our case, this does not hold in general because of the strange behaviour of the linear subspaces of a quadric. Conditions for this to hold are also discussed here.

Geometrically speaking we are going to study the Schubert calculus on the space of d-dimensional linear subspaces of a smooth quadric Q_{n} lying in the projective space ℙ^{n+1}. This variety is denoted by G(d, Q_{n}). It is a \frac{1}{2}(d + 1)(2n − 3d) dimensional subspace of G(d, ℙ^{n+1}), the Grassmann space of d-dimensional linear subspaces of the projective space ℙ^{n+1}. The correspondence between the spaces mentioned so far is as follows:

SO(n + 2)/P_{d+1} = G(d, Q_{n}) = A_{d+1}^{(n+2)}.

Throughout the article we let n = 2m or n = 2m + 1 and d is always a positive integer less than or equal to m. In section I we define certain points of ℙ^{n+1} as the skeleton points of Q_{n}. We define a flag using these skeleton points and interpret the definition of Schubert cells of G(d, Q_{n}) with respect to this flag. In section II we quote the classical intersection theorem of Hodge and Pedoe for comparison reasons. Section III gives the proof of our intersection theorem for G(d, Q_{n}). Since the geometry of smooth quadrics vary depending on the parity of their dimension, our arguments inevitably treat these two cases separately. In section IV we give explicit examples and discuss the converse of our triple intersection theorem.

Note that Hiller and Boe in [6] treated the case n = 2m + 1 and d = m and gave a Pieri type formula. A Giambelli type formula in this case was given by Pragacz in [9]. A simple and transparent proof of the main results of [6] can be found in [11]. Finally we refer the reader to a recent survey article [10], for recent developments.

The special Schubert cycle σ_{h}, 0 < h ≤ n − d, is the set of [d]-planes intersecting a given [n − d − h]-dimensional space lying on the quadric Q_{n}. The codimension of σ_{h} is h. For other definitions needed in the statement of our main result see section 3.

Main Theorem : For any two Schubert cycles Ω_{a_{0}···a_{d}} and Ω_{b_{0}···b_{d}} of A_{d+1}^{(n+2)} there exist integers λ_{0}, …, λ_{d+3} depending only on a_{0}, …, a_{d}, b_{0}, …, b_{d} and the parity of n such that for any special Schubert cycle σ_{h}, 0 < h ≤ n − d, if

1) dim_{ℂ}Ω_{a_{0}···a_{d}}+ dim_{ℂ}Ω_{b_{0}···b_{d}}+ dim_{ℂ}σ_{h} = 2 dim_{ℂ}A_{d+1}^{(n+2)} and

2) Ω_{a_{0}···a_{d}}Ω_{b_{0}···b_{d}}σ_{h} ≠ 0 then

(3) (n − d) − \frac{1}{2}d(d + 1) + e ≤ h + ∑_{i=0}^{d+3} λ_{i} ≤ n − d,

where e(Ω_{a_{0}···a_{d}}) is defined as the cardinality of the set {(a_{i}, a_{j})| i < j and a_{i} + a_{j} < n }, and e is e(Ω_{a_{0}···a_{d}}) + e(Ω_{b_{0}···b_{d}}).

The λ_{i}’s for the n = 2m case are given in lemma 6.1 and in sections 6.2, 6.3. The λ_{i}’s for the n = 2m + 1 case are given in lemma 7.1. A partial converse to this theorem is given in the last section, see theorem 13.

We refer to conditions (2) and (3) as MT(2) and MT(3) respectively in the forthcoming discussions.

Acknowledgements I thank Prof. İ. Dibağ for suggesting the problem and supplying material, and also for his encouragement at several stages. I also thank Prof. P. Pragacz for his numerous comments and generous help.

I. FLAGS A AND B IN Q_{n} AND SCHUBERT CELLS

1. Flags A and B in n = 2m case We first fix 2m + 2 skeleton points on Q_{2m} in section 1.1 and examine in sections 1.2 and 1.3 the dimensions of certain spaces constructed from skeleton points. Flags A and B are then constructed in section 1.4. Schubert cells will be constructed in section 3. They define homology cycles independent of the flags used, and hence are independent of the skeleton points chosen; this follows from [3] and [7]. 1.1 We choose and fix 2m + 2 points p_{0}, …, p_{2m+1} in Q_{2m}, called the skeleton points of Q_{2m}, as follows;

i) Choose p_{0} in Q_{2m} arbitrarily. ii) Once p_{0}, …, p_{k−1} in Q_{2m} are chosen with k ≤ m, choose p_{k} as any point in Q_{2m} which is not in the join of p_{0}, …, p_{k−1} but in the f-orthogonal of the join. (f-orthogonal means orthogonal with respect to the form Q^{c}(z_{1}, ..., z_{n}) = z_{1}^{2} + ⋯ + z_{n}^{2}, see [3, pp 501-502] for further details). In the notation from [3] we have

p_{k} ∈ {(p_{0} ∨ ⋯ ∨ p_{k−1})^{⊥_{f}} − (p_{0} ∨ ⋯ ∨ p_{k−1})} ∩ Q_{2m},

where we have used the notation ⊥_{f} to denote orthogonality with respect to the above form, (f-orthogonality). iii) Once p_{0}, …, p_{m} ∈ Q_{2m} are chosen, the remaining points are their complex conjugates, ordered as follows;

p_{2m+1−i} = c(p_{i}), i = 0, …, m

where c(·) is the complex conjugate.

1.2 Let I be a subset of I_{m} = {0, 1, ..., m}. Define S_{I} as the intersection of Q_{2m} with the join of all skeleton points p_{i} with i in I; S_{I} = (∨_{i∈I} p_{i}) ∩ Q_{2m}. Let \overline{I} denote the set of all integers of the form 2m + 1 − i with i in I. Then we have the following lemma.

Lemma 1.2 If I and J are two nonempty, disjoint subsets of I_{m} then;

i) S_{I} is a linear subspace of Q_{2m} and dim_{ℂ}S_{I} = #I − 1, where #I is the cardinality of I. ii) S_{J∪\overline{J}} is a smooth subquadric of Q_{2m} and dim_{ℂ}S_{J∪\overline{J}} = 2#J − 2. iii) S_{I∪J∪\overline{J}} is the join of S_{I} and S_{J∪\overline{J}} in Q_{2m} and dim_{ℂ}S_{I∪J∪\overline{J}} = 2#J + #I − 2.

□

1.3 For any nonempty subset L of I_{2m+1} define S_{L} as in 1.2. To find the dimension of S_{L} we construct two disjoint subsets I(L) and J(L) of I_{m} as follows:

I(L) = { i ∈ I_{m} | either i ∈ L or 2m + 1 − i ∈ L but not both }

J(L) = { i ∈ I_{m} | i ∈ L and 2m + 1 − i ∈ L }.

The following lemma on the dimension of S_{L} can now be proved using 1.2.

