Last updated: 14 October 2014
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 and There is little lost in assuming the polynomials 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 where 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 have degree and degree Mathematicians' intuition for a long time told them that the answer ought to be the product 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.|
|(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).|
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). 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.
By affine we mean the space of over a field We denote this space by by convention, By projective we mean the space of non-zero over a field where we identify and if there exists a scalar such that for Projective over is denoted by by convention, Identifying a point in with the point in we may think of as completed by the points "at infinity". Algebraic geometry in considers systems of (possibly nonhomogeneous) polynomial equations in unknowns. Algebraic geometry in considers systems of homogeneous polynomial equations in unknowns. Problems in may be translated to and from problems in by "homogenizing" or "dehomogenizing" the polynomials in question. A good description of this procedure may be found in [Ful1969].
A plane in is defined as the set of points which satisfy a system of homogeneous linear equations where is an matrix over We say the plane has projective dimension if has rank in this case is called a A in corresponds to a subspace of of affine dimension which we call a
Let denote the ring of polynomials in over a field of characteristic zero, and let be a subset of consisting of homogeneous polynomials. Consider the system of equations If we look for solutions only in we may not find any, or at any rate not enough; we therefore take an algebraically closed extension of and look for solutions in Following the classical theory, we take the field of rational functions over in a countably infinite number of indeterminates, and let be its algebraic closure. We then call the set a projective algebraic variety (abbreviated terminology: the unqualified term variety means some where The field is called the "universal" field; its relationship to the ground field is such that is contains all extensions of needed in the classical theory.
A variety is reducible over a field if it is the union of two distinct both distinct from A variety may be irreducible over a field but reducible over some extension field A variety which is irreducible over is called an absolutely irreducible variety. Any may be expressed, in an essentially unique way, as a finite, non-contractible union of irreducible ([HPe1953], X, 2, Thm. III and Th. IV).
Given a variety and a point in is said to be a generic point of over if:
|2)||Any form for which vanished on|
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 if and only if is irreducible over and|
|2)||given a point in and a field there is a unique having as a generic point.|
Let be an irreducible and let be a generic point of over The dimension of is defined as the maximum number of elements among which are algebraically independent over 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.
Let be planes in If have dimensions respectively, and then is non-empty. This is a simple fact of linear algebra. A fact of algebraic geometry is that the same statement holds true when and are arbitrary varieties in
Let be an irreducible of dimension the previous statement guarantees that has a non empty intersection with any For the case is different; some meet some don't. It turns out that is uniquely determined by the complex of 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 is a homogenious polynomial whose roots represent which meet Note that any in may be associated with a matrix of rank over such that iff Define as follows: Then represents the complex of which meet Let be a set of indeterminates, and the ring of polynomials in the with coefficients in It may be shown that the set is a principal ideal in The Cayley form is defined as the monic polynomial which generates An irreducible variety has a unique Cayley form, and is uniquely determined by its Cayley form ([HPe1953], X, 7, p.41).
The Cayley form of has the following properties:
|1)||if and only if|
|2)||Let the matrix of indeterminates be written where Then there exists a positive integer such that for any scalar and index|
Property 2) states that is homogeneous of the same degree in each set of variables The integer is called the order of
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.
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 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 be an unmixed variety of dimension with irreducible components having Cayley forms and order respectively. If is any sequence of positive integers, then will be called a Cayley form for For this extended definition, we still have:
|1)||if and only if|
We say that such a Cayley form has type
A Cayley form of type
Cayley forms of type constitute a system of varieties in the following sense. Let be the subset of consisting of polynomials which are homogeneous of degree in each set of indeterminates (i.e. property 2) above). is in fact a vector space; let its dimension be and let be a basis. With respect to this basis, any polynomial in (including all Cayley forms of type may be assigned coordinates which in turn can be regarded as a representative point of In this way, the set of Cayley forms of type may be identified with a subset of we denote this subset by is more than just a subset – in fact, it is a ([HPe1953], X, 8, Th. VII). Since Cayley forms represent varieties, the variety is called an algebraic system of varieties. Any subvariety of may also be referred to as an algebraic system of varieties. The geometrical statement that a variety may be continuously changed into a variety is now interpreted as saying that and belong to an irreducible algebraic system of varieties – i.e., and have Cayley forms both belonging to an irreducible component of some The algebraic systems play a major role in the development of the intersection calculus.
