Suppose we are given $X =\mathbb{P}^1_k\times_{k}\mathbb{P}^1_k$ with its canonical projections $\pi_i$ to $\mathbb{P}^1_k$. I'd like to show that $\pi_1^*\mathcal{O}(n)\otimes_{\mathcal{O}_X}\pi_1^*\mathcal{O}(m)$ is very ample if and only if both $n$ and $m$ are strictly positive.
I know that $\mathcal{O}(1)$ is ample, and that tensor products of ample line bundles are ample, but I don't think that $\pi_i^*\mathcal{O}(1)$ will be ample (if it was, the above statement would be false). Any tips on how to tackle this problem?
Look in Hartshorne, Ch. II, Section 7 (I think), especially the test for very ampleness in terms of separating points and tangent vectors. Applying this test should be pretty straightforward.
If we write $\mathcal O(m,n)$ for your tensor product of pull-backs, then e.g. $\mathcal O(1,1)$ gives the embedding of the product into $\mathbb P^3$ as a quadric, while of course $\mathcal O(1,0)$ and $\mathcal O(0,1)$ give the projections onto each of the $\mathbb P^1$ factors, which certainly aren't embeddings (and so these sheaves are not very ample).