Lemma 1.3 (n = 2m)

dim_{ℂ}S_{L} = \{ \begin{matrix}#L − 2 & if J(L) ≠ ∅, \\ #L − 1 & if J(L) = ∅. \end{matrix}

□

1.4 Flag A consists of a nested sequence of subvarieties

A_{0} ⊂ A_{1} ⊂ ⋯ ⊂ A_{m_{0}}, A_{m_{1}} ⊂ A_{m+1} ⊂ ⋯ ⊂ A_{2m} = Q_{2m}

of Q_{2m} such that A_{i} −A_{i−1} is an open cell of dimension i, [3, p 503]. Using the skeleton points introduced above we define a flag A where each A_{i} is defined as follows:

i) A_{i} = S_{{0,1,...,i}} for i = 0, ..., m − 1. ii) A_{m_{0}} = S_{{0,1,...,m}} and A_{m_{1}} = S_{{0,1,...,m−1,m+1}}. iii) A_{m+i} = S_{{0,1,...m+1+i}} for i = 1, ..., m.

Denote by V_{0} and V_{1} the two disjoint families of projective [m]-planes in Q_{2m}. We have arbitrarily labeled S_{{0,1,...,m}} as an element of V_{0}. Consequently S_{{0,1,...,m−1,m+1}} must belong to V_{1} regardless of m being odd or even. Together with a flag A we will consider its “dual” flag B:

B_{0} ⊂ B_{1} ⊂ ⋯ ⊂ B_{m_{0}}, B_{m_{1}} ⊂ ⋯ ⊂ B_{2m} = Q_{2m}.

For a discussion of dual flags on quadrics see [3, p 512]. Assuming m is even we define B_{i} as follows;

i) B_{i} = S_{{2m+1,2m,...,2m+1−i}} for i = 0, ..., m − 1. ii) B_{m_{0}} = S_{{2m+1,2m,...,m+2,m}} and B_{m_{1}} = S_{{2m+1,2m,...,m+1}} iii) B_{m+i} = S_{{2m+1,2m,...,m−i}} for i = 1, ..., m.

If however m is odd, then we redefine B_{m_{0}} and B_{m_{1}} as B_{m_{0}} = S_{{2m+1,...,m+1}} and B_{m_{1}} = S_{{2m+1,...,m+2,m}}.

2. Flags A and B in n = 2m + 1 case 2.1 The smooth quadric Q_{2m+1} in ℙ^{2m+2} can be realized as the intersection in ℙ^{2m+3} of Q_{2m+2} with a hyperplane H. With this in mind the geometric meaning of the skeleton points of Q_{2m+1} as defined below can be visualized as follows: construct a set of skeleton points p_{0}, ..., p_{2m+3} of Q_{2m+2} in ℙ^{2m+3} as explained in 1.1. The hyperplane H is then defined by identifying the coefficients of p_{m+1} with p_{m+2} in the join p_{0} ∨ … ∨ p_{2m+3}. The skeleton points of Q_{2m+1} are then obtained by renumbering the remaining points.

The skeleton points p_{0}, ..., p_{2m+2} of Q_{2m+1} are chosen in the following manner;

i) Choose p_{0} ∈ Q_{2m+1} arbitrarily. ii) For 0 < k < m, p_{k} is any point in Q_{2m+1} which is in the f-orthogonal of the join p_{0} ∨ … ∨ p_{k−1} but not in the join. iii) The complex conjugates of p_{0}, ..., p_{m} are also skeleton points with indices set as follows

p_{2m+2−i} = c(p_{i}), i = 0, ..., m.

iv) Choose p_{m+1} as any point in ℙ^{2m+2} which is f-orthogonal to p_{0} ∨ … ∨ p_{m} ∨ p_{m+2} ∨ … ∨ p_{2m+2}. It is easy to see that p_{m+1} is not a point of the quadric and that the points p_{0}, ..., p_{2m+2} span the whole space ℙ^{2m+2}.

2.2 Let L be a subset of I_{2m+2} = {0, ..., 2m + 2}. Define the subsets I(L) and J(L) of I_{m} as

I(L) = {i ∈ I_{m}| either i ∈ L or 2m + 2 − i ∈ L, but not both }
J(L) = {i ∈ I_{m}| i ∈ L and 2m + 2 − i ∈ L }.

Notice that neither of these sets can include m + 1. We further define a constant that depends on L;

ϵ = \{ \begin{matrix}0 & if m + 1 ∉ L, \\ 1 & if m + 1 ∈ L. \end{matrix}

We use this constant to determine the dimension of S_{L};

Lemma 2.2 (n = 2m + 1)

dim_{ℂ} S_{L} = \{ \begin{matrix}#L − 2 & if J(L) ≠ ∅, \\ #L − 1 − ϵ & if J(L) = ∅. \end{matrix}

Proof: It can be shown that S_{L} = (#I(L) − 1) + (2#J(L) − 2) + 1 + ϵ if J(L) ≠ ∅, and dim_{ℂ}S_{L} = #I(L) − 1 if J(L) = ∅. Combining these equalities with the fact that #L = #I(L) + 2#J(L) + ϵ yields the lemma. □

2.3 Flag A consists of a nested sequence

A_{0} ⊂ ⋯ ⊂ A_{2m+1} = Q_{2m+1}

where

i) A_{i} = S_{{0,...,i}} for i = 0, ..., m. ii) A_{m+i} = S_{{0,...,m+1+i}} for i = 1, ..., m + 1. In this case flag B is defined as

B_{0} ⊂ ⋯ ⊂ B_{2m+1} = Q_{2m+1}

where

i) B_{i} = S_{{2m+2,...,2m+2−i}} for i = 0, ..., m ii) B_{m+i} = S_{{2m+2,...,m+1−i}} for i = 1, ..., m + 1.

3. Schubert cells on A_{d+1}^{(n+2)} Reference for the spaces A_{s}^{(n)} and the Schubert cycles on them is [3]. Here we recall the basic definitions and results. First note that for d < [\frac{n}{2}] we can realize A_{d+1}^{(n+2)} as a (d + 1)(n − \frac{3}{2}d) dimensional subvariety of G(d,ℙ^{n+1}), the Grassmann variety of [d]-planes in ℙ^{n+1}. Any q ∈ A_{d+1}^{(2m+2)} can hence be considered as a [d]-plane, and using this interpretation we can define a sequence of subspaces in Q_{2m},

q_{0} ⊂ ⋯ ⊂ q_{m−1} ⊂ q_{m_{0}}, q_{m_{1}} ⊂ q_{m+1} ⊂ ⋯ ⊂ q_{2m}

where q_{i} = q ∩A_{i} if i = 0, 1, ..., m̂, ..., 2m and q_{m_{j}} = q ∩A_{m_{j}} for j = 0 or 1. The (closed) Schubert cell corresponding to the integers 0 ≤ a_{0} < ⋯ < a_{d} ≤ n, with a_{i} + a_{j} ≠ n for i < j, is defined as

Ω_{a_{0}···a_{d}} = {q ∈ A_{d+1}^{(2m+2)} | dim_{ℂ} q_{a_{i}} ≥ i }.

