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 ${\displaystyle G(k,V)}$  as the Grassmannian of ${\displaystyle k}$ -planes in a fixed ${\displaystyle n}$ -dimensional vector space ${\displaystyle V}$ , and ${\displaystyle A^{*}(G(k,V))}$  its Chow ring; note that sometimes the Grassmannian is denoted as ${\displaystyle G(k,n)}$  if the vector space isn't explicitly given. Associated to an arbitrary complete flag ${\displaystyle {\mathcal {V}}}$ 

${\displaystyle 0\subset V_{1}\subset \cdots \subset V_{n-1}\subset V_{n}=V}$ 

and a decreasing ${\displaystyle k}$ -tuple of integers ${\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{k})}$  where

${\displaystyle n-k\geq a_{1}\geq a_{2}\geq \cdots \geq a_{k}\geq 0}$ 

there are Schubert cycles (which are called Schubert cells when considering cellular homology instead of the Chow ring) ${\displaystyle \Sigma _{\mathbf {a} }({\mathcal {V}})\subset G(k,V)}$  defined as

${\displaystyle \Sigma _{\mathbf {a} }({\mathcal {V}})=\{\Lambda \in G(k,V):\dim(V_{n-k+i-a_{i}}\cap \Lambda )\geq i{\text{ for all }}i\geq 1\}}$ 

Since the class ${\displaystyle [\Sigma _{\mathbb {a} }({\mathcal {V}})]\in A^{*}(G(k,V))}$  does not depend on the complete flag, the class can be written as

${\displaystyle \sigma _{\mathbb {a} }:=[\Sigma _{\mathbb {a} }]\in A^{*}(G(k,V))}$ 

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 ${\displaystyle \mathbb {a} =(a_{1},\ldots ,a_{j},0,\ldots ,0)}$  the Schubert class ${\displaystyle \sigma _{(a_{1},\ldots ,a_{j},0,\ldots ,0)}}$  is typically denoted as just ${\displaystyle \sigma _{(a_{1},\ldots ,a_{j})}}$ . Also, the Schubert classes given by a single integer, ${\displaystyle \sigma _{a_{1}}}$ , 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 ${\displaystyle k}$ -plane ${\displaystyle \Lambda \subset V}$ : it will have only a zero intersection with ${\displaystyle V_{j}}$  for ${\displaystyle j\leq n-k}$ , whereas ${\displaystyle \dim(V_{j}\cap \Lambda )=i}$  for ${\displaystyle j=n-k+i\geq n-k}$ . For example, in ${\displaystyle G(4,9)}$ , a ${\displaystyle 4}$ -plane ${\displaystyle \Lambda }$  is the solution space of a system of five independent homogeneous linear equations. These equations will generically span when restricted to a subspace ${\displaystyle V_{j}}$  with ${\displaystyle j=\dim V_{j}\leq 5=9-4}$ , in which case the solution space (the intersection of ${\displaystyle V_{j}}$  with ${\displaystyle \Lambda }$ ) will consist only of the zero vector. However, once ${\displaystyle \dim(V_{j})+\dim(\Lambda )>n=9}$ , then ${\displaystyle V_{j}}$  and ${\displaystyle \Lambda }$  will necessarily have nonzero intersection. For example, the expected dimension of intersection of ${\displaystyle V_{6}}$  and ${\displaystyle \Lambda }$  is ${\displaystyle 1}$ , the intersection of ${\displaystyle V_{7}}$  and ${\displaystyle \Lambda }$  has expected dimension ${\displaystyle 2}$ , and so on.

The definition of a Schubert cycle states that the first value of ${\displaystyle j}$  with ${\displaystyle \dim(V_{j}\cap \Lambda )\geq i}$  is generically smaller than the expected value ${\displaystyle n-k+i}$  by the parameter ${\displaystyle a_{i}}$ . The ${\displaystyle k}$ -planes ${\displaystyle \Lambda \subset V}$  given by these constraints then define special subvarieties of ${\displaystyle G(k,n)}$ .^[1]

Properties

Inclusion

There is a partial ordering on all ${\displaystyle k}$ -tuples where ${\displaystyle \mathbb {a} \geq \mathbb {b} }$  if ${\displaystyle a_{i}\geq b_{i}}$  for every ${\displaystyle i}$ . This gives the inclusion of Schubert cycles

${\displaystyle \Sigma _{\mathbb {a} }\subset \Sigma _{\mathbb {b} }\iff a\geq b}$ 

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

Codimension formula

A Schubert cycle ${\displaystyle \Sigma _{\mathbb {a} }}$  has codimension

${\displaystyle \sum a_{i}}$ 

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

${\displaystyle i:G(k,n)\hookrightarrow G(k+1,n+1)}$ 

given by adding the extra basis element ${\displaystyle e_{n+1}}$  to each ${\displaystyle k}$ -plane, giving a ${\displaystyle (k+1)}$ -plane, has the property

