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 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 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 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 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 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 traingular matrices). Putting Schubert's system on a rigorous footing was 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 defined data. Denote the Grassmannian of -planes in a fixed -dimensional vector space  as , and its Chow ring as . Note that the Grassmannian is sometimes denoted  if the vector space isn't explicitly given or as  if the ambient space  and its -dimensional subspaces are replaced by their projectizations. Choosing an (arbitrary) complete flag 

to each weakly decreasing -tuple of integers , where

i.e., to each partition of weight
 

whose Young diagram fits into the rectangular one , 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 instead of the Chow ring. The latter are disjoint affine spaces, of dimension , whose union is . Since the homology class , called a Schubert class, does not depend on the choice of complete flag , 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. For a given sequence  with  the Schubert class  is usually just denoted . The Schubert classes given by a single integer , (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, the Schubert cells  and Schubert varieties are labelled differently, as  and , respectively,  where  is the complementary partition, to , with parts
,
whose Young diagram is the complement of the one for  within the  rectangular one (reversed, both horizontally and vertically). Another labelling convention for  and  is  and  
, respectively, where

is the multi-index defined by
 
or, alternatively, by
 . Explanation 
In order to explain the definition, consider a generic -plane . It will have only a zero intersection with  for , whereas
  for  
For example, in , a -plane  is the solution space of a system of five independent homogeneous linear equations. These equations will generically span when restricted to a subspace  with , in which case the solution space (the intersection of   with  ) will consist only of the zero vector. However, if ,  and  will necessarily have nonzero intersection. For example, the expected dimension of intersection of  and  is , the intersection of  and  has expected dimension , and so on. The definition of a Schubert variety states that the first value of  with  is generically smaller than the expected value  by the parameter . The -planes  given by these constraints then define special subvarieties of . Properties

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

showing an increase of the indices corresponds to an even greater specialization of subvarieties. Dimension formula 
A Schubert variety  has dimension equal to the weight 
 

of the partition . Alternatively, in the notational convention  indicated above, its codimension in  is the weight 
    

of the complementary partition  in the  dimensional rectangular Young diagram. This is stable under inclusions of Grassmannians. That is, the inclusion 
 
defined, for , by
 
has the property 

and the inclusion
 
defined by adding the extra basis element  to each -plane, giving a -plane, 
 

does as well

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

where ,  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 
Schubert classes  for partitions of any length  can be expressed as the determinant of a  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 

The intersection product between any pair of Schubert classes
 is given by

where  are the  Littlewood-Richardson coefficients. The Pieri formula is a special case of this, when  has length . 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 . We have the exact sequence of vector bundles over 

where  is the tautological bundle whose fiber, over any element  is the subspace  itself,    is the trivial vector bundle of rank ,  with  as fiber and  is the quotient vector bundle of rank , with  as fiber. The Chern classes of the bundles  and   are

where  is the partition whose Young diagram consists of a single column of length  and

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

One of the classical examples analyzed is the Grassmannian  since it parameterizes lines in . Using the Chow ring , Schubert calculus can be used to compute the number of lines on a cubic surface 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 
Recall that a line in  gives a dimension two subspace of , hence an element of . 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 . 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

The splitting formula then reads as the formal equation

where  and  for formal line bundles . The splitting equation gives the relations
 and . Since  can be viewed as the direct sum of formal line bundles

whose total Chern class is

it follows that

using the fact that
 and 

Since  is the top class, the integral is then

Therefore, there are  lines on a cubic surface. See also 

 Enumerative geometry
 Chow ring
 Intersection theory
 Grassmannian
 Giambelli's formula
 Pieri's formula
 Chern class
 Quintic threefold
 Mirror symmetry conjecture

References

 Summer school notes http://homepages.math.uic.edu/~coskun/poland.html
Phillip Griffiths and Joseph Harris (1978), Principles of Algebraic Geometry, Chapter 1.5
 

David Eisenbud and Joseph Harris (2016), "3264 and All That: A Second Course in Algebraic Geometry". Algebraic geometry
Topology of homogeneous spaces