product of projective schemes

Question

$X,Y$: projective schemes.

$L,M$:very ample invertible sheves on $X,Y$(resp.) and $\phi_1 :X \longrightarrow \mathbb{P}^r$, $\phi_2 :Y \longrightarrow \mathbb{P}^s$ are corresponding morphisms.

Consider $X\times Y$.

(1). I want to know corresponding(?) very ample invertible sheaf on $X\times Y$.

So I consider $\phi:X\times Y \longrightarrow \mathbb{P}^{rs+r+s}$ (composition of $\phi_1 \times \phi_2$ and Segre embedding).

Is it $Pr_1^*(L)\otimes Pr_2^*(M)?$ ($Pr_i$: i-th projection)

(2). Is any ideal sheaf of $X\times Y$ can be write $Pr_1^*(I_X)\otimes Pr_2^*(I_Y)?$ ($I_X$:ideal sheaf of $X$ & $I_Y$:ideal sheaf of $Y$)

I'm a beginner. I'm studying 'product of schemes'now. Any answer or comment would be a great help.

Answer 1

(1) Yes, that is the right very ample sheaf to look at. It is easy to see that it is very ample by noticing that a global section of $L$ times a global section of $M$ gives a global section of $pr_1^* L\otimes pr_2^* M$

(2) No. Take $X = Y$, consider the ideal sheaf of the diagonal. There is no reason to expect that you can write it this way, and usually you can't. For instance if $X$ and $Y$ are smooth of dimension at least $2$, it's easy to see that you can't.