We do not lose any generality by using only those Ω_{a_{0}···a_{d}}’s for which a_{i} + a_{j} ≠ n. This only avoids duplication, see [3, p 506].

The homology cycle represented by this cell, denoted by the same notation, is independent of the skeleton points used in its definition. The dimension of the cycle depends only on the Schubert symbol used;

dim_{ℂ} Ω_{a_{0}···a_{d}} = a_{0} + ⋯ + a_{d} − d(d + 1) + e

where

e = #{(a_{i}, a_{j})| i < j and a_{i} + a_{j} < n }.

In the above notation the special Schubert cycle σ_{h} appearing in the main theorem (see Introduction) can be expressed as

Ω_{n−d−h n−d+1···n} for 0 < h ≤ n − 2d, and

Ω_{n−d−h n−d···_{d+h}···n} for n − 2d < h ≤ n − d,

where d̂ + h means that d + h is to be omitted. If n−d−h = m, then we necessarily need to distinguish between m_{0} and m_{1}, but in the triple intersection arguments we do not need this distinction for the special Schubert cycles. The Schubert cycles for the odd dimensional case, A_{d+1}^{(2m+1)}, are defined similarly using the corresponding flag defined earlier. II RECALLING SOME DEFINITIONS AND RESULTS FROM STAN-DARD INTERSECTION THEORY The results of this section are classical, see for example [4], [7], [8]. We include this section with the sole purpose of comparing the main theorem of this paper with the classical triple intersection theorem on Grassmannian manifolds. 4 Summary Let 0 = V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{n+1} = ℂ^{n+1} be a nested sequence of vector subspaces of ℂ^{n+1} where dim_{ℂ} V_{i} = i for i = 0, ..., n + 1. If we define A_{i} = ℙ(V_{i+1}), the projectivization of V_{i+1}, for i = 0, ..., n, then

A_{0} ⊂ A_{1} ⊂ ⋯ ⊂ A_{n} = ℙ^{n}

is a cellular decomposition of ℙ^{n}. The variety of projective [d]-planes in ℙ^{n} is denoted by G(d, ℙ^{n}). The Schubert variety corresponding to the integers 0 ≤ a_{0} < ⋯ < a_{d} ≤ n is defined as

Ω_{a_{0}···a_{d}}^{c} = {q ∈ G(d, ℙ^{n}) | dim_{ℂ}(q ∩ A_{a_{i}}) ≥ i, i = 0, ..., d }.

Recall that the homology cycle represented by Ω_{a_{0}···a_{d}}^{c} is independent of the flag chosen and

dim_{ℂ} Ω_{a_{0}···a_{d}}^{c} = a_{0} + ⋯ + a_{d} − \frac{1}{2}d(d + 1).

The special Schubert cycle σ_{h}^{c} is defined to be the cycle Ω_{n−d−h n−d+1 ···n}^{c} and its codimension is h. Schubert cycles give a ℤ-basis of the cohomology ring of G(d, ℙ^{n}). As for the cohomology ring structure, we have equalities of the form

Ω_{a_{0}···a_{d}}^{c}Ω_{b_{0}···b_{d}}^{c} = ∑ α(a, b, c)Ω_{c_{0}···c_{d}}^{c}

where α(a, b, c) is an integer and the summation is over all Ω_{c_{0}···c_{d}}^{c}such that

dim_{ℂ} Ω_{c_{0}···c_{d}}^{c} = dim_{ℂ} Ω_{a_{0}···a_{d}}^{c} + dim_{ℂ} Ω_{b_{0}···b_{d}}^{c} − dim_{ℂ} G(d, ℙ^{n}).

One has

α(a, b, c) = Ω_{a_{0}···a_{d}}^{c}Ω_{b_{0}···b_{d}}^{c}Ω_{n+1−c_{d} ···n+1−c_{0}}^{c}.

The triple intersection theorem for G(d,ℙ^{n}) decides on the value of α(a, b, c) when c is the Schubert symbol for the dual of a special Schubert cycle. To be precise the theorem, [7, thm III, p 333], states that given Ω_{a_{0}···a_{d}}^{c} and Ω_{b_{0}···b_{d}}^{c} there exist integers λ_{0}^{c}, …, λ_{d+1}^{c} such that for any special Schubert cycle σ_{h}^{c}, if 1) dim_{ℂ} Ω_{a_{0}···a_{d}}^{c} + dim_{ℂ} Ω_{b_{0}···b_{d}}^{c} + dim_{ℂ} σ_{h}^{c} = 2 dim_{ℂ} G(d, ℙ^{n}) and

2) Ω_{a_{0}···a_{d}}^{c}Ω_{b_{0}···b_{d}}^{c}σ_{h}^{c} = 1 then

3) h + ∑_{i=1}^{d} λ_{i}^{c} = n − d. Conversely if (1) and (3) hold, then (2) holds. Here the λ_{i}^{c}’s are defined as

λ_{i}^{c} = max{0, n − a_{d−i} − b_{i−1} − 1}, i = 1, ..., d
λ_{0}^{c} = n − a_{d}
λ_{d+1}^{c} = n − b_{0}.

III TRIPLE INTERSECTION THEOREM FOR A_{d+1}^{(n+2)}

In section 5 we give a general argument which explains the role λ_{i}’s play in deriving the main theorem (MT). The values of λ_{i}’s for the case n = 2m are determined in section 6. The corresponding statements for the n = 2m + 1 case are listed without proof in section 7. Finally in section 8 we put all this together to establish the necessary conditions for having nonzero triple intersections.

5 General arguments for the n = 2m case

We start with two cycles Ω_{a_{0}···a_{d}} and Ω_{b_{0}···b_{d}} and we assume that the Schubert conditions for the former is expressed with respect to a flag A and that of the latter is expressed with respect to the corresponding dual flag B. Our arguments are independent of the choice of skeleton points used in the construction of the flags.

The two Schubert cycles Ω_{a_{0}···a_{d}} and Ω_{b_{0}···b_{d}} are disjoint unless a_{d−i}+b_{i} ≥ n for all i = 0, ..., d, hence we assume this throughout. Any point of the intersection Ω_{a_{0}···a_{d}} ∩Ω_{b_{0}···b_{d}} represents a [d]-plane lying inside Aa_{d−i} ∨ Bb_{i−1} for all i = 1, ..., d. Clearly this plane also lies in A_{a_{d}} and B_{b_{d}}, hence in the intersection

Λ = A_{a_{d}} ∩ (A_{a_{d−1}} ∨ B_{b_{0}}) ∩ ⋯ ∩ (A_{a_{0}} ∨ B_{b_{d−1}}) ∩ B_{b_{d}} ⊂ Q_{n}.

