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Wikipedia / 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 and, as such, is viewed as part of enumerative geometry. Giving it a more rigorous foundation was the aim of Hilbert's 15th problem. It is related to several more modern concepts, such as characteristic classes, and both its algorithmic aspects and applications remain of current interest. The term Schubert calculus is sometimes used to mean the enumerative geometry of linear subspaces of a vector space, which is roughly equivalent to describing the cohomology ring of Grassmannians. Sometimes it is used to mean the more general enumerative geometry of algebraic varieties that are homogenous spaces of simple Lie groups. Even more generally, Schubert calculus is sometimes understood as encompassing 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 further 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, consisting of associated cohomology classes, allows in particular the determination of cases in which the intersections of cells results in a finite set of points. A key result is that the Schubert cells (or rather, the classes of their Zariski closures, the Schubert cycles or Schubert varieties) span the whole cohomology ring.

The combinatorial aspects mainly arise in relation to computing intersections of Schubert cycles. 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 (as block triangular matrices).

Construction

Schubert calculus can be constructed using the Chow ring of the Grassmannian, where the generating cycles are represented by geometrically defined data.-dimensional subspaces are replaced by their projectivizations.) Choosing an (arbitrary) complete flag

, where

i.e., to each partition of weight

whose Young diagram, we associate a Schubert variety (or Schubert cycle, defined as

This is the closure, in the Zariski topology, of the Schubert cell

which is used when considering cellular homology.

may be given in terms of the dual complete flag

where

consisting of elements

, called a Schubert class, it can be written as

It can be shown that these classes are linearly independent and generate the Chow ring as their linear span. The associated intersection theory is called Schubert calculus, (i.e., a horizontal partition), are called special classes. Using the Giambelli formula below, all the Schubert classes can be generated from these special classes.

Other notational conventions

In some sources, is the complementary partition with parts

  1. ,

rectangular one (reversed, both horizontally and vertically).

is the multi-index defined by

are the pivot in reduced matricial echelon form.

Explanation

, whereas

, and so on.

.

Properties

Inclusion

. This gives the inclusion of Schubert varieties

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

Dimension formula

has codimension equal to the weight

is the weight

dimensional rectangular Young diagram.

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

, by

has the property

and the inclusion

-plane,

does as well

.

Intersection product

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

Pieri formula

given by

are the weights of the partitions. This 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

matrix having the special classes as entries.

This is known as the Giambelli formula. It has the same form as the first Jacobi-Trudi identity, expressing arbitrary Schur functions as determinants in terms of the complete symmetric functions.

For example,

and

General case

is given by

are the Littlewood-Richardson coefficients. The Pieri formula.

Relation with Chern classes

using the Chern classes of two natural vector bundles. We have the exact sequence

is the tautological bundle are

and

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

Gr(2,4)

, Schubert calculus can be used to compute 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

Lines on a cubic surface

if and only if. Therefore, the Euler class must be computed, which is given as

The splitting formula then reads as the formal equation

. The splitting equation gives the relations

  1. .

can be viewed as the direct sum of formal line bundles

whose total Chern class is

it follows that

using the fact that

is the top class, the integral is then

lines on a cubic surface.

See also

References