Schubert calculus

The term Schubert calculus refers to the intersection theory of Grassmannians. This is to honor Hermann Schubert, a German 18th century mathematician who solved a variety of problems in enumerative geometry by reducing them to the combinatorics and intersection theory of certain cycles on Grassmannians. These cycles are nowadays called Schubert cycles, their cycle classes are called Schubert classes.

We show how to define such classes in OSCAR.

To recall the definition of Schubert cycles on a Grassmannian $\mathrm{Gr}(k,n)$, we think of $\mathrm{Gr}(k, n)$ as the Grassmannian $\mathrm{Gr}(k, W)$ of $k$-dimensional subspaces of an $n$-dimensional $\mathbb{Q}$-vector space $W$.

A flag in $W$ is a strictly increasing sequence of linear subspaces

\[\{0\} \subset W_1 \subset \dots \subset W_{n-1} \subset W_n = W, \; \text{ with }\; \dim(W_i) = i.\]

Let such a flag $\mathcal{W}$ be given. For any sequence $a = (a_1, \ldots, a_k)$ of integers with $n-k \geq a_1 \geq \ldots \geq a_k \geq 0$, we define the Schubert cycle $\Sigma_a(\mathcal{W})$ by setting

\[\Sigma_a(\mathcal{W}):= \{ V \in \mathrm{Gr}(k, W)\mid \dim(W_{n-k+i-a_i} \cap V) \geq i, i = 1, \ldots, k \} \,.\]

Then we have:

• The Schubert cycle $\Sigma_a(\mathcal{W})$ is an irreducible subvariety of $\mathrm{Gr}(k, W)$ of codimension $|a| = \sum a_i$.
• Its cycle class $[\Sigma_a(\mathcal{W})]$ does not depend on the choice of the flag.

We define the Schubert class of $a = (a_1, \ldots, a_k)$ to be the cycle class

\[\sigma_a := [\Sigma_a(\mathcal{W})] \,.\]

The number of Schubert classes on the Grassmannian $\mathrm{Gr}(k, n)$ is equal to $\binom{n}{k}$.

Instead of $\sigma_a$, we write $\sigma_{a_1,\ldots,a_s}$ whenever $a = (a_1, \ldots, a_s, 0, \ldots, 0)$, and $\sigma_{p^i}$ whenever $a = (p, \ldots, p, 0, \ldots, 0) \in \mathbb Z^i \times \{0\}^{k-i}$.

The classes $\sigma_{1^i}$, $i = 1, \ldots, k$, and $\sigma_i$, $i = 1, \ldots, n-k$, are called special Schubert classes. They are closely related to the Chern classes of the tautological vector bundles on $\mathrm{Gr}(k, W)$. Recall:

• The tautological subbundle on $\mathrm{Gr}(k, W)$ is the vector bundle of rank $k$ whose fiber at $V \in \mathrm{Gr}(k, W)$ is the subspace $V \subset W$.
• The tautological quotient bundle on $\mathrm{Gr}(k, W)$ is the vector bundle of rank $(n-k)$ whose fiber at $V \in \mathrm{Gr}(k, W)$ is the quotient vector space $W/V$.

We denote these vector bundles by $S$ and $Q$, respectively.

The Chern classes of $S$ and $Q$ are

\[\operatorname{c}_i(S) = (-1)^i \sigma_{1^i} \; \text{ for } \; i = 1, \ldots, k\]

and

\[\operatorname{c}_i(Q) = \sigma_i \; \text{ for }\; i = 1, \ldots, n-k,\]

respectively.

All Schubert classes form a $\mathbb{Q}$-vector space basis of the Chow ring of $\mathrm{Gr}(k,n)$. The Chern classes of $S$ (the special Schubert classes $\sigma_{1^i}$, $i=1, \ldots, k$) form a minimal set of generators for the Chow ring of $\mathrm{Gr}(k,n)$ as a $\mathbb{Q}$-algebra. See [EH16] for the relations on these generators.