Recall that p_{0}, ..., p_{n+1} ∈ Q_{n} denote the skeleton points described in section 1.1. Using them we define auxiliary subsets of I_{n+1} = {0, 1, ..., n + 1}; L(0) = {r ∈ I_{n+1} | p_{r} ∈ A_{a_{d}}} L(i) = {r ∈ I_{n+1} | p_{r} ∈ Aa_{d−i} ∨ Bb_{i−1}}, i = 1, ..., d L(d + 1) = {r ∈ I_{n+1} | p_{r} ∈ A_{b_{d}}}. This is one of the key steps where we translate geometry into arithmetic. Observe in particular that A_{a_{d}} = S_{L(0)}, Aa_{d−i} ∨ Bb_{i−1} = S_{L(i)} for i = 1, ..., d, and B_{b_{d}} = S_{L(d+1)}. We can thus rewrite Λ as

Λ = S_{L(0)} ∩ S_{L(1)} ∩ ⋯ ∩ S_{L(d+1)}.

Furthermore if we let

L = L(0) ∩ L(1) ∩ ⋯ ∩ L(d + 1)

then clearly

Λ = S_{L}.

It is the dimension of S_{L} that we wish to calculate. For this we proceed as follows: we first calculate the cardinality of L(0), then with the intersection of each L(i) certain points of L(0) are left out, leaving us finally with only the points of L. Thus we define λ_{i}’s as

λ_{i} = #(I_{n+1} − L(i)),
= n + 2 − #L(i), i = 0, ..., d + 1.

Note that each λ_{i}, i = 1, ..., d, counts the number of skeleton points which do not belong to the sets Aa_{d−i} ∨ Bb_{i−1}, respectively. Moreover

λ_{0} = \{ \begin{matrix}n − a_{d} & if a_{d} > m \\ n − a_{d} + 1 & if a_{d} ≤ m \end{matrix}

and

λ_{d+1} = \{ \begin{matrix}n − b_{d} & if b_{d} > m \\ n − b_{d} + 1 & if b_{d} ≤ m. \end{matrix}

Normally the sum of these λ_{i}’s should correctly count the number of points left out while forming the intersection L(0)∩⋯∩L(d+1), but due to the geometric anomalities that occur in the middle dimension of smooth quadrics, the point p_{m} in the even dimensional case can be counted twice. To correct this oversight of λ_{0}, ..., λ_{d+1} we introduce λ_{d+2}, which is −1 when a certain combination of the Schubert conditions is present and 0 otherwise. We will need one more correction factor λ_{d+3} which will decide when a jump in dimension occurs as observed in lemmas 1.3 and 2.2.

6 Calculation of λ_{i}’s for the n = 2m case We now give a lemma with a table to calculate the λ_{i}’s using the a_{i}’s and b_{i}’s.

Lemma 6.1 When n = 2m the λ_{i}’s, i = 1, ..., d, are determined as in the table below:

1 ≤ i ≤ d a_{d−i} < m b_{i−1} ≤ m  λ_{i} = n − a_{d−i} − b_{i−1}
b_{i−1} > m a_{d−i} + b_{i−1} ≥ n  λ_{i} = 0
a_{d−i} + b_{i−1} < n λ_{i} = n − a_{d−i} − b_{i−1} − 1
a_{d−i} = m_{t} b_{i−1} = m_{t} m even λ_{i} = 1
m odd  λ_{i} = 0
b_{i−1} = m_{s} m even λ_{i} = 0
m odd  λ_{i} = 1
a_{d−i} > m b_{i−1} ≥ m  λ_{i} = 0.

Here s, t ∈ {0, 1} and s ≠ t. To find the λ_{i} corresponding to the case when a_{d−i} > m and b_{i−1} ≤ m we must observe that λ_{i} is a symmetric function of a_{d−i} and b_{i−1}. (Note that λ_{0} and λ_{d+1} were calculated in section 5.)

Proof \underline{Case 1} a_{d−i} < m, b_{i−1} ≤ m. L(i) = {0, 1, ..., a_{d−i}, n + 1, n, ..., n + 1 − b_{i−1}} ∈ I_{n+1}. Assume for the time being that a_{d−i} < b_{i−1} < m. Then the skeleton points missing from S_{L(i)} have indices a_{d−i} + 1, a_{d−i} + 2, ..., n − b_{i−1}, and there are (n − b_{i−1}) − (a_{d−i} + 1) + 1 = λ_{i} of them. Hence λ_{i} = n−a_{d−i} −b_{i−1} as claimed. If b_{i−1} = m, then depending on whether Bb_{i−1} is in V_{0} or in V_{1}, the element m + 1 of L(i) will be replaced by m, or vice versa depending on the parity of m. This changes L(i) but not #L(i) and hence λ_{i} still has the same value. Finally the argument is symmetric in a_{d−i} and b_{i−1}, and the assumption that one is less than the other is redundant. \underline{Case 2} a_{d−i} < m, b_{i−1} > m. If a_{d−i} + b_{i−1} ≥ n, then L(i) = I_{n+1} and λ_{i} = 0. If however a_{d−i} + b_{i−1} < n, then L(i) = {0, 1, ..., a_{d−i}, n + 1, n, ..., n − b_{i−1}} and consequently λ_{i} = n − a_{d−i} − b_{i−1} − 1. \underline{Case 3} a_{d−i} = m_{0}, b_{i−1} = m_{0}. If m is even, then L(i) = I_{n+1} − {m + 1}, and if m is odd then L(i) = I_{n+1}. Hence λ_{i} is 1 or 0 accordingly. \underline{Case 4} a_{d−i} = m_{0}, b_{i−1} = m_{1}. Similar to case 3. \underline{Case 5} a_{d−i} > m, b_{i−1} ≥ m. In this case a_{d−i} + b_{i−1} > n so L(i) = I_{n+1} and λ_{i} is 0.

□

Lemma 6.2 (Calculation of λ_{d+2} when n is even) Assume that there exist two numbers a_{i}, b_{j} with i + j > d − 1, such that a_{i} = m_{t}, b_{j} = m_{s} where t, s ∈ {0, 1}. Then;

For even m, λ_{d+2} = \{ \begin{matrix}−1 & ifs = t, \\ 0 & ifs ≠ t. \end{matrix} For odd m, λ_{d+2} = \{ \begin{matrix}0 & ifs = t, \\ −1 & ifs ≠ t. \end{matrix}

