Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 14 October 2014

Algebraic Geometry, Intersection Theory, and the Schubert Calculus

Enumerative Geometry

The Schubert Calculus is a branch of algebraic geometry essentially founded in 1874 by H. Schubert. Excellent accounts of this subject may be found in the articles by Kleiman and Laksov [KLa1972], R. P. Stanley [Sta1976], Kleiman [Kle1974], and Kaplansky [KapNONE]. The following italicized paragraphs are excerpted from [KLa1972] and [KapNONE].

In 1874 H. Schubert published his celebrated treatise, "Calculus of Enumerative Geometry". It dealt with finding the number of points, lines, planes, etc. satisfying certain geometric conditions, an important subject 100 years ago. Here is a typical enumerative problem: How many lines in 3-space, in general, intersect four given lines? Schubert would specialize the given four lines so as to make the first intersect the second and the third intersect the fourth. In this special case there are obviously two lines intersecting the four: the line joining the two points of intersection and the line of intersection of the two planes – one determined by the first two lines and the other by the second two. Now Schubert's "principle of conservation of number" states that there must be two solutions in the general case as well.

Very crudely put, Schubert's principle asserts that the number of solutions to an enumerative problem is invariant under "continuous change", provided multiplicities are counted suitably. As was often the case in early algebraic geometry, the methods of enumerative geometry were intuitive and rested on a weak foundation. However, the beauty of the subject inspired many mathematicians to develop rigorously the foundational material, such as topological and algebraic intersection theories.

Spurring the foundational work was the challenge of Hilbert's fifteenth problem: "... To establish rigorously and with exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position or conservation of number, by means of the enumerative calculus developed by him."

Today's interpretation of Hilbert's 15th problem is this: one is to give a suitable definition of intersection-multiplicities for algebraic varieties, and prove the expected properties.

We shall now motivate the discussion of multiplicities by taking up the first intersecting case: Bezout's theorem in the plane. Consider two curves in the plane. In the first instance, we mean by this the affine plane, so each is given by a polynomial in $x$ and $y\text{.}$ There is little lost in assuming the polynomials $f,g$ to be irreducible, and we do so (this makes the curves irreducible in the same sense that they are not expressible as finite unions of proper subcurves). If $f=cg$ where $c$ is a constant, then the two curves are the same, and we rule this out too. Now we are ready to ask: how many points of intersection are there?

Let $f$ have degree $m$ and $g$ degree $n\text{.}$ Mathematicians' intuition for a long time told them that the answer ought to be the product $mn\text{.}$ But there are three precautions that must be observed before this has a chance to be true. We illustrate each point by elementary example.

(1)	We must count intersections at infinity; in other words, the affine plane must be enlarged to the projective plane. Two parallel lines illustrate this.
(2)	The field must be algebraically closed. A non-intersecting line and circle illustrate this; the two points of intersection are imaginary. $$\begin{array}{c} \end{array}$$
(3)	We must, when appropriate, count multiple intersections. In the illustrated tangent to a circle, the point of tangency must be counted twice if we are to get the desired number of intersections (there are no further intersections at infinity or with complex coordinates). $$\begin{array}{c} \end{array}$$

Two touching circles illustrate all three difficulties at once. The circles are entitled to four points of intersection; the point of tangency must be counted twice, and in addition the circles share the circular points at infinity (which are imaginary and at infinity). $$\begin{array}{c} \end{array}$$ The first two of our three precautions need no further discussion; it is only multiplicity which needs attention.

When two lines or conics intersect, it is apparent at a glance what the multiplicity ought to be. In dealing with curves possessing singularities, or meeting with a high order of contact, the matter is not so simple, but for curves in the plane there is still no real difficulty. It is for algebraic varieties of higher dimension that a problem of considerable delicacy arises.

The first rigorous definition of intersection multiplicity was given by Lefschetz in 1924 using topological methods (relevant when the underlying field is the complex numbers). The purely algebraic theory was developed intuitively by the Italian schoool of C. Segre, Severi, Castelnuevo, Enriques, et. al., and given a rigorous footing by van der Waerden, Chow, Hodge, Weil, and many others (see [Kle1974]). In this section we offer an exposition of the algebraic intersection theory given in Methods of Algebraic Geometry ([HPe1953]), by Hodge and Pedoe. This text was chosen because it contains a thorough development of intersection theory, and a full discussion of the Schubert Calculus, and is accessible to the (tenacious) non-specialist.

Affine and Projective Space

By affine $n\text{-space}$ we mean the space of $n\text{-tuples}$ $(a_1,a_2,\dots ,a_n)$ over a field $K\text{.}$ We denote this space by $A^n\left(K\right)\text{;}$ by convention, $A^n=A^n\left(ℂ\right)\text{.}$ By projective $n\text{-space}$ we mean the space of non-zero $(n+1)\text{-tuples}$ $(a_0,a_1,\dots ,a_n)$ over a field $K,$ where we identify $(a_0,a_1,\dots ,a_n)$ and $(a_0^{\prime },a_1^{\prime },\dots ,a_n^{\prime })$ if there exists a scalar $c\ne 0$ such that $a_i^{\prime }=c{a}_i$ for $i=0,1,\dots ,n\text{.}$ Projective $n\text{-space}$ over $K$ is denoted by $P^n\left(K\right)\text{;}$ by convention, $P^n=P^n\left(ℂ\right)\text{.}$ Identifying a point $(a_1,a_2,\dots ,a_n)$ in $A^n$ with the point $(1,a_1,a_2,\dots ,a_n)$ in $P^n,$ we may think of $P^n$ as $A^n$ completed by the points $(0,a_1,\dots ,a_n)$ "at infinity". Algebraic geometry in $A^n$ considers systems of (possibly nonhomogeneous) polynomial equations in $n$ unknowns. Algebraic geometry in $P^n$ considers systems of homogeneous polynomial equations in $n+1$ unknowns. Problems in $A^n$ may be translated to and from problems in $P^n$ by "homogenizing" or "dehomogenizing" the polynomials in question. A good description of this procedure may be found in [Ful1969].