${\displaystyle i^{*}(\sigma _{\mathbb {a} })=\sigma _{\mathbb {a} }}$ 

Also, the inclusion

${\displaystyle j:G(k,n)\hookrightarrow G(k,n+1)}$ 

given by inclusion of the ${\displaystyle k}$ -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 ${\displaystyle \mathbb {b} =(b,0,\ldots ,0)}$ , there is an explicit formula of the product of ${\displaystyle \sigma _{b}}$  with an arbitrary Schubert class ${\displaystyle \sigma _{a_{1},\ldots ,a_{k}}}$  given by

${\displaystyle \sigma _{b}\cdot \sigma _{a_{1},\ldots ,a_{k}}=\sum _{\begin{matrix}|c|=|a|+b\\a_{i}\leq c_{i}\leq a_{i-1}\end{matrix}}\sigma _{\mathbb {c} }}$ 

Note ${\displaystyle |\mathbb {a} |=a_{1}+\cdots +a_{k}}$ . 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

${\displaystyle \sigma _{1}\cdot \sigma _{4,2,1}=\sigma _{5,2,1}+\sigma _{4,3,1}+\sigma _{4,2,1,1}}$ 

and

${\displaystyle \sigma _{2}\cdot \sigma _{4,3}=\sigma _{4,3,2}+\sigma _{4,4,1}+\sigma _{5,3,1}+\sigma _{5,4}+\sigma _{6,3}}$ 

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

${\displaystyle \sigma _{(a_{1},\ldots ,a_{k})}={\begin{vmatrix}\sigma _{a_{1}}&\sigma _{a_{1}+1}&\sigma _{a_{1}+2}&\cdots &\sigma _{a_{1}+k-1}\\\sigma _{a_{2}-1}&\sigma _{a_{2}}&\sigma _{a_{2}+1}&\cdots &\sigma _{a_{2}+k-2}\\\sigma _{a_{3}-2}&\sigma _{a_{3}-1}&\sigma _{a_{3}}&\cdots &\sigma _{a_{3}+k-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\\sigma _{a_{k}-k+1}&\sigma _{a_{k}-k+2}&\sigma _{a_{k}-k+3}&\cdots &\sigma _{a_{k}}\end{vmatrix}}}$ 

given by the determinant of a ${\displaystyle (k,k)}$ -matrix. For example,

${\displaystyle \sigma _{2,2}={\begin{vmatrix}\sigma _{2}&\sigma _{3}\\\sigma _{1}&\sigma _{2}\end{vmatrix}}=\sigma _{2}^{2}-\sigma _{1}\cdot \sigma _{3}}$ 

and

${\displaystyle \sigma _{2,1,1}={\begin{vmatrix}\sigma _{2}&\sigma _{3}&\sigma _{4}\\\sigma _{0}&\sigma _{1}&\sigma _{2}\\0&\sigma _{0}&\sigma _{1}\end{vmatrix}}}$ 

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 ${\displaystyle G(k,n)}$ . There is a sequence of vector bundles

${\displaystyle 0\to T\to {\underline {V}}\to Q\to 0}$ 

where ${\displaystyle {\underline {V}}}$  is the trivial vector bundle of rank ${\displaystyle n}$ , the fiber of ${\displaystyle T}$  over ${\displaystyle \Lambda \in G(k,n)}$  is the subspace ${\displaystyle \Lambda \subset V}$ , and ${\displaystyle Q}$  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

${\displaystyle c_{i}(T)=(-1)^{i}\sigma _{(1,\ldots ,1)}}$ 

where ${\displaystyle (1,\ldots ,1)}$  is an ${\displaystyle i}$ -tuple and

${\displaystyle c_{i}(Q)=\sigma _{i}}$ 

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

${\displaystyle A^{*}(G(k,n))={\frac {\mathbb {Z} [c_{1}(T),\ldots ,c_{k}(T),c_{1}(Q),\ldots ,c_{n-k}(Q)]}{(c(T)c(Q)-1)}}}$ 

G(2,4)

One of the classical examples analyzed is the Grassmannian ${\displaystyle G(2,4)}$  since it parameterizes lines in ${\displaystyle \mathbb {P} ^{3}}$ . Schubert calculus can be used to find the number of lines on a Cubic surface.