Some classic computations

We illustrate Schubert calculus with a few classic enumerative problems.

How many lines in $\mathbb P^3$ meet four general lines?

The Grassmannian of lines in $\mathbb P^3$ is $\mathrm{Gr}(2,4)$. The Schubert class $\sigma_1$ represents lines meeting a given line. The answer is obtained by integrating $\sigma_1^4$:

julia> G = abstract_grassmannian(2, 4)
AbstractVariety of dim 4

julia> s1 = schubert_class(G, 1)
-c[1]

julia> integral(s1^4)
2

How many lines lie on a cubic surface in $\mathbb P^3$?

To count lines on a cubic surface in $\mathbb P^3$, we consider the zero locus of a section of $\operatorname{Sym}^3 S^*$ on $\mathrm{Gr}(2,4)$. Here $S$ is the tautological subbundle, and a line lies on the surface if and only if the section vanishes on it. The answer is the top Chern class of $\operatorname{Sym}^3 S^*$:

julia> G = abstract_grassmannian(2, 4);

julia> S, Q = tautological_bundles(G);

julia> integral(top_chern_class(symmetric_power(dual(S), 3)))
27

How many lines lie on a quintic threefold in $\mathbb P^4$?

The Grassmannian of lines in $\mathbb{P}^4$ is $\mathrm{Gr}(2,5)$. To count lines on the general quintic threefold, we consider the zero locus of a section of $\operatorname{Sym}^5 S^*$:

julia> G = abstract_grassmannian(2, 5)
AbstractVariety of dim 6

julia> S = tautological_bundles(G)[1];

julia> integral(top_chern_class(symmetric_power(dual(S), 5)))
2875

Functions

schubert_class(G::AbstractVariety, lambda::Int...)
schubert_class(G::AbstractVariety, lambda::Vector{Int})
schubert_class(G::AbstractVariety, lambda::Partition)

julia> G = abstract_grassmannian(2, 4)
AbstractVariety of dim 4

julia> s0 = schubert_class(G, 0)
1

julia> s1 = schubert_class(G, 1)
-c[1]

julia> s2 = schubert_class(G, 2)
c[1]^2 - c[2]

julia> s11 = schubert_class(G, [1, 1])
c[2]

julia> s21 = schubert_class(G, [2, 1])
-c[1]*c[2]

julia> s22 = schubert_class(G, [2, 2])
c[2]^2

julia> # Pieri's formula:

julia> s1*s1 == s2 + s11
true

julia> s1*s2 == s1*s11 == s21
true

julia> s1*s21 == s2*s2 == s22
true

julia> G = abstract_grassmannian(2, 5)
AbstractVariety of dim 6

julia> s3 = schubert_class(G, 5-2)
-c[1]^3 + 2*c[1]*c[2]

julia> s3^2 == point_class(G)
true

julia> Q = tautological_bundles(G)[2]
AbstractBundle of rank 3 on AbstractVariety of dim 6

julia> chern_class(Q, 3)
-c[1]^3 + 2*c[1]*c[2]

julia> G = abstract_grassmannian(2, 4)
AbstractVariety of dim 4

julia> s1 = schubert_class(G, 1)
-c[1]

julia> s1 == schubert_class(G, [1, 0])
true

julia> integral(s1^4)
2

schubert_classes(G::AbstractVariety)

schubert_classes(G::AbstractVariety, m::Int)

julia> G = abstract_grassmannian(2,4)
AbstractVariety of dim 4

julia> schubert_classes(G)
5-element Vector{Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}}:
 [1]
 [-c[1]]
 [c[1]^2 - c[2], c[2]]
 [-c[1]*c[2]]
 [c[2]^2]

julia> basis(G)
5-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [c[1]]
 [c[2], c[1]^2]
 [c[1]*c[2]]
 [c[2]^2]

julia> schubert_classes(G, 2)
2-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 c[1]^2 - c[2]
 c[2]