A plane $L$ in $P^n\left(K\right)$ is defined as the set of points $(\cancel{a_0}{t}_0,\cancel{a_1}{t}_1,\dots ,\cancel{a_n}{t}_n)$ which satisfy a system of homogeneous linear equations $AX=0,$ where $A$ is an $r\text{-by-}(n+1)$ matrix over $K\text{.}$ We say the plane $L$ has projective dimension $d$ if $A$ has rank $n-d\text{;}$ in this case $L$ is called a $d\text{-plane}\text{.}$ A $d\text{-plane}$ in $P^n\left(K\right)$ corresponds to a subspace of $A^{n+1}\left(K\right)$ of affine dimension $d+1,$ which we call a $(d+1)\text{-subspace.}$

Projective Algebraic Varieties

Let $k[x_0,x_1,\dots ,x_n]$ denote the ring of polynomials in $(n+1)\text{-indeterminates}$ over a field $k$ of characteristic zero, and let $S$ be a subset of $k[x_0,x_1,\dots ,x_n]$ consisting of homogeneous polynomials. Consider the system of equations $$f(x_0,\dots ,x_n)=0,f\in S\text{.}$$ If we look for solutions only in $P^n\left(k\right),$ we may not find any, or at any rate not enough; we therefore take an algebraically closed extension $K$ of $k,$ and look for solutions in $P_n\left(K\right)\text{.}$ Following the classical theory, we take the field of rational functions over $k$ in a countably infinite number of indeterminates, and let $K$ be its algebraic closure. We then call the set $$V\left(S\right)=\{x\prime \in P_n\left(K\right)\hspace{0.17em}|\hspace{0.17em}f\left(x\prime \right)=0\hspace{0.17em}\text{for all}\hspace{0.17em}f\in S\}$$ a $(k,K)$ projective algebraic variety (abbreviated terminology: $k\text{-variety).}$ the unqualified term variety means some $k\prime \text{-variety,}$ where $k<k\prime <K\text{.}$ The field $K$ is called the "universal" field; its relationship to the ground field $k$ is such that is contains all extensions of $k$ needed in the classical theory.

Reducibility

A variety $V$ is reducible over a field $k$ if it is the union of two distinct $k\text{-varieties,}$ both distinct from $V\text{.}$ A variety may be irreducible over a field $k,$ but reducible over some extension field $k\prime \text{.}$ A variety which is irreducible over $K$ is called an absolutely irreducible variety. Any $k\text{-variety}$ may be expressed, in an essentially unique way, as a finite, non-contractible union of irreducible $k\text{-varieties}$ ([HPe1953], X, 2, Thm. III and Th. IV).

Generic Point

Given a variety $V$ and a point $\xi$ in $P^n\left(K\right),$ $\xi$ is said to be a generic point of $V$ over $k$ if:

1)	$\xi \in V,$
2)	Any form $f\in k[x_0,\dots ,x_n],$ for which $f\left(\xi \right)=0,$ vanished on $V\text{.}$

This definition is due to van der Waerden. It provides a rigorous basis for the intuitive concept of a "general" point of a variety. Any algebraic property which holds for a generic point of a variety should also hold for any particular point of the variety; this is the content of part 2) of the definition. In [HPe1953] it is shown that:

1)	a variety has a generic point over $k$ if and only if $V$ is irreducible over $k\text{;}$ and
2)	given a point $\xi$ in $P^n\left(K\right)$ and a field $k,$ there is a unique $k\text{-variety}$ $V$ having $\xi$ as a generic point.

Dimension

Let $V$ be an irreducible $k\text{-variety,}$ and let $\xi =(1,{\xi }_1,{\xi }_2,\dots ,{\xi }_n)$ be a generic point of $V$ over $k\text{.}$ The dimension of $V$ is defined as the maximum number of elements among ${\xi }_1,{\xi }_2,\dots ,{\xi }_n$ which are algebraically independent over $k\text{.}$ An irreducible variety of dimension zero consists of a finite number of points ([HPe1953], X, 4, Th. V). The dimension of a reducible variety is defined to be the maximum of the dimensions of its irreducible components. A variety whose irreducible components all have the same dimension is called unmixed or said to have pure dimension.

Cayley Form

Let $L_1,L_2$ be planes in $P^n\left(K\right)\text{.}$ If $L_1,L_2$ have dimensions $d_1,d_2,$ respectively, and $d_1+d_2\ge 0,$ then $L_1\wedge L_2$ is non-empty. This is a simple fact of linear algebra. A fact of algebraic geometry is that the same statement holds true when $L_1$ and $L_2$ are arbitrary varieties in $P^n\left(K\right)\text{.}$

Let $V$ be an irreducible $k\text{-variety}$ of dimension $d\text{;}$ the previous statement guarantees that $V$ has a non empty intersection with any $(n-d)\text{-plane.}$ For $(n-d-1)\text{-planes,}$ the case is different; some meet $V,$ some don't. It turns out that $V$ is uniquely determined by the complex of $(n-d-1)\text{-planes}$ which it meets. This geometric statement may be given a rigorous algebraic basis by means of the Cayley form of a variety.

The Cayley form of $V$ is a homogenious polynomial whose roots represent $(n-d-1)\text{-planes}$ which meet $V\text{.}$ Note that any $(n-d-1)\text{-plane}$ $L$ in $P^n\left(K\right)$ may be associated with a $(d+1)\text{-by-}(n+1)$ matrix $U$ of rank $d+1$ over $K$ such that $x\in L$ iff $Ux=0\text{.}$ Define $L\left(V\right)$ as follows: $$L\left(V\right)=\left\{U\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& U\hspace{0.17em}\text{is a}\hspace{0.17em}(d+1)\text{-by-}(n+1)\hspace{0.17em}\text{matrix over}\hspace{0.17em}K,\\ \text{2)}& \text{rank}\left(U\right)=d+1,\\ \text{3)}& Ux=0\hspace{0.17em}\text{for some}\hspace{0.17em}x\in V\end{array}\}\text{.}$$ Then $L\left(V\right)$ represents the complex of $(n-d-1)\text{-planes}$ which meet $V\text{.}$ Let $\left\{x_{ij}\hspace{0.17em}\right|\hspace{0.17em}0\le i\le d,0\le j\le n\}$ be a set of $(d+1)·(n+1)$ indeterminates, and $k\left[x_{ij}\right]$ the ring of polynomials in the $x_{ij}$ with coefficients in $k\text{.}$ It may be shown that the set $$U=\{f\in k\left[x_{ij}\right]\hspace{0.17em}|\hspace{0.17em}f\left(U\right)=0\hspace{0.17em}\text{for all}\hspace{0.17em}U\in L\left(V\right)\}$$ is a principal ideal in $k\left[x_{ij}\right]\text{.}$ The Cayley form is defined as the monic polynomial which generates $U\text{.}$ An irreducible variety has a unique Cayley form, and is uniquely determined by its Cayley form ([HPe1953], X, 7, p.41).

The Cayley form $F$ of $V$ has the following properties:

1)	$F\left(U\right)=0$ if and only if $U\in L\left(V\right)\text{;}$
2)	Let the matrix of indeterminates $X=\left(x_{ij}\right)$ be written $(X_0,X_1,\dots ,X_d),$ where $X_i=(x_{i0},x_{i1},\dots ,x_{in})\text{.}$ Then there exists a positive integer $g$ such that for any scalar $t$ and index $i$ $(0\le i\le d),$ $$F(X_0,X_1,\dots ,X_{i-1},t{X}_i,X_{i+1},\dots ,X_d)=t^g·F\left(X\right)\text{.}$$

Property 2) states that $F$ is homogeneous of the same degree $g$ in each set of variables $X_0,X_2,\dots ,X_d\text{.}$ The integer $g$ is called the order of $V\text{.}$

The idea of representing a curve in projective 3-space by the complex of lines which it meets is due to Cayley. Hodge and Pedoe cite this as the reason for their terminology. The Cayley form was first defined and extensively used by van der Waerden and Chow; it is sometimes called the Chow form. It provides a mechanism for translating classical geometrical arguments of the Italian school into algebraic form. In addition, the Cayley form may be used to "coordinatize" varieties, as discussed below.

