Schubert calculus

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 k-planes in a fixed n-dimensional vector space V as \mathbf{Gr}(k,V), and its Chow ring as A^*(\mathbf{Gr}(k,V)). (Note that the Grassmannian is sometimes denoted \mathbf{Gr}(k,n) if the vector space isn't explicitly given or as \mathbb{G}(k-1,n-1) if the ambient space V and its k-dimensional subspaces are replaced by their projectivizations.) Choosing an (arbitrary) complete flag ::\mathcal{V} = (V_1 \subset \cdots \subset V_{n-1} \subset V_n = V), \quad \dim{V}_i = i, \quad i=1 , \dots , n,

to each weakly decreasing k-tuple of integers \mathbf{a} = (a_1,\ldots, a_k), where

i.e., to each partition of weight :: |\mathbf{a}| = \sum_{i=1}^k a_i,

whose Young diagram fits into the k \times (n-k) rectangular one for the partition (n-k)^k, we associate a Schubert variety (or Schubert cycle) \Sigma_{\mathbf{a}}(\mathcal{V}) \subset \mathbf{Gr}(k,V), defined as ::\Sigma_{\mathbf{a}}(\mathcal{V}) = { w \in \mathbf{Gr}(k,V) : \dim (V_{n-k +i - a_i} \cap w) \geq i \text{ for } i = 1, \dots, k }.

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 |\mathbf{a}|, whose union is \mathbf{Gr}(k,V).

An equivalent characterization of the Schubert cell X_\mathbf{a}(\mathcal{V}) may be given in terms of the dual complete flag

: \tilde{\mathcal{V}}= (\tilde{V}_1 \subset \tilde{V}_2 \cdots \subset \tilde{V}_n = V),

where : \tilde{V}i := V_n\backslash V{n-i}, \quad i=1, \dots, n \quad (V_0:= \emptyset).

Then X_{\mathbf{a}}(\mathcal{V}) \subset \mathbf{Gr}(k,V) consists of those k-dimensional subspaces w \subset V that have a basis (\tilde{W}_1, \dots, \tilde{W}_k) consisting of elements :: \tilde{W}i \in \tilde{V}{k+a_i -i +1 }, \quad i=1, \dots, k

of the subspaces {\tilde{V}{k+a_i -i +1}}{i=1, \dots, k}.

Since the homology class [\Sigma_{\mathbf{a}}(\mathcal{V})] \in A^(\mathbf{Gr}(k,V)), called a Schubert class, does not depend on the choice of complete flag \mathcal{V}, it can be written as ::\sigma_{\mathbf{a}} := [\Sigma_{\mathbf{a}}] \in A^(\mathbf{Gr}(k,V)). 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 \mathbf{a} = (a_1,\ldots, a_j, 0, \ldots, 0) with a_j0 the Schubert class \sigma_{(a_1,\ldots, a_j,0,\ldots,0)} is usually just denoted \sigma_{(a_1,\ldots, a_j)}. The Schubert classes given by a single integer \sigma_{a_1}, (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 X_\mathbf{a} and Schubert varieties \Sigma_\mathbf{a} are labelled differently, as S_\lambda and \bar{S}\lambda, respectively, where \lambda is the complementary partition to \mathbf{a} with parts :: \lambda_i := n-k-a{k-i+1} , whose Young diagram is the complement of the one for \mathbf{a} within the k \times (n-k) rectangular one (reversed, both horizontally and vertically).

Another labelling convention for X_\mathbf{a} and \Sigma_\mathbf{a} is C_L and \bar{C}L, respectively, where L= (L_1, \dots, L_k) \subset (1, \dots, n) is the multi-index defined by :: L_i := n-k - a_i +i=\lambda{k-i+1} +i.

The integers (L_1, \dots, L_k) are the pivot locations of the representations of elements of X_{\mathbf{a}} in reduced matricial echelon form.

Explanation

In order to explain the definition, consider a generic k-plane w \subset V. It will have only a zero intersection with V_j for j \leq n-k, whereas :: \dim(V_{j} \cap w) = i for j = n-k +i \geq n-k. For example, in \mathbf{Gr}(4,9), a 4-plane w is the solution space of a system of five independent homogeneous linear equations. These equations will generically span when restricted to a subspace V_j with j=\dim V_j \leq 5=9-4, in which case the solution space (the intersection of V_j with w) will consist only of the zero vector. However, if \dim(V_j) + \dim(w) n=9, V_j and w will necessarily have nonzero intersection. For example, the expected dimension of intersection of V_6 and w is 1, the intersection of V_7 and w has expected dimension 2, and so on.

