The Segre embedding, and why $\PP^m \times \PP^n$ is projective

The Segre embedding identifies $\PP^m\times\PP^n$ with the rank-one tensor locus, cut out by the $2\times2$ minors.Suggest a slogan

The map \[ \psi: \PP^m \times \PP^n \to \PP^{(m+1)(n+1)-1}, \qquad ([x_i],[y_j]) \mapsto [x_i y_j] \] is well defined and injective, and its image is the closed set cut out by the $2 \times 2$ minors \[ z_{ij} z_{kl} - z_{il} z_{kj} = 0 . \] A product of projective varieties is therefore projective.

For $m = n = 1$ there is one minor, and the image is the quadric surface \[ z_{00} z_{11} - z_{01} z_{10} = 0 \] in $\PP^3$. The two rulings of that quadric are the images of $\ts{p} \times \PP^1$ and $\PP^1 \times \ts{q}$, and they make it a surface whose Picard group is $\ZZ^2$ rather than $\ZZ$.

The minors say the matrix $(z_{ij})$ has rank one, which is the coordinate-free statement: the Segre image is the locus of rank-one tensors in $\PP(V \tensor W)$.

Is $\PP^1 \times \PP^1$ a projective variety?

What equations cut out the Segre image?

Show that the Segre variety is the categorical product of projective varieties $X \subseteq \PP^n$ and $Y \subseteq \PP^m$.

What is the graph $\Gamma_f$ of a morphism $f \colon X \to B$ of quasi-projective varieties?

Give a precise definition of a morphism of projective varieties, in terms of homogeneous coordinates locally.

Describe the image of $[x:y] \mapsto [x^2 : xy : y^2]$, and give equations for the Veronese surface and its ambient $\PP^N$.

What is a determinantal variety?

Dependencies

D-CP2MHProjective varieties and homogeneous idealsuses

What links to this

Affine or projective