Systems of Varieties

A geometrical argument widely used by the Italian school was that of "continuous variation" of a variety. Early justifications of this kind of argument took the ground field $k$ to be the complex numbers, which permitted topological reasoning. The purely algebraic theory developed in [HPe1953] uses an extended definition of the Cayley form, described as follows.

Let $V$ be an unmixed variety of dimension $d$ with irreducible components $V_1,V_2,\dots ,V_k$ having Cayley forms $F_1,F_2,..,F_k$ and order $g_1,g_2,\dots ,g_k,$ respectively. If $a_1,a_2,\dots ,a_k$ is any sequence of positive integers, then $$F=\prod _{i=1}^k{\left(F_i\right)}^{a_i}$$ will be called a Cayley form for $V\text{.}$ For this extended definition, we still have:

1)	$F\left(U\right)=0$ if and only if $U\in L\left(V\right),$
2)	$F(X_0,\dots ,t{X}_i,\dots ,X_d)=t^g·F(X_0,\dots ,X_d),$ where $g=\sum _{i=1}^k{a}_i{g}_i\text{.}$

We say that such a Cayley form has type $(d,n,g)\text{.}$ A Cayley form of type (d,n,g) uniquely defines a point set variety of dimension $d$ (which might be irreducible), but a point set variety may be associated with many Cayley forms of type (d,n,g) (depending on the choice of integers $a_1,a_2,\dots ,a_k\text{).}$

Cayley forms of type $(d,n,g)$ constitute a system of varieties in the following sense. Let $S_{d,n,g}$ be the subset of $K\left[x_{ij}\right]$ consisting of polynomials which are homogeneous of degree $g$ in each set of indeterminates $X_0,X_1,\dots ,X_d$ (i.e. property 2) above). $S_{d,n,g}$ is in fact a vector space; let its dimension be $D+1,$ and let $\left\{m_{\alpha }\hspace{0.17em}\right|\hspace{0.17em}\alpha =0,1,2,\dots ,D\}$ be a basis. With respect to this basis, any polynomial in $S_{d,n,g}$ (including all Cayley forms of type $(d,n,g)\text{)}$ may be assigned coordinates $(c_0,c_1,\dots ,c_D),$ which in turn can be regarded as a representative point of $P^D\left(K\right)\text{.}$ In this way, the set of Cayley forms of type $(d,n,g)$ may be identified with a subset of $P^D\left(K\right)\text{;}$ we denote this subset by $V_{d,n,g}\text{.}$ $V_{d,n,g}$ is more than just a subset – in fact, it is a $k\text{-variety}$ ([HPe1953], X, 8, Th. VII). Since Cayley forms represent varieties, the variety $V_{d,n,g}$ is called an algebraic system of varieties. Any subvariety of $V_{d,n,g}$ may also be referred to as an algebraic system of varieties. The geometrical statement that a variety $V$ may be continuously changed into a variety $V\prime$ is now interpreted as saying that $V$ and $V\prime$ belong to an irreducible algebraic system of varieties – i.e., $V$ and $V\prime$ have Cayley forms both belonging to an irreducible component of some $V_{d,n,g}\text{.}$ The algebraic systems $V_{d,n,g}$ play a major role in the development of the intersection calculus.

Simple Points and Tangent Spaces

Let $V$ be an irreducible variety of dimension $d$ in $P^n\left(K\right),$ and let $$f_i(x_0,\dots ,x_n)=0,(i=1,2,\dots ,n)$$ be a set of defining equations for $V\text{.}$ If $x\prime$ is a point of $V,$ then the Jacobian matrix $$J_{x\prime }=\left|\frac{\partial f_i}{\partial x_j}\right|{|}_{x=x\prime }$$ has rank at most $n-d,$ and there are points of $V$ for which the rank is exactly $n-d$ ([HPe1953], X, 14, Th. I). The spaces $\left\{x\hspace{0.17em}\right|\hspace{0.17em}{J}_{x\prime }·x=0\}$ contains $x\prime ,$ and is called the tangent space to $V$ at $x\prime \text{.}$ Points of $V$ for which the Jacobian has full rank are called simple points of $V\text{;}$ points which are not simple are called singular. "Most" points of a variety $V$ are simple; the singular points form a proper subvariety of $V\text{.}$

Algebraic Correspondences

$R\text{-way}$ Space

An $r\text{-way}$ space $P^{n_1,\dots ,n_r}\left(K\right)$ is the direct product of projective spaces $P^{n_1}\left(K\right),P^{n_2}\left(K\right),\dots ,P^{n_r}\left(K\right)\text{.}$ The theory of algebraic varieties in ordinary, or 1-way projective space, can easily be extended to give a theory of algebraic varieties in $P^{n_1,\dots ,n_r}\left(K\right)\text{.}$ Let $x^i$ be the $(n_i+1)\text{-tuple}$ $(x_0^i,x_1^i,\dots ,x_{n_i}^i),$ and let $F$ be a set of polynomials in $k[x^1,x^2,\dots ,x^r]$ which are homogeneous in each set of indeterminates $(x_0^i,x_1^i,\dots ,x_{n_i}^i)\text{.}$ The zero-set of $F$ is called a $k\text{-variety}$ in $r\text{-way}$ space. The notions of reducibility and generic point are defined in a straightforward manner, analogous to the definitions for 1-way space.

Specializations

If ${\xi }_i\in P^{n_i}\left(K\right)$ and $\xi =({\xi }_1,{\xi }_2,\dots ,{\xi }_r)$ is a generic point of an irreducible variety in $r\text{-way}$ space, then a point $x\prime =(x_1^{\prime },x_2^{\prime },\dots ,x_r^{\prime })$ is called a specialization of $\xi$ if $f\left(\xi \right)=0$ implies $f\left(x\prime \right)=0$ for any polynomial $f\text{.}$ If $$x\prime =(x_1^{\prime },x_2^{\prime },\dots ,x_r^{\prime })$$ is a specialization of $$\xi =({\xi }_1,{\xi }_2,\dots ,{\xi }_r),$$ then clearly $(x_1^{\prime },x_2^{\prime },\dots ,x_s^{\prime })$ is a specialization of $({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_s^{\prime })\text{.}$ A useful fact is the converse: if $(x_1^{\prime },x_2^{\prime },\dots ,x_s^{\prime })$ is a specialization of $({\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_s^{\prime }),$ then there exists $x_{s+1}^{\prime },x_{s+2}^{\prime },\dots ,x_r^{\prime }$ such that $(x_1^{\prime },\dots ,x_s^{\prime },x_{s+1}^{\prime },\dots ,x_r^{\prime })$ is a specialization of $({\xi }_1,{\xi }_2,\dots ,{\xi }_r)\text{.}$

Correspondences

An $r\text{-way}$ correspondence is defined as a variety in $r\text{-way}$ space. If $V$ is an $r\text{-way}$ correspondence in $P^{n_1,\dots ,n_r}\left(K\right)$ and $(x_1^{\prime },\dots ,x_r^{\prime })$ is a point of $V,$ then the points $x_1^{\prime },\dots ,x_r^{\prime }$ are said to be related by (or correspond according to) $V\text{.}$ In applications, a 2-way correspondence (often called, simply, a correspondence) is the most important.