Proof: For general indices x and z let a_{d−x} = m_{t} and b_{z−1} = m_{s} where t, s ∈ {0, 1}. If x = z then the middle dimension complications are already incorporated into the considerations leading to the calculation of λ_{x}. If however x ≠ z then a complication will arise in the intersection L(x) ∩ L(z), and we intend to correct this with λ_{d+2}. First assume x > z; then b_{x−1} > b_{z−1} = m and λ_{x} will be zero since a_{d−x} + b_{x−1} ≥ n. Similarly a_{d−z} > a_{d−x} = m and λ_{z} is also zero. In this case L(x) ∩ L(z) = I_{n+1}, and λ_{x} + λ_{z} correctly counts the number of missing skeleton points. Next assume that x < z; then a_{d−z} < a_{d−x} = m and b_{x−1} < b_{z−1} = m, which in turn gives λ_{z} = m − a_{d−z} and λ_{x} = m − b_{x−1} according to the previous lemma. Assume now that m is even. When s ≠ t the spaces A_{a_{d−x}} and B_{b_{z−1}} do not have a point in common and again λ_{x} + λ_{z} correctly counts the number of missing skeleton points from the intersection L(x) ∩ L(z). However if t = s, then the spaces A_{a_{d−x}} and B_{b_{z−1}} share a point. Without loss of generality assume that t is such that A_{a_{d−x}} ∩ B_{b_{z−1}} = p_{m+1}. This shows that the sets of skeleton points that are left out by L(x) and L(z) both contain the point p_{m+1}, i.e. λ_{x} and λ_{z} both count p_{m+1}. Hence the number of skeleton points left out by L(x) ∩ L(z) is λ_{x} + λ_{z} − 1. This correction factor is λ_{d+2}. If m is odd we argue similarly. Thus when x < z we let i = d − x and j = z − 1 to obtain the statement of the lemma.

□.

6.3 We are now in a position to calculate dim_{ℂ}S_{L} in terms of λ_{i}’s. This is where we need the correction factor λ_{d+3} which registers the shift in dimension due to lemma 1.3. First we observe that

#L = #L(0) − (λ_{1} + ⋯ + λ_{d+2})
= (n + 2 − λ_{0}) − (λ_{1} + ⋯ + λ_{d+2})
= n − (λ_{0} + ⋯ + λ_{d+2}) + 2.

On the other hand

dim_{ℂ} S_{L} = \{ \begin{matrix}#L − 2 & ifJ(L) ≠ ∅ \\ #L − 1 & ifJ(L) = ∅ \end{matrix}

Therefore define λ_{d+3} as

λ_{d+3} = \{ \begin{matrix}0 & ifJ(L) ≠ ∅ \\ −1 & ifJ(L) = ∅. \end{matrix}

Then, we finally have

dim_{ℂ} S_{L} = n − (λ_{0} + ⋯ + λ_{d+3}).

To calculate λ_{d+3} we must observe that J(L) will be empty if either {0, 1, ..., m} or {n + 1, n, ..., m + 1} is disjoint from L, i.e. if either of these sets is ignored by the intersection L(0)∩L(1)∩⋯∩L(d+1). We therefore define an algorithm which checks if this is the case.

ALGORITHM: Define the following subintervals of I_{m}:

I(0) = \{ \begin{matrix}I_{m} & if a_{d} ≤ m \\ \{j ∈ I_{m}| j < n − a_{d} \} & if a_{d} > m. \end{matrix}

I(d + 1) = \{ \begin{matrix}I_{m} & if b_{d} ≤ m \\ \{j ∈ I_{m}| j < n − b_{d} \} & if b_{d} > m. \end{matrix}

For i = 1, ..., d define I(i) as

I(i) = \{ \begin{matrix}\{j ∈ I_{m} | j > min\{a_{d−i}, b_{i−1}\}\} & if a_{d−i}, b_{i−1} ≤ m \\ \{j ∈ I_{m} | a_{d−i} < j < n − b_{i−1} \} & if a_{d−i} < m < b_{i−1} \\ \{j ∈ I_{m} | b_{i−1} < j < n − a_{d−i} \} & if a_{d−i} > m > b_{i−1}. \\ ∅ & otherwise. \end{matrix}

CONCLUSION OF THE ALGORITHM: (n = 2m)

λ_{d+3} = \{ \begin{matrix}−1 & if ∪_{i=0}^{d+1} I(i) = I_{m}, \\ 0 & otherwise. \end{matrix}

This completes the calculations of the λ_{i}’s in n = 2m case.

7 The λ_{i}’s for the n = 2m + 1 case

In this section we give without proof the corresponding statements for the case n = 2m + 1. We also remind that λ_{0} and λ_{d+1} were calculated in section 5 (regardless of the parity of n). Lemma 7.1 When n = 2m+1 the λ_{i}’s i = 1, ..., d are determined as in the table below:

1 ≤ i ≤ d a_{d−i} < m b_{i−1} ≤ m  λ_{i} = n − a_{d−i} − b_{i−1}
b_{i−1} > m a_{d−i} + b_{i−1} ≥ n  λ_{i} = 0
a_{d−i} + b_{i−1} < n λ_{i} = n − a_{d−i} − b_{i−1} − 1
a_{d−i} = m b_{i−1} = m  λ_{i} = 1
a_{d−i} > m b_{i−1} > m  λ_{i} = 0.

Once again we remind that λ_{i} is a symmetric function of a_{d−i} and b_{i−1}.

λ_{d+2} = 0 when n is odd: Recall that we need this correction factor when A_{m} and B_{m} share a point which the other λ_{i}’s fail to count. But when n is odd, then A_{m} is always disjoint from B_{m}, hence the other λ_{i}’s do their job correctly. λ_{d+3} when n is odd is calculated using the same algorithm as before except that we need the following modification. CONCLUSION OF THE ALGORITHM: (n = 2m + 1)

λ_{d+3} = \{ \begin{matrix}−1 & if ∪_{i=0}^{d+1} I(i) = I_{m}, and m + 1 ∉ L \\ 0 & otherwise. \end{matrix}

8 Completion of the proof of the main theorem We will describe the inequalities of the main theorem for the case n = 2m. The arguments for the n = 2m + 1 case follow very closely the proof given here using this time the λ_{i}’s defined for the odd dimensional case, and we leave it to the reader. 8.1 We have shown that all the [d]-spaces that are represented by points of Ω_{a_{0}···a_{d}} ∩ Ω_{b_{0}···b_{d}} lie in the n−(λ_{0} +⋯+λ_{d+3}) dimensional subvariety S_{L} of Q_{2m}. These [d]-spaces also belong to σ_{h} if they intersect a certain [n − d − h]-dimensional space in Q_{2m} which belongs to a flag used in the description of σ_{h}. Generically this intersection is empty if (n − d − h) + (n − ∑ λ_{i}) < n. i.e. for nonempty intersection we must have h + (λ_{0} + ⋯ + λ_{d+3}) ≤ n − d. This proves the second inequality of the main theorem.