Let be an irreducible variety of dimension in and let be a set of defining equations for If is a point of then the Jacobian matrix has rank at most and there are points of for which the rank is exactly ([HPe1953], X, 14, Th. I). The spaces contains and is called the tangent space to at Points of for which the Jacobian has full rank are called simple points of points which are not simple are called singular. "Most" points of a variety are simple; the singular points form a proper subvariety of
An space is the direct product of projective spaces The theory of algebraic varieties in ordinary, or 1-way projective space, can easily be extended to give a theory of algebraic varieties in Let be the and let be a set of polynomials in which are homogeneous in each set of indeterminates The zero-set of is called a in space. The notions of reducibility and generic point are defined in a straightforward manner, analogous to the definitions for 1-way space.
If and is a generic point of an irreducible variety in space, then a point is called a specialization of if implies for any polynomial If is a specialization of then clearly is a specialization of A useful fact is the converse: if is a specialization of then there exists such that is a specialization of
An correspondence is defined as a variety in space. If is an correspondence in and is a point of then the points are said to be related by (or correspond according to) In applications, a 2-way correspondence (often called, simply, a correspondence) is the most important.
Let be a correspondence in Associated with is a variety in and a variety in defined as follows: is called the object variety of and is called the image variety of If is irreducible, so are and (the converse is false). If we define For we define in a similar fashion:
([HPe1953], XI, 6, Th. I) Suppose is a correspondence whose object variety is irreducible. Let be a generic point of and any specialization of If has pure dimension then any irreducible component of has dimension at least
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 in having the following properties:
|1)||is determined by equations|
|2)||The object variety of is determined by the equations and is irreducible;|
|3)||The variety corresponding to a generic point of is unmixed.|
Given this formalism, the generic point represents the variable data of a geometric problem, and represents the general solution. The basic theory of normal problems explains the connection between properties of the general solution and properties of particular solutions For example, if has dimension then has dimension at least (Theorem 3.1). If consists of points, then also consists of 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 of a normal problem as a non-contractible union of irreducible varieties: By definition, the multiplicity of each component of the generic solution is one. Now let be a specialization of and write the particular solution as a union of irreducible varieties: The problem is to define the multiplicities of the components so that expected theorems hold true; e.g., if the solutions and are finite sets, then the number of points in is the same as the number of points in counting multiplicities. Intuitively, a solution has multiplicity greater than one when is specialized in a way such that two or more of the generic solution components specialize to an identical variety. We proceed with the formal discussion of this phenomenon. By assumption, each component of is irreducible and has dimension Let component have Cayley form and order and set Then is a Cayley form which belongs to the algebraic system In fact, is a generic point of some irreducible component of The point is in turn a generic point of some variety (correspondence) in 2-way space This new correspondence relates an element of the object variety of to a Cayley form of the solution variety Some qualification is necessary because even though any specialization of may be completed to a specialization of is not necessarily a Cayley form of It can be shown that for any specialization of but in some cases the inclusion may be strict (e.g., whenever there is more than one specialization of which corresponds to Suppose, however, that is a Cayley form for i.e. Write as a product where is the Cayley form for the irreducible component of Then we define the multiplicity of as
This definition of multiplicity is made for a solution only if there exists a unique specialization of corresponding to
([HPe1953], XI, 7, Th. I and Th. IV) The following conditions guarantee that multiplicities can be defined for components of
|1)||is a variety of pure dimension|
|2)||is a simple point of the object variety of|
We can now sketch how the theory of normal problems is used to develop an intersection calculus for the subvarieties of a nonsingular, variety (a full treatment is given in [HPe1953], XII). Let denote the free abelian group generated by the irreducible, subvarieties of The notation (or if the dimension needs emphasis) will be used to denote the element of corresponding to an irreducible subvariety A general element of can be written as or
Given and we wish to define the intersection product If we define Otherwise, let the irreducible components of be denoted by We shall define as the sum where the are multiplicities obtained from a suitably chosen normal problem. To formulate the normal problem, we require the following:
|1)||and intersect properly (i.e. the intersection is a pure variety of dimension|
There exists an irreducible algebraic system with generic Cayley form such that
Given these conditions, we consider the normal problem based on the correspondence Corresponding to the specialization of is the particular solution Conditions 1) and 2) i) guarantee that multiplicities may be defined for the irreducible components of (Theorem 3.2). is then defined as To ensure that this product is well-defined, one must verify the same multiplicities are obtained if some other algebraic system 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 and satisfying 1) and 2). We define by linear extension, whenever all basis products are define. The intersection-product so defined is commutative, associative, and distributive ([HPe1953], XII, 7, Th. I).