Let $C$ be a correspondence in $P^{m,n}\left(K\right)\text{.}$ Associated with $C$ is a variety $M$ in $P^m\left(K\right)$ and a variety $N$ in $P^n\left(K\right),$ defined as follows: $$\begin{array}{rcl}{\displaystyle M}& =& {\displaystyle \{x\prime \in P^m\left(K\right)\hspace{0.17em}|\hspace{0.17em}(x\prime ,y\prime )\in C\hspace{0.17em}\text{for some}\hspace{0.17em}y\prime \hspace{0.17em}\text{in}\hspace{0.17em}{P}^n\left(K\right)\},}\\ {\displaystyle N}& =& {\displaystyle \{y\prime \in P^n\left(K\right)\hspace{0.17em}|\hspace{0.17em}(x\prime ,y\prime )\in C\hspace{0.17em}\text{for some}\hspace{0.17em}x\prime \hspace{0.17em}\text{in}\hspace{0.17em}{P}^m\left(K\right)\}\text{.}}\end{array}$$ $M$ is called the object variety of $C,$ and $N$ is called the image variety of $C\text{.}$ If $C$ is irreducible, so are $M$ and $N$ (the converse is false). If $x\prime \in M,$ we define $$N\left(x\prime \right)=\{y\prime \in N\hspace{0.17em}|\hspace{0.17em}(x\prime ,y\prime )\in C\}\text{.}$$ For $y\prime \in N,$ we define $M\left(y\prime \right)$ in a similar fashion: $$M\left(y\prime \right)=\{x\prime \in M\hspace{0.17em}|\hspace{0.17em}(x\prime ,y\prime )\in C\}\text{.}$$

([HPe1953], XI, 6, Th. I) Suppose $C$ is a correspondence whose object variety $M$ is irreducible. Let $\xi$ be a generic point of $M,$ and $x\prime$ any specialization of $\xi \text{.}$ If $N\left(\xi \right)$ has pure dimension $d,$ then any irreducible component of $N\left(x\prime \right)$ has dimension at least $d\text{.}$

Normal Problems

The following quote is taken from [HPe1953]: "In most problems which are of significance in geometry, the data involve certain elements which can be varied, and what is sought is a solution of the problem when the variable elements are chosen as generally as possible; from the solution of the general problem deductions regarding the solutions of particular cases are then made."

Such problems are formally referred to as normal problems. A normal problem consists of a non-void correspondence $C$ in $P^{m,n}\left(K\right)$ having the following properties:

1)	$C$ is determined by equations $$\begin{array}{rcl}{\displaystyle f_i(x,y)}& =& {\displaystyle 0,i=1,\dots ,r,}\\ {\displaystyle g_i(x,y)}& =& {\displaystyle 0,i=1,\dots ,s\text{.}}\end{array}$$
2)	The object variety $M$ of $C$ is determined by the equations $g_i\left(x\right)=0$ $\text{(}i=1,2,\dots ,s\text{),}$ and is irreducible;
3)	The variety $N\left(\xi \right)$ corresponding to a generic point of $M$ is unmixed.

Given this formalism, the generic point $\xi$ represents the variable data of a geometric problem, and $N\left(\xi \right)$ represents the general solution. The basic theory of normal problems explains the connection between properties of the general solution $N\left(\xi \right)$ and properties of particular solutions $N\left(x\prime \right)\text{.}$ For example, if $N\left(\xi \right)$ has dimension $b,$ then $N\left(x\prime \right)$ has dimension at least $b$ (Theorem 3.1). If $N\left(\xi \right)$ consists of $p$ points, then $N\left(x\prime \right)$ also consists of $p$ points – usually. This last statement is Schubert's Principle of Conservation of Number. The historic controversy surrounding the Principle arose from this difficulty in handling "multiple" solutions.

The notion of multiplicity is developed in [HPe1953] as follows. Write the general solution $N\left(\xi \right)$ of a normal problem as a non-contractible union of irreducible varieties: $$N\left(\xi \right)=V_1\cup V_2\cup \cdots \cup V_k\text{.}$$ By definition, the multiplicity of each component of the generic solution is one. Now let $x\prime$ be a specialization of $\xi ,$ and write the particular solution $N\left(x\prime \right)$ as a union of irreducible varieties: $$N\left(x\prime \right)=V_1^{\prime }\cup V_2^{\prime }\cup \cdots \cup V_i^{\prime }\text{.}$$ The problem is to define the multiplicities of the components $V_i^{\prime }$ so that expected theorems hold true; e.g., if the solutions $N\left(\xi \right)$ and $N\left(x\prime \right)$ are finite sets, then the number of points in $N\left(\xi \right)$ is the same as the number of points in $N\left(x\prime \right),$ counting multiplicities. Intuitively, a solution has multiplicity greater than one when $\xi$ is specialized in a way such that two or more of the generic solution components $V_1,V_2,\dots ,V_k$ specialize to an identical variety. We proceed with the formal discussion of this phenomenon. By assumption, each component of $N\left(\xi \right)$ is irreducible and has dimension $d\text{.}$ Let component $V_i$ have Cayley form $F_i$ and order $g_i,$ and set $$F=\prod _{i=1}^k{F}_i\text{.}$$ Then $F$ is a Cayley form which belongs to the algebraic system $V_{d,n,g}$ $\text{(}g=\sum _{i=1}^k{g}_i\text{).}$ In fact, $F$ is a generic point of some irreducible component $k$ of $V_{d,n,g}\text{.}$ The point $(\xi ,F)$ is in turn a generic point of some variety (correspondence) in 2-way space $P^{n,D}\left(K\right)\text{.}$ This new correspondence relates an element $x\prime$ of the object variety of $C$ to a Cayley form of the solution variety $N\left(x\prime \right)\text{.}$ Some qualification is necessary because even though any specialization $x\prime$ of $\xi$ may be completed to a specialization $(x\prime ,F\prime )$ of $(\xi ,F),$ $F\prime$ is not necessarily a Cayley form of $N\left(x\prime \right)\text{.}$ It can be shown that for any specialization $x\prime$ of $\xi ,$ $V\left(F\prime \right)<N\left(x\prime \right),$ but in some cases the inclusion may be strict (e.g., whenever there is more than one specialization of $F$ which corresponds to $x\prime \text{).}$ Suppose, however, that $F\prime$ is a Cayley form for $N\left(x\prime \right),$ i.e. $V\left(F\prime \right)=N\left(x\prime \right)\text{.}$ Write $F\prime$ as a product $\prod _{i=1}^l{\left(F_i^{\prime }\right)}^{a_i},$ where $F_i^{\prime }$ is the Cayley form for the irreducible component $V_i^{\prime }$ of $N\left(x\prime \right)\text{.}$ Then we define the multiplicity of $V_i^{\prime }$ as $a_i\text{.}$

