In mathematics, Schubert calculus[1] 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.
The objects introduced by Schubert are the Schubert cells,[2] which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag.
The intersection theory[3] 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.
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).
Schubert calculus can be constructed using the Chow ring [3] of the Grassmannian, where the generating cycles are represented by geometrically defined data.
Choosing an (arbitrary) complete flag to each weakly decreasing
, where i.e., to each partition of weight whose Young diagram fits into the
, defined as This is the closure, in the Zariski topology, of the Schubert cell[1][2] which is used when considering cellular homology instead of the Chow ring.
, called a Schubert class, does not depend on the choice of complete flag
The associated intersection theory is called Schubert calculus.
, (i.e., a horizontal partition), are called special classes.
is the solution space of a system of five independent homogeneous linear equations.
These equations will generically span when restricted to a subspace
This gives the inclusion of Schubert varieties showing an increase of the indices corresponds to an even greater specialization of subvarieties.
, by has the property and the inclusion defined by adding the extra basis element
The intersection product was first established using the Pieri and Giambelli formulas.
It has the same form as the first Jacobi-Trudi identity, expressing arbitrary Schur functions
For example, and The intersection product between any pair of Schubert classes
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
, Schubert calculus can be used to compute the number of lines on a cubic surface.
[4] The Chow ring has the presentation and as a graded Abelian group[6] it is given by Recall that a line in
to get the number of points where the generic section vanishes on
must be computed, which is given as The splitting formula then reads as the formal equation where
can be viewed as the direct sum of formal line bundles whose total Chern class is it follows that using the fact that Since