Let $X$ and $Y$ be normal varieties with $D$ and $E$ Cartier divisors on $X$ and $Y$, respectively. Let $(D,E)$ denote the divisor $\pi\_X^\*(D)+\pi\_Y^\*(E)$ on the product $X\times Y$, where $\pi\_X$ and $\pi\_Y$ are the projections from $X\times Y$.


There is a natural inclusion $H^0(X,\mathcal{O}(D))\otimes H^0(Y,\mathcal{O}(E))\to H^0(X\times Y,\mathcal{O}((D,E)))$ induced by the inclusion $K(X)\otimes K(Y)\to K(X\times Y)$.


Under what conditions is this inclusion an isomorphism? This holds for example if $X$ and $Y$ are toric. 



Global sections for divisors on products of varieties
This is always true, it follows from the Künneth formula for coherent sheaves (see [Kunneth formula for sheaf cohomology of varieties](https://mathoverflow.net/questions/34673/kunneth-formula-for-sheaf-cohomology-of-varieties)).