The definition of a Schubert variety states that the first value of j with \dim(V_{j} \cap w) \geq i is generically smaller than the expected value n-k +i by the parameter a_i . The k-planes w \subset V given by these constraints then define special subvarieties of \mathbf{Gr}(k,n).

Properties

Inclusion

There is a partial ordering on all k-tuples where \mathbf{a} \geq \mathbf{b} if a_i \geq b_i for every i. This gives the inclusion of Schubert varieties :: \Sigma_{\mathbf{a}} \subset \Sigma_{\mathbf{b}} \iff \mathbf{a} \geq \mathbf{b},

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

Dimension formula

A Schubert variety \Sigma_{\mathbf{a}} has codimension equal to the weight ::|\mathbf{a}|=\sum a_i

of the partition \mathbf{a}. Alternatively, in the notational convention S_\lambda indicated above, its dimension in \mathbf{Gr}(k,n) is the weight :: |\lambda|= \sum_{i=1}^k\lambda_i = k(n-k)-|\mathbf{a}|.

of the complementary partition \lambda \subset (n-k)^k in the k \times (n-k) dimensional rectangular Young diagram.

This is stable under inclusions of Grassmannians. That is, the inclusion :: i_{(k, n)}:\mathbf{Gr}(k, \mathbf{C}^n) \hookrightarrow \mathbf{Gr}(k, \mathbf{C}^{n+1}), \quad \mathbf{C}^n =\text{span}{e_1, \dots, e_n} defined, for w\in\mathbf{Gr}(k, \mathbf{C}^n) , by

has the property

and the inclusion :: \tilde{i}{(k,n)}: \mathbf{Gr}(k,n) \hookrightarrow \mathbf{Gr}(k+1,n+1) defined by adding the extra basis element e{n+1} to each k-plane, giving a (k+1)-plane, ::\tilde{i}{(k, n)}: w \mapsto w \oplus\mathbf {C} e{n+1} \subset \mathbf{C}^n \oplus \mathbf {C} e_{n+1} =\mathbf{C}^{n+1}

does as well ::\tilde{i}{(k,n)}^*(\sigma{\mathbf{a}}) = \sigma_{\mathbf{a}}.

Thus, if X_{\mathbf{a}} \subset \mathbf{Gr}k(n) and \Sigma{\mathbf{a}} \subset \mathbf{Gr}k(n) are a cell and a subvariety in the Grassmannian \mathbf{Gr}k(n), they may also be viewed as a cell X{\mathbf{a}} \subset \mathbf{Gr}\tilde{k}(\tilde{n}) and a subvariety \Sigma_{\mathbf{a}} \subset \mathbf{Gr}\tilde{k}(\tilde{n}) within the Grassmannian \mathbf{Gr}\tilde{k}(\tilde{n}) for any pair (\tilde{k}, \tilde{n}) with \tilde{k} \geq k and \tilde{n}-\tilde{k} \geq n-k.

Intersection product

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

Pieri formula

In the special case \mathbf{b} = (b,0,\ldots, 0), there is an explicit formula of the product of \sigma_b with an arbitrary Schubert class \sigma_{a_1,\ldots, a_k} given by ::\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_{\mathbf{c}},

where |\mathbf{a}| = a_1 + \cdots + a_k, |\mathbf{c}| = c_1 + \cdots + c_k 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, ::\sigma_1 \cdot \sigma_{4,2,1} = \sigma_{5,2,1} + \sigma_{4,3,1} + \sigma_{4,2,1,1}.

and ::\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 \sigma_{\mathbf{a}} for partitions of any length \ell(\mathbf{a})\leq k can be expressed as the determinant of a (k \times k) matrix having the special classes as entries. :: \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}

This is known as the Giambelli formula. It has the same form as the first Jacobi-Trudi identity, expressing arbitrary Schur functions s_{\mathbf{a}} as determinants in terms of the complete symmetric functions {h_j := s_{(j)}}.

For example, ::\sigma_{2,2} = \begin{vmatrix} \sigma_2 & \sigma_3 \ \sigma_1 & \sigma_2 \end{vmatrix} = \sigma_2^2 - \sigma_1\cdot\sigma_3

and ::\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}.

