coordinate free Segre embedding

Question

Is there a coordinate free description of the Segre embedding ? Is there a relation between $\mathbb P(W)$, $\mathbb P(V)$ and $\mathbb P(V \oplus W)$ ? I don't find a reference for this.

Answer 1

The Segre embedding is simply the projective version of the map $V \times W \to V \otimes W$.

For answer to your other question, there are two maps $i : V \to V \oplus W$ and $j : W \to V \oplus W$ which embedd $\mathbb P(V)$ in $\mathbb P(V \oplus W)$ and similarly for $\mathbb P(W)$.

I think the book of Harris, A first course in algebraic geometry, is a good reference