8.2 We rewrite the dimension condition (1) of the main theorem and rearrange it to obtain

a_{d} + b_{d} + ∑_{i=1}^{d}(a_{d−i} + b_{i−1} + 1) − d − (d + 1)(n + \frac{1}{2}d) + e = h (∗)

where e is as given in the statement of the theorem. Recall that a_{d} = n − λ_{0} and b_{d} = n − λ_{d+1}. For a_{d−i} + b_{i−1} + 1, i = 1, ..., d, we have four cases to consider. We list these cases first and then examine them; Case 1: a_{d−i} + b_{i−1} + 1 = n − λ_{i} if either “a_{d−i} < m, b_{i−1} > m and a_{d−i} + b_{i−1} < n” or “a_{d−i} > m, b_{i−1} < m and a_{d−i} + b_{i−1} < n.” Case 2: a_{d−i} + b_{i−1} + 1 = n − λ_{i} + 1 if either “a_{d−i} < m, b_{i−1} ≤ m” or “a_{d−i} ≤ m, b_{i−1} < m.” Case 3: a_{d−i} + b_{i−1} + 1 = n − λ_{i} + 2 if a_{d−i} = b_{i−1} = m_{t}, t = 0 or 1, when m is even. When m is odd the same expression for λ_{i} holds if a_{d−i} = m_{t}, b_{i−1} = m_{s}, t,s∈ {0, 1} and s ≠ t. Case 4: a_{d−i} + b_{i−1} + 1 ≥ n − λ_{i} + 1 if a_{d−i} + b_{i−1} ≥ n. We now examine these cases. If case 1 holds for all i = 1, ..., d, then no a_{i} or b_{j} is m so λ_{d+2} = 0. Since either a_{d−i} or b_{i−1} is greater than m, then the interval I(i) does not contain the integer m, for i = 1, ..., d. Hence λ_{d+3} = 0, and

a_{d} + b_{d} + ∑_{i=1}^{d}(a_{d−i} + b_{i−1} + 1) ≥ n − λ_{0} + n − λ_{d+1} + ∑_{i=1}^{d}(n − λ_{i}) − λ_{d+2} − λ_{d+3}. (∗∗)

If case 2 holds only once, and the rest is case 1, then there is a single occurrence of m among a_{0}, ..., a_{d}, b_{0}, ..., b_{d}, and hence λ_{d+2} = 0. Assume either a_{d−k} or b_{k−1} is ≤ m. Then

a_{d−k} + b_{k−1} + 1 = n − λ_{k} − λ_{d+3},

hence (∗∗) holds. If case 3 holds, say when i = k, then

a_{d−k} + b_{k−1} + 1 = n − λ_{k} + 2 ≥ n − λ_{k} − λ_{d+2} − λ_{d+3},

hence (∗∗) holds. If case 4 holds at least once and the rest is case 1, we can have at most one occurrence of m, so λ_{d+2} = 0. If case 4 holds for i = k,

a_{d−k} + b_{k−1} + 1 ≥ n − λ_{k} + 1 ≥ n − λ_{k} − λ_{d+3}

and (∗∗) holds. In any other combination of cases from 1 to 4 the inequality (∗∗) is easily seen to hold. Substituting (∗∗) into (∗) we obtain

(n − d) − \frac{1}{2}d(d + 1) + e ≤ h + ∑_{i=0}^{d+3} λ_{i}

which completes the proof.

□

IV EXAMPLES In this section we use the notation G(d, Q_{n}) to denote the subvariety of the Grassmannian manifold consisting of the [d]-planes in the smooth quadric Q_{n}. Due to the representation theorem of Dibağ [3, p 501] we have A_{d}^{(n)} ≃ G(d − 1, Q_{n−2}). The notation for Schubert varieties is explained in section 3.

NOTE: In the following intersection-product tables Schubert cycles appearing in the intersection are given without multiplicities. e.g. in Table 1, Ω_{14} · Ω_{2_{0}4} is given as Ω_{12_{1}}, Ω_{03} and Ω_{14} · Ω_{2_{0}3} is given as Ω_{02_{1}}, meaning that Ω_{14} · Ω_{2_{0}4} = c_{1}Ω_{12_{1}} + c_{2}Ω_{03} and Ω_{14} · Ω_{2_{0}3} = c_{3}Ω_{02_{1}}, where c_{1}, c_{2} and c_{3} are nonzero integers which we omit. For example, in the products involving special Schubert varieties, the multiplicities in the examples below are 1, 2 or 4 as Pragacz (private communication) points out.

9 Cohomology ring structure of A_{2}^{(6)} ≃ G(1, Q_{4}) We give the homology intersection structure. The 0-dimensional cycle Ω_{01} and the 5dimensional cycle Ω_{34} are dual, Ω_{01}Ω_{34} = 1; and we omit them in table 1. The numbers in the rightmost column are homological dimension.

10 Cohomology ring structure of A_{3}^{(6)} ≃ G(2, Q_{4}) A_{3}^{(6)} consists of two isomorphic connected components V_{0}, V_{1}, say. The dimension of each component is 3 and planes from different components do not generically intersect, see [5, p 735]. For example Ω_{12_{0}4}Ω_{02_{0}3} = 1 but Ω_{12_{0}4}Ω_{02_{1}3} = 0. In general Ω_{a_{0}a_{1}a_{2}}Ω_{b_{0}b_{1}b_{2}} = 0 if both 2_{0} and 2_{1} appear in the set of indices {a_{0}, ..., b_{2}}. For this reason we give in table 2 the homology intersection table for one of the components only. The table for the other component is identical. All the 2’s appearing in the table are either all 2_{0}, for the component V_{0}, or all 2_{1}, for the component V_{1}, hence we omit this labeling. 11 Cohomology ring structure of A_{2}^{(7)} ≃ G(1, Q_{5}) The Hasse diagram for the Schubert cycles of A_{2}^{(7)} is given in table 4 with the dimensions given on the right hand column. Intersection products are given in table 3.

Ω_{2_{0}3}  Ω_{2_{1}3}  Ω_{14}  Ω_{2_{0}4}  Ω_{2_{1}4}
Ω_{02_{0}}  0  0  0  1  0  1
Ω_{02_{1}}  0  0  0  0  1  1
Ω_{12_{0}}  1  0  0  Ω_{02_{0}}  0  2
Ω_{12_{1}}  0  1  0  0  Ω_{02_{1}}  2
Ω_{03}  0  0  1  Ω_{02_{1}}  Ω_{02_{0}}  2
Ω_{2_{0}3}  0  Ω_{02_{0}}, Ω_{02_{1}}  Ω_{02_{1}}  Ω_{12_{1}}  Ω_{12_{1}}, Ω_{03} 3
Ω_{2_{1}3} Ω_{02_{0}}, Ω_{02_{1}}  0  Ω_{02_{0}}  Ω_{12_{0}}, Ω_{03}  Ω_{12_{0}}  3
Ω_{14}  Ω_{02_{1}}  Ω_{02_{0}}  Ω_{02_{0}}, Ω_{02_{1}} Ω_{12_{1}}, Ω_{03} Ω_{12_{0}}, Ω_{03} 3
Ω_{2_{0}4}  Ω_{12_{1}}  Ω_{12_{0}}, Ω_{03} Ω_{12_{1}}, Ω_{03}  Ω_{2_{0}3}  Ω_{14}  4
Ω_{2_{1}4} Ω_{12_{1}}, Ω_{03}  Ω_{12_{0}}  Ω_{12_{0}}, Ω_{03}  Ω_{14}  Ω_{2_{1}3}  4

