Schubert calculus

In some sources[1][2], the Schubert cells ${\displaystyle X_{\mathbf {a} }}$ and Schubert varieties ${\displaystyle \Sigma _{\mathbf {a} }}$are labelled differently, as ${\displaystyle C_{\lambda }}$ and ${\displaystyle {\bar {C}}_{\lambda }}$, respectively, where ${\displaystyle \lambda }$ is the complementary partition, to ${\displaystyle \mathbf {a} }$, with parts

${\displaystyle \lambda _{i}=n-a_{k-i+1}}$,

whose Young diagram is the complement of the one for ${\displaystyle \mathbf {a} }$ within the ${\displaystyle k\times (n-k)}$ rectangular one (reversed, both horizontally and vertically).

Another labelling convention for ${\displaystyle X_{\mathbf {a} }}$ and ${\displaystyle \Sigma _{\mathbf {a} }}$ is ${\displaystyle S_{L}}$ and ${\displaystyle {\bar {S}}_{L}}$, respectively, where ${\displaystyle L=(L_{1},\dots ,L_{k})\subset (1,\dots ,n)}$ is the multi-index defined by

${\displaystyle L_{i}:=n-k-\lambda _{i}+i}$

or, alternatively, by

${\displaystyle L_{i}:=n-k-a_{i}+i}$.

In order to explain the definition, consider a generic ${\displaystyle k}$-plane ${\displaystyle w\subset V}$. It will have only a zero intersection with ${\displaystyle V_{j}}$ for ${\displaystyle j\leq n-k}$, whereas

${\displaystyle \dim(V_{j}\cap w)=i}$ for ${\displaystyle j=n-k+i\geq n-k.}$

For example, in ${\displaystyle \mathbf {Gr} (4,9)}$, a ${\displaystyle 4}$-plane ${\displaystyle w}$ 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 ${\displaystyle w}$) will consist only of the zero vector. However, if ${\displaystyle \dim(V_{j})+\dim(w)>n=9}$, ${\displaystyle V_{j}}$ and ${\displaystyle w}$ will necessarily have nonzero intersection. For example, the expected dimension of intersection of ${\displaystyle V_{6}}$ and ${\displaystyle w}$ is ${\displaystyle 1}$, the intersection of ${\displaystyle V_{7}}$ and ${\displaystyle w}$ has expected dimension ${\displaystyle 2}$, and so on.

The definition of a Schubert variety states that the first value of ${\displaystyle j}$ with ${\displaystyle \dim(V_{j}\cap w)\geq i}$ is generically smaller than the expected value ${\displaystyle n-k+i}$ by the parameter ${\displaystyle a_{i}}$. The ${\displaystyle k}$-planes ${\displaystyle w\subset V}$ given by these constraints then define special subvarieties of ${\displaystyle \mathbf {Gr} (k,n)}$.[4]

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

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

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

A Schubert variety ${\displaystyle \Sigma _{\mathbf {a} }}$ has codimension ${\displaystyle \sum a_{i}}$, which is stable under inclusions of Grassmannians. That is, the inclusion

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

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

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

Also, the inclusion

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

given by inclusion of the ${\displaystyle k}$-plane has the same pullback property.

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

In the special case ${\displaystyle \mathbf {b} =(b,0,\ldots ,0)}$, 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

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

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

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

and

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

Schubert classes ${\displaystyle \sigma _{\mathbf {a} }}$ for partitions of any length ${\displaystyle \ell (\mathbf {a} )}$ can be expressed as the determinant of a ${\displaystyle (k\times k)}$ matrix having the special classes as entries.

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

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

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

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

The intersection product between any pair of Schubert classes ${\displaystyle \sigma _{\mathbf {a} },\sigma _{\mathbf {b} }}$ is given by

${\displaystyle \sigma _{\mathbf {a} }\sigma _{\mathbf {b} }=\sum _{\mathbf {c} }c_{\mathbf {a} \mathbf {b} }^{\mathbf {c} }\sigma _{\mathbf {c} },}$

where ${\displaystyle \{c_{\mathbf {a} \mathbf {b} }^{\mathbf {c} }\}}$ are the Littlewood-Richardson coefficients.[5] The Pieri formula is a special case of this, when ${\displaystyle \mathbf {b} =(b,0,\dots ,0)}$ has length ${\displaystyle \ell (\mathbf {b} )=1}$.

The Chow ring has the presentation

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

and as a graded Abelian group[6] it is given by

${\displaystyle {\begin{aligned}A^{0}(\mathbf {Gr} (2,4))&=\mathbb {Z} \cdot 1\\A^{2}(\mathbf {Gr} (2,4))&=\mathbb {Z} \cdot \sigma _{1}\\A^{4}(\mathbf {Gr} (2,4))&=\mathbb {Z} \cdot \sigma _{2}\oplus \mathbb {Z} \cdot \sigma _{1,1}\\A^{6}(\mathbf {Gr} (2,4))&=\mathbb {Z} \cdot \sigma _{2,1}\\A^{8}(\mathbf {Gr} (2,4))&=\mathbb {Z} \cdot \sigma _{2,2}\\\end{aligned}}}$

Recall that a line in ${\displaystyle \mathbb {P} ^{3}}$ gives a dimension two subspace of ${\displaystyle \mathbb {A} ^{4}}$, hence an element of ${\displaystyle \mathbb {G} (1,3)\cong \mathbf {Gr} (2,4)}$. 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 ${\displaystyle s\in \Gamma (\mathbb {G} (1,3),{\text{Sym}}^{3}(T^{*}))}$. 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

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

The splitting formula then 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 viewed as the direct sum of formal line 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

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

it follows that

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

using the fact that

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

Since ${\displaystyle \sigma _{2,2}}$ is the top class, the integral is then

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

Therefore there are ${\displaystyle 27}$ lines on a cubic surface.