To overcome the difficulty of undefined products, we equip with an equivalence relation ("virtual equivalence") together with definitions of the sum and product of equivalence classes. The relation is defined in a "bootstrapping" fashion: first we define an equivalence relation ("narrow equivalence"), then a relation ("wide equivalence") in terms of and finally in terms of Let be the set of positive elements in i.e. an element such that all and some Identify an element belonging to with the Cayley form where is the Cayley form of Narrow equivalence is defined on as the smallest equivalence relation containing the set Wide equivalence is defined on as follows: If define the positive part of as Define the negative part of as Virtual equivalence is defined on as follows:
If let denote the virtual equivalence class containing We define the sum and product of virtual equivalence classes as follows: where and 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 and contain representatives whose product is defined is guaranteed by Chow's "moving lemma" ([HPe1953], XII, 10, Th. II).
The collection of virtual equivalence classes of will be denoted by is a free abelian group with the single basis element is a free abelian group with single basis element where is any point of The intersection ring of denoted by is the graded, commutative ring where sums and products are defined in the obvious way. Many geometrical properties of the variety are reflected in the algebraic structure of the ring Our narrow interest here concerns the enumerative properties of a most important result is the following ([HPe1953], XII, p.193; [KLa1972], p.1070).
If are subvarieties of which intersect properly in a finite number of points (counted with multiplicity), then We mention that an empty intersection is a proper intersection, in which case
Let be a projective in (or equivalently, a of Each such plane is an irreducible variety of dimension and order 1; the Cayley forms of such planes constitute the algebraic system The system will be called the Grassmann variety of in projective (or the Grassmann variety of of affine and will be denoted by or if is understood1.
The system is contained in a projective space of dimension and is an irreducible, nonsingular variety of projective dimension ([HPe1953], XIV, 1). Let be a in with basis matrix A point of corresponding to the Cayley form of is given by the compound of ([HPe1953], XIV, 1). Elements of are called Plucker coordinates; they satisfy a set of quadratic polynomial equations, called the Plucker relations, which define the variety (see [KLa1972]).
In section 2, the set 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 The symbol will be used both for the collection of of and for the corresponding linear subvariety of is an irreducible subvariety of projective dimension where
The Schubert varieties play a fundamental role in describing the structure of the intersection ring of The equivalence is called a Schubert cycle and will be denoted by where and This notation is consistant since if then for (this is indirectly shown in [HPe1953] in the proof of the following theorem).
([HPe1953], XIV, Th. I – "the basis theorem") The Schubert cycles freely generated 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 The rules show that the special Schubert cycles generated as a ring.
(Pieri's formula) where the sum ranges over all satisfying
(The determinataal formula) where by convention if or
Another useful rule, the duality theorem, gives the product of Schubert cycles of complementary grade. Two grades of are complementary if their sum equals
(The duality theorem) If and are Schubert cycles of complementary grade, then
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 Use the determinantal formula to write Use Pieri's formula repeatedly to obtain We could now apply the duality theorem and obtain The general intersection theory would then guarantee that three Schubert varieties of type 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 to the proof of which is delayed till the next section.
Let be the map from onto defined by and linear extension. Then is a ring isomorphism.
Recall the definition of the set given in Section 2: The intersection theory of may be used to obtain the following result.
If is an algebraically closed subfield of then
We show that
Let be three
respectively, such that
and intersect properly. Now regard
and continue to intersect properly.
Either or If then consists of a finite, nonzero number of points (when counted with multiplicity), and from Theorem 3.3 we have If let be a point in and let be the subvariety of given by the intersection of with generic passing through Then meets properly in a finite, nonzero number of points (again, counted with multiplicity). Hence so
Now suppose that Let be any three of type respectively. If this would state that and have proper intersection containing no points, which implies that a contradiction. Hence is non-empty. To show it is enough to show this intersection contains the Plucker coordinates of a which is a subspace of The variety consists of points satisfying the quadratic Plucker equations and a set of linear equations. Since are towers in the linear equations defining have all their coefficients in the field Hence the system of equations defining the intersection has all its coefficients in the algebraically closed field Since a solution exists in the extension field a solution must exist in
From this point on we assume that is algebraically closed.
If there exists such that
|2)||is a non-zero multiple of|
Let Since the product is nonzero, and so as in the proof of 3.9 there is some such that By the basis theorem, where clearly is nonzero only if Then Hence for some is non-zero multiple of and the corollary follows.
If then and so
A Schubert cycle of type has projective dimension ([HPe1953], XIV, 3); since has projective dimension a Schubert cycle of type has grade Plugging in the grades of and we get the inequality Rearranging, or
Theorem 3.11 provides a useful characterization of To examine the conjecture that 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.
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.
This is an excerpt from Steven Andrew Johnson's 1979 dissertation The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices.