Schubert calculus

Initially the definition looks a little bit awkward. Given a generic ${\displaystyle k}$-plane ${\displaystyle \Lambda \subset V}$ it will have only a zero intersection with ${\displaystyle V_{i}}$ for ${\displaystyle i\leq n-k}$ and ${\displaystyle \dim(V_{n-k+i}\cap \Lambda )=i}$ for ${\displaystyle i>n-k}$. For example, in ${\displaystyle G(4,9)}$ given a ${\displaystyle 4}$-plane ${\displaystyle \Lambda }$, this is cut out by a system of five linear equations. The ${\displaystyle 2}$-plane ${\displaystyle V_{2}}$ is not guaranteed to intersect at anywhere other than the origin since there are five free parameters it could live in. Furthermore, once ${\displaystyle \dim(V_{i})+\dim(\Lambda )>n}$, then they necessarily intersect. This means, the expected dimension of intersection of ${\displaystyle V_{5}}$ and ${\displaystyle \Lambda }$ should have dimension ${\displaystyle 1}$, the intersection of ${\displaystyle V_{6}}$ and ${\displaystyle \Lambda }$ should have dimension ${\displaystyle 2}$, and so on. These cycles then define special subvarieties of ${\displaystyle G(k,n)}$.

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.

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.

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

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

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

Schubert classes with tuples of length two or more can be described as a determinental 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

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

The Chow ring has the presentation

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

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]

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.