This definition of multiplicity is made for a solution $N\left(x\prime \right)$ only if there exists a unique specialization $F\prime$ of $F$ corresponding to $x\prime \text{.}$

([HPe1953], XI, 7, Th. I and Th. IV) The following conditions guarantee that multiplicities can be defined for components of $N\left(x\prime \right)\text{:}$

1)	$N\left(x\prime \right)$ is a variety of pure dimension $b\text{;}$
2)	$x\prime$ is a simple point of the object variety of $C\text{.}$

The Intersection Calculus

Intersection Product

We can now sketch how the theory of normal problems is used to develop an intersection calculus for the subvarieties of a nonsingular, $n\text{-dimensional}$ variety $V$ (a full treatment is given in [HPe1953], XII). Let $F_d\left(V\right)$ denote the free abelian group generated by the irreducible, $d\text{-dimensional}$ subvarieties of $V\text{.}$ The notation $\bar{V}$ (or ${\bar{V}}_d,$ if the dimension $d$ needs emphasis) will be used to denote the element of $F_d\left(V\right)$ corresponding to an irreducible subvariety $V\text{.}$ A general element of $F_d\left(V\right)$ can be written as $$\sum _{i=1}^k{a}_i·{\bar{V}}^{\left(i\right)}$$ or $$\sum _{i=1}^k{a}_i·{\bar{V}}_d^{\left(i\right)}\text{.}$$

Given ${\bar{V}}_a\in F_a\left(V\right)$ and ${\bar{V}}_b\in F_b\left(V\right),$ we wish to define the intersection product ${\bar{V}}_a·{\bar{V}}_b\text{.}$ If $a+b-n<0,$ we define ${\bar{V}}_a·{\bar{V}}_b=0\text{.}$ Otherwise, let the irreducible components of $V_a\wedge V_b$ be denoted by $V^{\left(1\right)},V^{\left(2\right)},\dots ,V^{\left(k\right)}\text{.}$ We shall define ${\bar{V}}_a·{\bar{V}}_b$ as the sum $\sum _{i=1}^k{a}_i·{\bar{V}}^{\left(i\right)},$ where the $a_i\text{'s}$ are multiplicities obtained from a suitably chosen normal problem. To formulate the normal problem, we require the following:

1)

$V_a$ and $V_b$ intersect properly (i.e. the intersection is a pure variety of dimension $a+b-n\text{);}$

2)

There exists an irreducible algebraic system $S$ with generic Cayley form $F_b$ such that i) The Cayley form $F_b$ of $V_b$ is a simple point of $S\text{;}$ ii) The variety $V\left(F^{*}\right)$ meets $V_a$ simply. This means that $V\left(F^{*}\right)$ meets $V_a$ properly; each point of $V\left(F^{*}\right)\wedge V_a$ is a simple point of $V\left(F^{*}\right)$ and $V_a\text{;}$ and for each point of $V\left(F^{*}\right)\wedge V_a,$ the tangent spaces to $V\left(F^{*}\right)$ and $V_a$ meet properly.

Given these conditions, we consider the normal problem based on the correspondence $$C=\left\{(F,x)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& F\in S,\\ \text{2)}& x\in V_a\wedge V\left(F^{*}\right)\end{array}\}\text{.}$$ Corresponding to the specialization $F_b$ of $F^{*}$ is the particular solution $$N\left(F_b\right)=V_a\wedge V\left(F_b\right)=V_a\wedge V_b\text{.}$$ Conditions 1) and 2) i) guarantee that multiplicities $a_1,a_2,\dots ,a_k$ may be defined for the irreducible components of $V_a\wedge V_b$ (Theorem 3.2). ${\bar{V}}_a·{\bar{V}}_b$ is then defined as $$\sum _{i=1}^k{a}_i·{\bar{V}}^{\left(i\right)}\text{.}$$ To ensure that this product is well-defined, one must verify the same multiplicities are obtained if some other algebraic system $S\prime$ is used; this can be done ([HPe1953], XII, 8, Th. I). Actually, the approach in [HPe1953] is to first define the intersection product in terms of a particular normal problem (chosen for its tractability), and then to show any other normal problem based on condition 2) gives an equivalent definition.

So far our definition applies to basis elements ${\bar{V}}_a$ and ${\bar{V}}_b$ satisfying 1) and 2). We define $$(\sum _{i=1}^k{a}_i·{\bar{V}}^{\left(i\right)})·(\sum _{j=1}^l{b}_j·{\bar{V}}^{\left(j\right)})$$ by linear extension, whenever all basis products are define. The intersection-product so defined is commutative, associative, and distributive ([HPe1953], XII, 7, Th. I).

Equivalence Relations

To overcome the difficulty of undefined products, we equip $F_d\left(V\right)$ with an equivalence relation $\stackrel{V}{=}$ ("virtual equivalence") together with definitions of the sum and product of equivalence classes. The relation $\stackrel{V}{=}$ is defined in a "bootstrapping" fashion: first we define an equivalence relation $\stackrel{N}{=}$ ("narrow equivalence"), then a relation $\stackrel{W}{=}$ ("wide equivalence") in terms of $\stackrel{N}{=},$ and finally $\stackrel{V}{=}$ in terms of $\stackrel{W}{=}\text{.}$ Let $F_d^{+}\left(V\right)$ be the set of positive elements in $F_d\left(V\right),$ i.e. an element $\sum _{i=1}^k{a}_i·{\bar{V}}_d^{\left(i\right)}$ such that all $a_i\ge 0$ and some $a_i>0\text{.}$ Identify an element $\sum _{i=1}^k{a}_i·{\bar{V}}_d^{\left(i\right)}$ belonging to $F_d^{+}\left(V\right)$ with the Cayley form $\prod _{i=1}^k{F}_i^{a_i},$ where $F_i$ is the Cayley form of $V_d^{\left(i\right)}\text{.}$ Narrow equivalence $\stackrel{N}{=}$ is defined on $F_d^{+}\left(V\right)$ as the smallest equivalence relation containing the set $$\left\{(X,Y)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& {\displaystyle X=\sum _{i=1}^k{a}_i{\bar{U}}^{\left(i\right)},}\\ \text{2)}& {\displaystyle Y=\sum _{j=1}^l{b}_j{\bar{V}}^{\left(j\right)},}\\ \text{3)}& \text{There exists an irreducible algebraic system containing}\\ & \text{the Cayley forms associated with}\hspace{0.17em}X\hspace{0.17em}\text{and}\hspace{0.17em}Y\end{array}\}\text{.}$$ Wide equivalence $\stackrel{W}{=}$ is defined on $F_d^{+}\left(V\right)$ as follows: $$X\stackrel{NW}{=}Y⟺\text{there exists}\hspace{0.17em}Z\in F_d^{*}\left(V\right)\hspace{0.17em}\text{such that}\hspace{0.17em}X+Z\stackrel{WN}{=}Y+Z\text{.}$$ If $X=\sum _{i=1}^k{a}_i·{\bar{V}}_d^{\left(i\right)}$ define $p\left(X\right),$ the positive part of $X,$ as $\sum _{a_i\ge 0}{a}_i·{\bar{V}}_d^{\left(i\right)}\text{.}$ Define $n\left(X\right),$ the negative part of $X,$ as $p(-X)\text{.}$ Virtual equivalence $\stackrel{V}{=}$ is defined on $F_d\left(V\right)$ as follows: $$X\stackrel{V}{=}Y⟺p\left(X\right)-n\left(Y\right)\stackrel{W}{=}p\left(Y\right)-n\left(X\right)\text{.}$$

