Schubert calculus

In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties. Even more generally, “Schubert calculus” is often understood to encompass the study of analogous questions in generalized cohomology theories.

The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety.

The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring.

In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (by block matrix).

Putting Schubert's system on a rigorous footing is Hilbert's fifteenth problem.

Construction

Schubert calculus can be constructed using the Chow ring of the Grassmannian where the generating cycles are represented by geometrically meaningful data.[1] Denote as the Grassmannian of -planes in a fixed -dimensional vector space , and its Chow ring; note that sometimes the Grassmannian is denoted as if the vector space isn't explicitly given. Associated to an arbitrary complete flag

and a decreasing -tuple of integers where

there are Schubert cycles (which are called Schubert cells when considering cellular homology instead of the Chow ring) defined as

Since the class does not depend on the complete flag, the class can be written as

which are called Schubert classes. It can be shown these classes generate the Chow ring, and the associated intersection theory is called Schubert calculus. Note given a sequence the Schubert class is typically denoted as just . Also, the Schubert classes given by a single integer, , are called special classes. Using the Giambeli formula below, all of the Schubert classes can be generated from these special classes.

Explanation of definition

Initially the definition looks a little bit awkward. Given a generic -plane it will have only a zero intersection with for and for . For example, in given a -plane , this is cut out by a system of five linear equations. The -plane is not guaranteed to intersect at anywhere other than the origin since there are five free parameters it could live in. Furthermore, once , then they necessarily intersect. This means, the expected dimension of intersection of and should have dimension , the intersection of and should have dimension , and so on. These cycles then define special subvarieties of .

Inclusion

There is a partial ordering on all -tuples where if for every . This gives the inclusion of Schubert cycles

showing an increase of the indices corresponds to an even greater specialization of subvarieties.

Codimension formula

A Schubert cycle has codimension

which is stable under inclusions of Grassmannians. That is, the inclusion

given by adding the extra basis element to each -plane, giving a -plane, has the property

Also, the inclusion

given by inclusion of the -plane has the same pullback property.

Intersection product

The intersection product was first established using the Pieri and Giambelli formulas.

Pieri formula

In the special case , there is an explicit formula of the product of with an arbitrary Schubert class given by

Note . This formula is called the Pieri formula and can be used to determine the intersection product of any two Schubert classes when combined with the Giambelli formula. For example

and

Giambelli formula

Schubert classes with tuples of length two or more can be described as a determinental equation using the classes of only one tuple. The Giambelli formula reads as the equation

given by the determinant of a -matrix. For example,

and

Relation with Chern classes

There is an easy description of the cohomology ring, or the Chow ring, of the Grassmannian using the Chern classes of two natural vector bundles over the grassmannian . There is a sequence of vector bundles

where is the trivial vector bundle of rank , the fiber of over is the subspace , and is the quotient vector bundle (which exists since the rank is constant on each of the fibers). The Chern classes of these two associated bundles are

where is an -tuple and

The tautological sequence then gives the presentation of the Chow ring as

G(2,4)

One of the classical examples analyzed is the Grassmannian since it parameterizes lines in . Schubert calculus can be used to find the number of lines on a Cubic surface.

Chow ring

The Chow ring has the presentation

and as a graded Abelian group it is given by

[2]

Lines on a cubic surface

This Chow ring can be used to compute the number of lines on a cubic surface.[1] Recall a line in gives a dimension two subspace of , hence . Also, the equation of a line can be given as a section of . Since a cubic surface is given as a generic homogeneous cubic polynomial, this is given as a generic section . Then, a line is a subvariety of if and only if the section vanishes on . Therefore, the Euler class of can be integrated over to get the number of points where the generic section vanishes on . In order to get the Euler class, the total Chern class of must be computed, which is given as

Then, the splitting formula reads as the formal equation

where and for formal line bundles . The splitting equation gives the relations

and .

Since can be read as the direct sum of formal vector bundles

whose total Chern class is

hence

using the fact

and

Then, the integral is

since is the top class. Therefore there are lines on a cubic surface.

See also

References

  1. 3264 and All That (PDF). pp. 132, section 4.1, 200, section 6.2.1.
  2. Katz, Sheldon. Enumerative Geometry and String Theory. p. 96.
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