Table 1: Intersection products for A_{2}^{(6)}

Ω_{012} Ω_{023} Ω_{124} Ω_{234}
Ω_{012} 0  0  0  1  0
Ω_{023} 0  0  1 Ω_{023}  1
Ω_{124} 0  1 Ω_{023} Ω_{124}  2
Ω_{234} 1 Ω_{023} Ω_{124} Ω_{234}  3

Table 2: Intersection products for A_{3}^{(6)}

Ω_{15}  Ω_{24}  Ω_{25}  Ω_{34}  Ω_{35}  dim
Ω_{02}  0  0  0  0  1  1
Ω_{03}  0  0  1  0  Ω_{02}  2
Ω_{12}  0  0  0  1  Ω_{02}  2
Ω_{04}  1  0  Ω_{02}  0  Ω_{03}  3
Ω_{13}  0  1  Ω_{02}  Ω_{02}  Ω_{03}, Ω_{12} 3
Ω_{15}  Ω_{02}  Ω_{02}  Ω_{03}, Ω_{12}  Ω_{03}  Ω_{04}, Ω_{15} 4
Ω_{24}  Ω_{02}  Ω_{02}  Ω_{03}, Ω_{12} Ω_{03}, Ω_{12} Ω_{04}, Ω_{13} 4
Ω_{25} Ω_{03}, Ω_{12} Ω_{03}, Ω_{12} Ω_{04}, Ω_{13} Ω_{04}, Ω_{13} Ω_{15}, Ω_{24} 5
Ω_{34}  Ω_{03}  Ω_{03}, Ω_{12} Ω_{04}, Ω_{13}  Ω_{13}  Ω_{24}  5
Ω_{35} Ω_{04}, Ω_{13} Ω_{04}, Ω_{13} Ω_{15}, Ω_{24}  Ω_{24}  Ω_{25}, Ω_{34} 6

Table 3: Intersection products for A_{2}^{(7)}

Ω_{45}  7
Ω_{35}  6
Ω_{25}  Ω_{34}  5
Ω_{15}  Ω_{24}  4
Ω_{04}  Ω_{13}  3
Ω_{03}  Ω_{12}  2
Ω_{02}  1
Ω_{01}  0
Table 4: Hasse diagram for A_{2}^{(7)}

12 Examples of triple intersections We verify the necessity of the condition (3) in MT by some examples.

1) Ω_{14}Ω_{14}Ω_{2_{0}4} ≠ ∅ in A_{2}^{(6)} ≃ G(1, Q_{4}) e = 0, h = 1, n = 4, d = 1, m = 2.

a_{1} = 4  λ_{0} = n − a_{1} = 0  Section 5
a_{0} = 1 b_{0} = 1 λ_{1} = n − a_{0} − b_{0} = 2 Lemma 6.1
b_{1} = 4 λ_{2} = n − b_{1} = 0  Section 5
λ_{3} = 0  Lemma 6.2
λ_{4} = 0  Algorithm 6.3

In this case MT(3) holds with 2 ≤ 3 ≤ 3, showing among other things that the upper bound of h + ∑_{i=0}^{d+3} λ_{i} cannot be improved.

2) Ω_{124}Ω_{023}Ω_{234} ≠ ∅ in one component of A_{3}^{(6)} ≃ G(2, Q_{4}). e = 1 + 2 + 0, h = 0, n = 4 d = 2, m = 2.

a_{2} = 4  λ_{0} = n − a_{2} = 0  Section 5
a_{1} = 2 b_{0} = 0 λ_{1} = n − a_{1} − b_{0} = 2 Lemma 6.1
a_{0} = 1 b_{1} = 2 λ_{2} = n − a_{0} − b_{1} = 1 Lemma 6.1
b_{2} = 3 λ_{3} = n − b_{2} = 1  Section 5
λ_{4} = −1  Lemma 6.2
λ_{5} = −1  Algorithm 6.3

Hence MT(3) is satisfied as 2 ≤ 2 ≤ 2, showing also that the lower bound cannot be improved either. Note also that if Ω_{124} and Ω_{023} are taken in different components of A_{3}^{(6)} then λ_{4} = 0 and (3) of MT is not satisfied, implying that the above intersection is zero, which also follows from the fact that [2]-planes of different families in Q_{4} do not generically intersect. Hence A_{2_{s}} and B_{2_{t}} cannot have a line in common for a generic choice of flags.

3) Ω_{34}Ω_{34}Ω_{15} = 0 in A_{2}^{(7)}≃G(1, Q_{5}). e = 0, h = 3, n = 5, d = 1, m = 2.

a_{1} = 4  λ_{0} = n − a_{1} = 1 Section 5
a_{0} = 3 b_{0} = 3 λ_{1} = 0  Lemma 7.1
b_{1} = 4 λ_{2} = n − b_{1} = 1 Section 5
λ_{3} = 0  Section 7
λ_{4} = 0  Algorithm 7

h + ∑_{i=0}^{4} λ_{i} = 5 ≰ n − d. In this example the algebra predicts that the cycles will not intersect, and indeed we can check from table 4 that (Ω_{34}Ω_{34})Ω_{15} = Ω_{13}Ω_{15} = 0.

4) We show that MT(3) is not sufficient; Ω_{12_{1}}Ω_{2_{1}4}Ω_{2_{0}4} = 0 in A_{2}^{(6)} ≃ G(1, Q_{4}). e = 1 + 0 + 0, h = 1, n = 4, d = 1, m = 2.

a_{1} = 2_{1}  λ_{0} = n − a_{1} + 1 = 3 Section 5
a_{0} = 1 b_{0} = 2_{1} λ_{1} = n − a_{0} − b_{0} = 1 Lemma 6.1
b_{1} = 4 λ_{2} = n − b_{1} = 0  Section 5
λ_{3} = −1  Lemma 6.2
λ_{4} = −1  Algorithm 6.3

In this case MT(3) is satisfied with equality holding on both sides, 3 ≤ 3 ≤ 3, hence MT(3) alone is not sufficient for MT(2).

13. Sufficiency of MT(3)

We start this section by analyzing the last example of the previous section. Using the notation of section 5, all the lines in Q_{4} which simultaneously belong to the Schubert cells Ω_{12_{1}} and Ω_{2_{1}4} lie in the space S_{L} where L = {0, 1, 3}. S_{L} is hence a [2]-plane which belongs to V_{1}. We want these lines also to belong to the Schubert cell Ω_{2_{0}4}, i.e. we want to know if there is a line in S_{L} which intersects a [2_{0}]-plane, an element of V_{0}. Since in Q_{4} elements of V_{1} do not generically intersect elements of V_{0} there is no such line in S_{L}. This explains why MT(3) alone is not sufficient for MT(2). But in this particular example there is some relief(!); using the commutativity of intersection we can write Ω_{12_{1}}Ω_{2_{1}4}Ω_{2_{0}4}=Ω_{12_{1}}Ω_{2_{0}4}Ω_{2_{1}4}, and we try our main theorem on this new order of intersection: Ω_{12_{1}}Ω_{2_{0}4}Ω_{2_{1}4}=0 in A_{2}^{(6)} ≃ G(1, Q_{4}) e = 1, h = 1, n = 4, d = 1, m = 2.

a_{1} = 2_{1}  λ_{0} = n − a_{1} + 1 = 3 Section 5
a_{0} = 1 b_{0} = 2_{0} λ_{1} = n − a_{0} − b_{0} = 1 Lemma 6.1
b_{1} = 4 λ_{2} = n − b_{1} = 0  Section 5
λ_{3} = 0  Lemma 6.2
λ_{4} = −1  Algorithm 6.3

h + ∑_{i=0}^{4} λ_{i} = 4 ≰ n − d = 3. Hence the algebra tells us that the intersection is zero.