Intersection Ring

If $X\in F_d\left(V\right),$ let $\left[X\right]$ denote the virtual equivalence class containing $X\text{.}$ We define the sum and product of virtual equivalence classes as follows: $$\begin{array}{rcl}{\displaystyle \left[X\right]+\left[Y\right]}& =& {\displaystyle [X+Y],}\\ {\displaystyle \left[X\right]·\left[Y\right]}& =& {\displaystyle [X\prime ·Y\prime ],}\end{array}$$ where $\left[X\right]=\left[X\prime \right],$ $\left[Y\right]=\left[Y\prime \right],$ and $X\prime ·Y\prime$ is defined.

It can be shown that these operations are well-defined, commutative, associative, and distributive ([HPe1953], XII, 10). The fact that any two virtual classes $\left[X\right],$ and $\left[Y\right]$ contain representatives $X\prime ,Y\prime$ whose product $X\prime ·Y\prime$ is defined is guaranteed by Chow's "moving lemma" ([HPe1953], XII, 10, Th. II).

The collection of virtual equivalence classes of $F_{n-d}\left(V\right)$ will be denoted by $A_d\left(V\right)\text{.}$ $A_0\left(V\right)$ is a free abelian group with the single basis element $\left[\bar{V}\right]\text{.}$ $A_n\left(V\right)$ is a free abelian group with single basis element $\left[\bar{P}\right],$ where $P$ is any point of $V\text{.}$ The intersection ring of $V,$ denoted by $A\left(V\right),$ is the graded, commutative ring $$\sum _{i=0}^{\infty }{A}_i\left(V\right),$$ where sums and products are defined in the obvious way. Many geometrical properties of the variety $V$ are reflected in the algebraic structure of the ring $A\left(V\right)\text{.}$ Our narrow interest here concerns the enumerative properties of $V\text{;}$ a most important result is the following ([HPe1953], XII, p.193; [KLa1972], p.1070).

If $V_a,V_b,\dots ,V_h$ are subvarieties of $V$ which intersect properly in a finite number of points $P$ (counted with multiplicity), then $$\left[{\bar{V}}_a\right]·\left[{\bar{V}}_b\right]·\cdots ·\left[{\bar{V}}_h\right]=r·\left[\bar{P}\right]\text{.}$$ We mention that an empty intersection is a proper intersection, in which case $r=0\text{.}$

Intersection Theory of the Grassmann Variety

Plucker Coordinates

Let $L$ be a projective $(d-1)\text{-plane}$ in $P^{n-1}\left(K\right)$ (or equivalently, a $d\text{-subspace}$ of $A^n\left(K\right)\text{).}$ Each such plane is an irreducible variety of dimension $d-1$ and order 1; the Cayley forms of such planes constitute the algebraic system $V_{d-1,n-1,1}\text{.}$ The system $V_{d-1,n-1,1}$ will be called the Grassmann variety of $(d-1)\text{-planes}$ in projective $(n-1)\text{-space}$ (or the Grassmann variety of $d\text{-subspaces}$ of affine $n\text{-space),}$ and will be denoted by $G_{d,n}\left(K\right),$ or $G_{d,n}$ if $K$ is understood¹.

The system $G_{d,n}$ is contained in a projective space of dimension $N=\left(\genfrac{}{}{0ex}{}{n}{d}\right)-1,$ and is an irreducible, nonsingular variety of projective dimension $(n-d)d$ ([HPe1953], XIV, 1). Let $L$ be a $d\text{-subspace}$ in $A^n\left(K\right)$ with $d\text{-by-}n$ basis matrix $B\text{.}$ A point of $G_{d,n}$ corresponding to the Cayley form of $L$ is given by the $d^{\text{th}}$ compound of $B$ ([HPe1953], XIV, 1). Elements of $G_{d,n}$ are called Plucker coordinates; they satisfy a set of quadratic polynomial equations, called the Plucker relations, which define the variety $G_{d,n}$ (see [KLa1972]).

In section 2, the set $$\mathrm{\Omega }\left(A\right)=\left\{L\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& L\hspace{0.17em}\text{is a}\hspace{0.17em}d\text{-subspace in}\hspace{0.17em}{A}^n\left(K\right),\\ \text{2)}& \text{dim}(L\wedge A_i)\ge i,\hspace{0.17em}1\le i\le d\end{array}\}$$ was introduced and called a Schubert variety. This terminology is appropriate, because condition 2) of the definition may be expressed as a system of homogeneous linear equations in the Plucker coordinates of the $d\text{-subspace}$ $L\text{.}$ The symbol $\mathrm{\Omega }\left(A\right)$ will be used both for the collection of $d\text{-subspaces}$ of $A^n\left(K\right)$ and for the corresponding linear subvariety of $G_{d,n}\text{.}$ $\mathrm{\Omega }\left(A\right)$ is an irreducible subvariety of projective dimension $\sum _{i=1}^d(a_i-i),$ where $a_i=\text{dim}\left(A_i\right)\text{.}$

Intersection Ring of $G_{d,n}$

The Schubert varieties play a fundamental role in describing the structure of the intersection ring $A\left(G_{d,n}\right)$ of $G_{d,n}\text{.}$ The equivalence $\left[\mathrm{\Omega }\left(A\right)\right]$ is called a Schubert cycle and will be denoted by $\mathrm{\Omega }\left(a\right),$ where $a\in Q_{d,n}$ and $a_i=\text{dim}\left(A_i\right)\text{.}$ This notation is consistant since if $\left[\mathrm{\Omega }\left(A\right)\right]=\left[\mathrm{\Omega }\left(B\right)\right],$ then $\text{dim}\left(A_i\right)=\text{dim}\left(B_i\right)$ for $i=1,2,\dots ,d$ (this is indirectly shown in [HPe1953] in the proof of the following theorem).

([HPe1953], XIV, Th. I – "the basis theorem") The Schubert cycles $\mathrm{\Omega }\left(a\right),$ $a\in Q_{d,n},$ freely generated $A\left(G_{d,n}\right)$ as an abelian group.

The Schubert calculus essentially consists of the basis theorem and two classical rules – Pieri's formula and the determinantal formula (or Fiambelli's formula) – used to compute products in $A\left(G_{d,n}\right)\text{.}$ The rules show that the special Schubert cycles $$\sigma \left(h\right)=\mathrm{\Omega }(n-d+1-h,n-d+2,n-d+3,n-d+4,\dots ,n-1,n)$$ generated $A\left(G_{d,n}\right)$ as a ring.

