Products Segre and Veronese Embeddings and Grassmannians — Examples

For $([s:t],[u:v])\in {\mathbf{P}}^1\times {\mathbf{P}}^1$, Segre gives $[z_{00}:z_{01}:z_{10}:z_{11}]=[su:sv:tu:tv]\mathrm{.}$ The sole $2\times 2$ minor is $z_{00}{z}_{11}-z_{01}{z}_{10}$, so the image is the quadric surface $V_{+}(z_{00}{z}_{11}-z_{01}{z}_{10})\subseteq {\mathbf{P}}^3$. On $z_{00}\mathrm{\ne }0$, its inverse sends $[z]$ to $([z_{00}:z_{10}],[z_{00}:z_{01}])$; the other three charts give the analogous formulas.

The quadratic Veronese map is $[s:t]\mapsto [s^2:st:t^2]=[Z_0:Z_1:Z_2]$. Its image has equation $Z_0{Z}_2-Z_1^2=0$. The line $a{Z}_0+b{Z}_1+c{Z}_2=0$ pulls back to the binary quadratic $a{s}^2+bst+c{t}^2=0$, illustrating the hyperplane-section correspondence.

A line in ${\mathbf{P}}^3$ is a $2$-plane in $k^4$. With Plucker coordinates $p_{01},p_{02},p_{03},p_{12},p_{13},p_{23}$, the unique Plucker relation is $p_{01}{p}_{23}-p_{02}{p}_{13}+p_{03}{p}_{12}=0.$ Thus $\mathrm{Gr}(2,4)$ is the Klein quadric in ${\mathbf{P}}^5$.

If $\dim V=n$, the only $0$-plane is $\{0\}$ and the only $n$-plane is $V$, so both $\mathrm{Gr}(0,V)$ and $\mathrm{Gr}(n,V)$ are one point. Their Plucker spaces are $\mathbf{P}({\mathrm{\Lambda }}^{0}V)=\mathbf{P}(k)$ and $\mathbf{P}({\mathrm{\Lambda }}^{n}V)$ respectively, again one point. There is no $r$-dimensional subspace when $r<0$ or $r>n$, so the stipulated Grassmannian is empty in those cases.

For every field $k$, the Zariski topology of ${\mathbf{A}}_k^1\times {\mathbf{A}}_k^1$ equals the product of the two factor Zariski topologies.

In ${\mathbf{P}}^n\times ({\mathbf{P}}^n{)}^{\vee }$, write a point as $[x_0:\cdots :x_n]$ and a hyperplane as $[a_0:\cdots :a_n]$. Incidence is the closed bilinear equation ${\sum }_{i=0}^n{a}_i{x}_i=0.$ For fixed $[a]$ its fibre is that hyperplane ${\mathbf{P}}^{n-1}$; for fixed $[x]$ its fibre is the projective space of hyperplanes through $[x]$, also ${\mathbf{P}}^{n-1}$ when $n\ge 1$. When $n=0$, the unique point of ${\mathbf{P}}^0$ lies on no hyperplane: the incidence locus and both fibres are empty.

The tensor product of two domains over every field is a domain.

For $([s:t],[u:v])\in {\mathbf{P}}^1\times {\mathbf{P}}^1$, the bidegree $(2,3)$ coordinates are the twelve products $s^{2-i}{t}^i{u}^{3-j}{v}^j(0\le i\le 2,0\le j\le 3)\mathrm{.}$ Scaling the two representatives by $\lambda ,\mu$ scales every coordinate by ${\lambda }^2{\mu }^3$. Up to the target coordinate permutation determined by the chosen Veronese monomial orderings, this is $\sigma \circ ({\nu }_{1,2}\times {\nu }_{1,3})$, hence is the stated closed embedding.