The key questions for the sufficiency of MT(3) are the following:

i) Is S_{L} big enough to intersect a generic [n − d − h]-plane? (this is the condition imposed by σ_{h}) ii) Is S_{L} big enough to contain a [d]-plane at all? The first of these questions gives rise to the familiar necessary condition of MT(3)

h + ∑_{i=0}^{4} λ_{i} ≤ n − d. (∗)

This condition is also sufficient for an affirmative answer to (i) when n is odd, or when h + ∑_{i=0}^{4} λ_{i} ≠ m in case n = 2m. While S_{L} is sufficiently large to intersect a generic [n − d − h]-plane, it may not be large enough to contain any [d]-plane. And even it does contain some [d]-planes we may not conclude that any of these [d]-planes also satisfies the given Schubert conditions. However if dimS_{L} < m, when n = 2m, then S_{L} is an [n − ∑_{i=0}^{d+3} λ_{i}]-plane, and the inequality (∗) guarantees that S_{L} intersects a generic [n − d − h]-plane in Q_{2m}. If furthermore S_{L} is large enough to contain a [d]-plane, i.e. if dimS_{L} = n − ∑_{i=0}^{d+3} λ_{i} ≥ d, then we can conclude that Ω_{a_{0}···a_{d}}Ω_{b_{0}···b_{d}}σ_{h} ≠ ∅.

We collect these arguments in the following theorem. Assume here that Ω_{a_{0}···a_{d}}, Ω_{b_{0}···b_{d}} and σ_{h} are as given in the statement of the main theorem.

Theorem 13 The condition MT(3) is sufficient for having a nontrivial triple intersection, Ω_{a_{0}···a_{d}}Ω_{b_{0}···b_{d}}σ_{h} ≠ 0, if one of the following conditions holds:

i) λ_{d+3} = −1 and ∑_{i=0}^{d+3} λ_{i} > m, when n = 2m or ii) λ_{d+3} = −1 when n = 2m + 1. □

Note that when λ_{d+3} = −1 then J(L) = ∅ and in that case S_{L} is an [n−∑_{i=0}^{d+3} λ_{i}]-plane. In the even dimensional case we want to exclude the case when ∑_{i=0}^{d+3} λ_{i} = m since the cases m = m_{0} or m = m_{1} are different (see section 3). If, for example, λ_{d+3} = −1, n − ∑_{i=0}^{d+3} λ_{i} = m_{s} and n − d − h = m_{t}, then MT(3) is sufficient for MT(2) when

i) s = t and m is even, or ii) s ≠ t and m is odd.

When n is odd on the other hand, we do not have such middle dimension complications and λ_{d+3} = −1 is enough to assure the sufficiency of MT(3).

Now applying theorem 13 to the example 4 of section 12, we find that MT(3) holds, λ_{d+3} = −1 but ∑_{i=0}^{d+3} λ_{i} ≯ m, so as Theorem 13 above predicts, MT(2) does not hold.

It is important to observe that λ_{d+3} = −1 is not a necessary condition of MT(2). Hence if MT(3) holds but λ_{d+3} = 0, then we can conclude nothing about the triple intersection. Compare the following two examples for this purpose. In example 1 of section 12 MT(3) holds, λ_{d+3} = 0 but MT(2) also holds. In G(1, Q_{6}) on the other hand, if we consider Ω_{04}Ω_{45}Ω_{46} we see that MT(3) holds, and λ_{d+3} = 0, but this intersection is zero, i.e. MT(2) does not hold.

These two examples show us that when λ_{d+3} = 0 the inequalities of MT(3) do not necessarily imply MT(2). However when λ_{d+3} = −1 and ∑_{i=0}^{d+3} λ_{i} > m then MT(3) safely implies MT(2), as it does in the following example.

In A_{2}^{(8)} ≃ G(1, Q_{6}) consider Ω_{13_{0}}Ω_{45}Ω_{46}. e = 1, h = 1, n = 6, d = 1, m = 3.

a_{1} = 3_{0}  λ_{0} = n − a_{1} + 1 = 4  Section 5
a_{0} = 1 b_{0} = 4 λ_{1} = n − a_{0} − b_{0} − 1 = 0 Lemma 6.1
b_{1} = 5 λ_{2} = n − b_{1} = 1  Section 5
λ_{3} = 0  Lemma 6.2
λ_{4} = −1  Algorithm 6.3

Here MT(3) holds with 5 ≤ 5 ≤ 5. We also have ∑_{i=0}^{d+3} λ_{i} = 4 > 3 = m and λ_{4} = −1. Hence we conclude from these algebraic considerations that Ω_{13_{0}}Ω_{45}Ω_{46}≠ 0.

References

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[3] Dibağ,İ., Topology of the complex varieties A_{s}^{(n)}, Journal of Differential Geometry 11 (1976), 499-520.

[4] Fulton, W., Intersection Theory, Springer-Verlag 1984.

[5] Griffiths, P., & Harris, J., Principles of Algebraic Geometry, Wiley N.Y. 1978.

[6] Hiller, H., & Boe, B., Pieri formulas for SO_{2n+1}/U_{n} and Sp_{n}/U_{n}, Advances in Math., 62 (1986) 49-67.

[7] Hodge, W.V.D., & Pedoe, D., Methods of Algebraic Geometry, vol. II, Cambridge University Press (1968).

[8] Kleiman, S., and Laksov, D., Schubert Calculus, Amer. Math. Monthly, 79 (1972) 1061-1082.

[9] Pragacz, P., Algebro-Geometric Applications of Schur S- and Q- Polynomials, in Topics in Invariant Theory-Séminaire d’Algèbre Dubreil-Malliavin 1989-1990, Springer Lecture Notes in Math. no: 1478 (1991), 130-191.

[10] Pragacz, P., Geometric Applications of Symmetric Polynomials, Max-Planck Institut für Mathematik, Bonn, Preprint (1992).

[11] Pragacz, P., and Ratajski, J., Pieri for Isotropic Grassmannians; The Operator Approach, Max-Planck Institut für Mathematik, Bonn, Preprint (1992).

Sinan Sertöz Bilkent University Department of Mathematics 06533 Ankara,Turkey

e-mail: sertoz@fen.bilkent.edu.tr