General case

The intersection product between any pair of Schubert classes \sigma_{\mathbf{a}}, \sigma_{\mathbf{b}} is given by :: \sigma_{\mathbf{a}} \sigma_{\mathbf{b}} =\sum_{\mathbf{c}}c^\mathbf{c}{\mathbf{a} \mathbf{b}}\sigma{\mathbf{c}},

where {c^\mathbf{c}_{\mathbf{a} \mathbf{b}}} are the Littlewood-Richardson coefficients. The Pieri formula is a special case of this, when \mathbf{b}=(b,0, \dots, 0) has length \ell(\mathbf{b})=1.

Relation with Chern classes

There is an easy description of the cohomology ring, or the Chow ring, of the Grassmannian \mathbf{Gr}(k,V) using the Chern classes of two natural vector bundles over \mathbf{Gr}(k,V). We have the exact sequence of vector bundles over \mathbf{Gr}(k,V) ::0 \to T \to \underline{V} \to Q \to 0

where T is the tautological bundle whose fiber, over any element w \in \mathbf{Gr}(k, V) is the subspace w \subset V itself, , \underline{V}:= \mathbf{Gr}(k,V) \times V is the trivial vector bundle of rank n, with V as fiber and Q is the quotient vector bundle of rank n-k, with V/w as fiber. The Chern classes of the bundles T and Q are

where (1)^i is the partition whose Young diagram consists of a single column of length i and

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

Gr(2,4)

One of the classical examples analyzed is the Grassmannian \mathbf{Gr}(2,4) since it parameterizes lines in \mathbb{P}^3. Using the Chow ring A^*(\mathbf{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 :: \begin{align} A^0(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot 1 \ A^2(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot \sigma_1 \ A^4(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot \sigma_2 \oplus \mathbb{Z} \cdot \sigma_{1,1}\ A^6(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot\sigma_{2,1} \ A^8(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot\sigma_{2,2} \ \end{align}

Lines on a cubic surface

Recall that a line in \mathbb{P}^3 gives a dimension 2 subspace of \mathbb{A}^4, hence an element of \mathbb{G}(1,3) \cong \mathbf{Gr}(2,4). Also, the equation of a line can be given as a section of \Gamma(\mathbb{G}(1,3), T^). Since a cubic surface X is given as a generic homogeneous cubic polynomial, this is given as a generic section s \in \Gamma(\mathbb{G}(1,3),\text{Sym}^3(T^)). A line L \subset \mathbb{P}^3 is a subvariety of X if and only if the section vanishes on [L] \in \mathbb{G}(1,3). Therefore, the Euler class of \text{Sym}^3(T^) can be integrated over \mathbb{G}(1,3) to get the number of points where the generic section vanishes on \mathbb{G}(1,3). In order to get the Euler class, the total Chern class of T^ must be computed, which is given as

The splitting formula then reads as the formal equation ::\begin{align} c(T^*) &= (1 + \alpha)(1 + \beta) \ &= 1 + \alpha + \beta + \alpha\cdot\beta \end{align},

where c(\mathcal{L}) = 1+\alpha and c(\mathcal{M}) = 1 + \beta for formal line bundles \mathcal{L},\mathcal{M}. The splitting equation gives the relations ::\sigma_1 = \alpha + \beta and \sigma_{1,1} = \alpha\cdot\beta.

Since \text{Sym}^3(T^) can be viewed as the direct sum of formal line bundles ::\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

it follows that ::\begin{align} 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{align}

using the fact that ::\sigma_{1,1}\cdot \sigma_1^2 = \sigma_{2,1}\sigma_1 = \sigma_{2,2} and \sigma_{1,1}\cdot \sigma_{1,1} = \sigma_{2,2}.

Since \sigma_{2,2} is the top class, the integral is then ::\int_{\mathbb{G}(1,3)}27\sigma_{2,2} = 27.

Therefore, there are 27 lines on a cubic surface.

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".

References

1. (1972). "Schubert Calculus". American Mathematical Society.
2. Fulton, William. (1997). "Young Tableaux. With Applications to Representation Theory and Geometry, Chapt. 9.4". Cambridge University Press.
3. (1998). "Intersection Theory". [[Springer-Verlag]].
4. "3264 and All That".
5. Fulton, William. (1997). "Young Tableaux. With Applications to Representation Theory and Geometry, Chapt. 5". Cambridge University Press.
6. Katz, Sheldon. "Enumerative Geometry and String Theory".