In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Schubert calculus is a branch of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
introduced in the nineteenth century by
Hermann Schubert
__NOTOC__
Hermann Cäsar Hannibal Schubert (22 May 1848 – 20 July 1911) was a German mathematician.
Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite nu ...
, in order to solve various counting problems of
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
(part of
enumerative geometry
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
History
The problem of Apollonius is one of the earliest examp ...
). It was a precursor of several more modern theories, for example
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es, 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
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
.
The objects introduced by Schubert are the Schubert cells, which are
locally closed In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied:
* E is the intersection of an open set and a closed set in X.
* For each point x\in E ...
sets in a
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
defined by conditions of
incidence of a linear subspace in projective space with a given
flag
A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
. For details see
Schubert variety In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using line ...
.
The
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
of these cells, which can be seen as the product structure in the
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
of the Grassmannian of associated
cohomology class
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
es, 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
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
, which is a
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
, to the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
that acts on it, similar questions are involved in the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle ...
and classification of
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
s (by
block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
).
Putting Schubert's system on a rigorous footing is
Hilbert's fifteenth problem
Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous foundation.
Introduction
Schubert calculus is the ...
.
Construction
Schubert calculus can be constructed using the
Chow ring
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-c ...
of the
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
where the generating cycles are represented by geometrically meaningful data.
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
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, once
, then
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 cycle 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 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 determinantal 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
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...
.
Chow ring
The Chow ring has the presentation
and as a graded Abelian group it is given by
Lines on a cubic surface
This Chow ring can be used to compute the number of lines on a cubic surface.
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
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of ...
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
*
Enumerative geometry
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
History
The problem of Apollonius is one of the earliest examp ...
*
Chow ring
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-c ...
*
Intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
*
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
*
Giambelli's formula In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes in terms of special Schubert classes, or Schur functions in terms of complete symmetric functions.
It states
:\displaystyle \sigma_\lambda= \det(\si ...
*
Pieri's formula In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function.
In terms of Schur func ...
*
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
*
Quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
*
Mirror symmetry conjecture
In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromo ...
References
* Summer school notes http://homepages.math.uic.edu/~coskun/poland.html
*
Phillip Griffiths
Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
and
Joseph Harris (1978), ''Principles of Algebraic Geometry'', Chapter 1.5
*
*
*{{eom, id=S/s130080, first=Frank, last= Sottile
*
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously serve ...
and
Joseph Harris (2016), "3264 and All That: A Second Course in Algebraic Geometry".
Algebraic geometry
Topology of homogeneous spaces