(Pieri's formula) $$\mathrm{\Omega }\left(a\right)·\sigma \left(h\right)=\sum \mathrm{\Omega }\left(b\right),$$ where the sum ranges over all $B\in Q_{d,n}$ satisfying

i)	$1\le b_1\le a_1<b_2\le a_2<b_3\cdots <b_d\le a_d,$ and
ii)	${\displaystyle \sum _{i=1}^d{a}_i}={\displaystyle \sum _{i=1}^d{b}_i}+h\text{.}$

(The determinataal formula) $$\mathrm{\Omega }\left(a\right)=\text{det}\left(\sigma (n-d-a_i+j)\right),$$ where by convention $\sigma \left(h\right)=0$ if $h<0$ or $h>n-d\text{.}$

Another useful rule, the duality theorem, gives the product of Schubert cycles of complementary grade. Two grades of $A\left(G_{d,n}\right)$ are complementary if their sum equals $(n-d)·d\text{.}$

(The duality theorem) If $\mathrm{\Omega }\left(a\right)$ and $\mathrm{\Omega }\left(b\right)$ are Schubert cycles of complementary grade, then $$\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)=\{\begin{array}{ll}\mathrm{\Omega }(1,2,3,\dots ,d),& \text{if}\hspace{0.17em}a=\bar{b},\\ 0,& \text{if}\hspace{0.17em}a\ne \bar{b}\text{.}\end{array}$$

The rules 3.4, 3.5, and 3.6 provide a method for expressing the product of any two Schubert cycles as a sum of Schubert cycles. As an example, consider the product $$\mathrm{\Omega }(2,4,6)·\mathrm{\Omega }(2,4,6)\text{in}A\left(G_{d,n}\right)\text{.}$$ Use the determinantal formula to write $$\begin{array}{rcl}{\displaystyle \mathrm{\Omega }(2,4,6)}& =& {\displaystyle \text{det}\mid \begin{array}{ccc}\sigma (3-2+1)& \sigma (3-2+2)& \sigma (3-2+3)\\ \sigma (3-4+1)& \sigma (3-4+2)& \sigma (3-4+3)\\ \sigma (3-6+1)& \sigma (3-6+2)& \sigma (3-6+3)\end{array}\mid }\\ & =& {\displaystyle \text{det}\mid \begin{array}{ccc}\sigma \left(2\right)& \sigma \left(3\right)& \sigma \left(4\right)\\ \sigma \left(0\right)& \sigma \left(1\right)& \sigma \left(2\right)\\ 0& 0& 0\end{array}\mid }\\ & =& {\displaystyle (\sigma \left(2\right)\sigma \left(1\right)-\sigma \left(0\right)\sigma \left(3\right))·\sigma \left(0\right)}\\ & =& {\displaystyle \sigma \left(2\right)\sigma \left(1\right)-\sigma \left(3\right)\text{.}}\end{array}$$ Use Pieri's formula repeatedly to obtain $$\begin{array}{rcl}{\displaystyle \mathrm{\Omega }(2,4,6)·\mathrm{\Omega }(2,4,6)}& =& {\displaystyle \mathrm{\Omega }(2,4,6)·(\sigma \left(2\right)\sigma \left(1\right)-\sigma \left(3\right))}\\ & =& {\displaystyle \mathrm{\Omega }(2,4,6)\sigma \left(2\right)\sigma \left(1\right)-\mathrm{\Omega }(2,4,6)\sigma \left(3\right)}\\ & =& {\displaystyle (\mathrm{\Omega }(1,3,6)+\mathrm{\Omega }(1,4,5)+\mathrm{\Omega }(2,3,5))·\sigma \left(1\right)-\mathrm{\Omega }(1,3,5)}\\ & =& {\displaystyle \mathrm{\Omega }(1,2,6)+\mathrm{\Omega }(1,3,5)+\mathrm{\Omega }(1,3,5)+\mathrm{\Omega }(1,3,5)+\mathrm{\Omega }(2,3,4)-\mathrm{\Omega }(1,3,5)}\\ & =& {\displaystyle 2·\mathrm{\Omega }(1,3,5)+\mathrm{\Omega }(1,2,6)+\mathrm{\Omega }(2,3,4)\text{.}}\end{array}$$ We could now apply the duality theorem and obtain $$\mathrm{\Omega }(2,4,6)·\mathrm{\Omega }(2,4,6)·\mathrm{\Omega }(2,4,6)=2·\mathrm{\Omega }(1,2,3)\text{.}$$ The general intersection theory would then guarantee that three Schubert varieties of type $(2,4,6)$ intersect either in one point (which has multiplicity one two?), two points (if the intersection is simple), or infinitely many points (if the intersection is not proper).

At this point we mention a useful theorem relating $A\left(G_{d,n}\right)$ to $A\left(G_{n-d,n}\right),$ the proof of which is delayed till the next section.

Let $h$ be the map from $A\left(G_{d,n}\right)$ onto $A\left(G_{n-d,n}\right)$ defined by $$h\left(\mathrm{\Omega }\left(a\right)\right)=\mathrm{\Omega }\left(\stackrel{\sim }{a}\right)$$ and linear extension. Then $h$ is a ring isomorphism.

Characterization of $V_d^n\left(k\right)$

Recall the definition of the set $V_d^n\left(k\right)$ given in Section 2: $$V_d^n\left(k\right)=\left\{(a,b,c)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& a,b,c\in Q_{d,n},\\ \text{2)}& \text{For any three}\hspace{0.17em}(d,n,k)\text{-towers}\hspace{0.17em}A,B,C\text{, of type}\hspace{0.17em}a,b,c\text{, respectively,}\\ & \mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)\ne \varnothing \end{array}\}\text{.}$$ The intersection theory of $G_{d,n}\left(K\right)$ may be used to obtain the following result.