Chow ring

The Chow ring has the presentation

${\displaystyle A^{*}(G(2,4))={\frac {\mathbb {Z} [\sigma _{1},\sigma _{1,1},\sigma _{2}]}{((1-\sigma _{1}+\sigma _{1,1})(1+\sigma _{1}+\sigma _{2})-1)}}}$ 

and as a graded Abelian group it is given by

${\displaystyle {\begin{aligned}A^{0}(G(2,4))&=\mathbb {Z} \cdot 1\\A^{2}(G(2,4))&=\mathbb {Z} \cdot \sigma _{1}\\A^{4}(G(2,4))&=\mathbb {Z} \cdot \sigma _{2}\oplus \mathbb {Z} \cdot \sigma _{1,1}\\A^{6}(G(2,4))&=\mathbb {Z} \cdot \sigma _{2,1}\\A^{8}(G(2,4))&=\mathbb {Z} \cdot \sigma _{2,2}\\\end{aligned}}}$ ^[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 ${\displaystyle \mathbb {P} ^{3}}$  gives a dimension two subspace of ${\displaystyle \mathbb {A} ^{4}}$ , hence ${\displaystyle \mathbb {G} (1,3)\cong G(2,4)}$ . Also, the equation of a line can be given as a section of ${\displaystyle \Gamma (\mathbb {G} (1,3),T^{*})}$ . Since a cubic surface ${\displaystyle X}$  is given as a generic homogeneous cubic polynomial, this is given as a generic section ${\displaystyle s\in \Gamma (\mathbb {G} (1,3),{\text{Sym}}^{3}(T^{*}))}$ . Then, a line ${\displaystyle L\subset \mathbb {P} ^{3}}$  is a subvariety of ${\displaystyle X}$  if and only if the section vanishes on ${\displaystyle [L]\in \mathbb {G} (1,3)}$ . Therefore, the Euler class of ${\displaystyle {\text{Sym}}^{3}(T^{*})}$  can be integrated over ${\displaystyle \mathbb {G} (1,3)}$  to get the number of points where the generic section vanishes on ${\displaystyle \mathbb {G} (1,3)}$ . In order to get the Euler class, the total Chern class of ${\displaystyle T^{*}}$  must be computed, which is given as

${\displaystyle c(T^{*})=1+\sigma _{1}+\sigma _{1,1}}$ 

Then, the splitting formula reads as the formal equation

${\displaystyle {\begin{aligned}c(T^{*})&=(1+\alpha )(1+\beta )\\&=1+\alpha +\beta +\alpha \cdot \beta \end{aligned}}}$ 

where ${\displaystyle c({\mathcal {L}})=1+\alpha }$  and ${\displaystyle c({\mathcal {M}})=1+\beta }$  for formal line bundles ${\displaystyle {\mathcal {L}},{\mathcal {M}}}$ . The splitting equation gives the relations

${\displaystyle \sigma _{1}=\alpha +\beta }$  and ${\displaystyle \sigma _{1,1}=\alpha \cdot \beta }$ .

Since ${\displaystyle {\text{Sym}}^{3}(T^{*})}$  can be read as the direct sum of formal vector bundles

${\displaystyle {\text{Sym}}^{3}(T^{*})={\mathcal {L}}^{\otimes 3}\oplus ({\mathcal {L}}^{\otimes 2}\otimes {\mathcal {M}})\oplus ({\mathcal {L}}\otimes {\mathcal {M}}^{\otimes 2})\oplus {\mathcal {M}}^{\otimes 3}}$ 

whose total Chern class is

${\displaystyle c({\text{Sym}}^{3}(T^{*}))=(1+3\alpha )(1+2\alpha +\beta )(1+\alpha +2\beta )(1+3\beta )}$ 

hence

${\displaystyle {\begin{aligned}c_{4}({\text{Sym}}^{3}(T^{*}))&=3\alpha (2\alpha +\beta )(\alpha +2\beta )3\beta \\&=9\alpha \beta (2(\alpha +\beta )^{2}+\alpha \beta )\\&=9\sigma _{1,1}(2\sigma _{1}^{2}+\sigma _{1,1})\\&=27\sigma _{2,2}\end{aligned}}}$ 

using the fact

${\displaystyle \sigma _{1,1}\cdot \sigma _{1}^{2}=\sigma _{2,1}\sigma _{1}=\sigma _{2,2}}$  and ${\displaystyle \sigma _{1,1}\cdot \sigma _{1,1}=\sigma _{2,2}}$ 

Then, the integral is

${\displaystyle \int _{\mathbb {G} (1,3)}27\sigma _{2,2}=27}$ 

since ${\displaystyle \sigma _{2,2}}$  is the top class. Therefore there are ${\displaystyle 27}$  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

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.

• Summer school notes http://homepages.math.uic.edu/~coskun/poland.html
• Phillip Griffiths and Joseph Harris (1978), Principles of Algebraic Geometry, Chapter 1.5
• Kleiman, Steven (1976). "Rigorous foundations of Schubert's enumerative calculus". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. Vol. XXVIII.2. American Mathematical Society. pp. 445–482. ISBN 0-8218-1428-1.
• Steven Kleiman and Dan Laksov (1972). "Schubert calculus" (PDF). American Mathematical Monthly. 79: 1061–1082. doi:10.2307/2317421.
• Sottile, Frank (2001) [1994], "Schubert calculus", Encyclopedia of Mathematics, EMS Press
• David Eisenbud and Joseph Harris (2016), "3264 and All That: A Second Course in Algebraic Geometry".