If $k$ is an algebraically closed subfield of $K,$ then $$V_d^n\left(k\right)=\left\{(a,b,c)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& a,b,c\in Q_{d,n},\\ \text{2)}& \mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)\ne 0\end{array}\}\text{.}$$

		Proof.
	Suppose $(a,b,c)\in V_d^n\left(k\right)\text{.}$ We show that $\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)\ne 0\text{.}$ Let $A,B,C$ be three $(d,n,k)\text{-towers}$ of type $a,b,c,$ respectively, such that $\mathrm{\Omega }\left(A\right),\mathrm{\Omega }\left(B\right),$ and $\mathrm{\Omega }\left(C\right)$ intersect properly. Now regard $A,B,C$ as $(d,n,K)\text{-towers;}$ then $\mathrm{\Omega }\left(A\right),\mathrm{\Omega }\left(B\right),$ and $\mathrm{\Omega }\left(C\right)$ continue to intersect properly. By assumption, Thus, $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ is non-empty. Let its dimension be $h\text{.}$ Either $h=0$ or $h>0\text{.}$ If $h=0,$ then $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ consists of a finite, nonzero number of points $r$ (when counted with multiplicity), and from Theorem 3.3 we have $\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)=r·\mathrm{\Omega }(1,2,3,\dots ,d)\ne 0\text{.}$ If $h>0,$ let $p$ be a point in $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ and let $V$ be the subvariety of $G_{d,n}\left(K\right)$ given by the intersection of $G_{d,n}$ with $h$ generic $(N-1)\text{-planes}$ passing through $p\text{.}$ Then $V$ meets $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ properly in a finite, nonzero number of points $r$ (again, counted with multiplicity). Hence $$\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)·\left[\bar{V}\right]=r·\mathrm{\Omega }(1,2,\dots ,d)\ne 0,$$ so $\mathrm{\Omega }\left(A\right)\cancel{\wedge }·\mathrm{\Omega }\left(B\right)\cancel{\wedge }·\mathrm{\Omega }\left(C\right)\ne 0\text{.}$ Now suppose that $\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)\ne 0\text{.}$ Let $A,B,C$ be any three $(d,n,k)\text{-towers}$ of type $a,b,c$ respectively. If $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)=\varnothing ,$ this would state that $\mathrm{\Omega }\left(A\right),\mathrm{\Omega }\left(B\right),$ and $\mathrm{\Omega }\left(C\right)$ have proper intersection containing no points, which implies that $\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)\ne 0,$ a contradiction. Hence $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ is non-empty. To show $(a,b,c)\in V_d^n\left(k\right),$ it is enough to show this intersection contains the Plucker coordinates of a $d\text{-space}$ which is a subspace of $A^n\left(k\right)\text{.}$ The variety $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ consists of points satisfying the quadratic Plucker equations and a set of linear equations. Since $A,B,C$ are towers in $A^n\left(k\right),$ the linear equations defining $\mathrm{\Omega }\left(A\right)\wedge \mathrm{\Omega }\left(B\right)\wedge \mathrm{\Omega }\left(C\right)$ have all their coefficients in the field $k\text{.}$ Hence the system of equations defining the intersection has all its coefficients in the algebraically closed field $k\text{.}$ Since a solution exists in the extension field $K,$ a solution must exist in $k\text{.}$ $\square$

From this point on we assume that $k$ is algebraically closed.

If $(a\prime ,b\prime ,c\prime )\in V_d^n,$ there exists $(a,b,c)\in V_d^n$ such that

1)	$(a,b,c)\le (a\prime ,b\prime ,c\prime ),$ and
2)	$\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)$ is a non-zero multiple of $\mathrm{\Omega }(1,2,\dots ,d)\text{.}$

		Proof.
	Let $$\mathrm{\Omega }\left(a\prime \right)·\mathrm{\Omega }\left(b\prime \right)·\mathrm{\Omega }\left(c\prime \right)=\sum _e{n}_e·\mathrm{\Omega }\left(e\right)\text{.}$$ Since $(a\prime ,b\prime ,c\prime )\in V_d^n,$ the product is nonzero, and so as in the proof of 3.9 there is some $e\prime$ such that $$\mathrm{\Omega }\left(a\prime \right)·\mathrm{\Omega }\left(b\prime \right)·\mathrm{\Omega }\left(c\prime \right)·\mathrm{\Omega }\left(\bar{e}\prime \right)=n_{e\prime }·\mathrm{\Omega }(1,2,\dots ,d)\ne 0\text{.}$$ By the basis theorem, $$\mathrm{\Omega }\left(c\prime \right)·\mathrm{\Omega }\left(\bar{e}\prime \right)=\sum _f{m}_f·\mathrm{\Omega }\left(f\right),$$ where clearly $m_f$ is nonzero only if $f\le c\prime \text{.}$ Then $$\sum _f{m}_f·\mathrm{\Omega }\left(a\prime \right)·\mathrm{\Omega }\left(b\prime \right)·\mathrm{\Omega }\left(f\right)=n_{e\prime }·\mathrm{\Omega }(1,2,\dots ,d)\ne 0\text{.}$$ Hence for some $f\prime \le c\prime ,$ $$\mathrm{\Omega }\left(a\prime \right)·\mathrm{\Omega }\left(b\prime \right)·\mathrm{\Omega }\left(f\prime \right)$$ is non-zero multiple of $\mathrm{\Omega }(1,2,\dots ,d),$ and the corollary follows. $\square$

$$(a,b,c)\in V_d^n\Rightarrow \sum (a,b,c)\ge 2(n+1)d-\frac{d(d+1)}{2}\text{.}$$

		Proof.
	If $(a,b,c)\in V_d^n,$ then $\mathrm{\Omega }\left(a\right)·\mathrm{\Omega }\left(b\right)·\mathrm{\Omega }\left(c\right)\ne 0,$ and so $$\text{grade}\hspace{0.17em}\mathrm{\Omega }\left(a\right)+\text{grade}\hspace{0.17em}\mathrm{\Omega }\left(b\right)+\text{grade}\hspace{0.17em}\mathrm{\Omega }\left(c\right)\cancel{\le }=\text{grade}\hspace{0.17em}\mathrm{\Omega }(0,1,2,\dots ,d)\text{.}$$ $\square$

A Schubert cycle of type $a$ has projective dimension $$\sum _{i=1}^d{a}_i-\frac{(d+1)d}{2}$$ ([HPe1953], XIV, 3); since $G_{d,n}$ has projective dimension $(n-d)d,$ a Schubert cycle of type $a$ has grade $$(n-d)d-(\sum _{i=1}^d{a}_i-\frac{(d+1)d}{\cancel{d}2}\text{.})$$ Plugging in the grades of $\mathrm{\Omega }\left(a\right),\mathrm{\Omega }\left(a\right),\mathrm{\Omega }\left(c\right),$ and $\mathrm{\Omega }(1,2,\dots ,d),$ we get the inequality $$3(n-d)(d-(\sum _{i=1}^d{a}_i+b_i+c_i)+\frac{\cancel{1}3d(d+1)}{2})\le (n-d)d-\sum _{i=1}^{d}i+\frac{d(d+1)}{2}\text{.}$$ Rearranging, $$\sum _{i=1}^d{a}_i+b_i+c_i\ge 2(n\cancel{+1}-d)d+\frac{3d(d+1)}{2}$$ or $$\sum _{i=1}^d(a,b,c)\ge 2(n+1)d-\frac{d(d+1)}{2}\text{.}$$

Theorem 3.11 provides a useful characterization of $V_d^n\text{.}$ To examine the conjecture that $V_d^n=T_d^n,$ we require yet another characterization, based on the Littlewood-Richardson rule for multiplying Schur functions in the algebra of symmetric polynomials. This topic is developed in Section 4.

Footnotes

¹ Our notation, which displays the affine dimensions of the underlying linear spaces, is a personal preference, especially convenient in the overall setting of this work.

Notes and References

This is an excerpt from Steven Andrew Johnson's 1979 dissertation The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices.

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