Can blow-up of a surface be a product of two curves?

Question

Is there any smooth projective surface $S$ over $k=\bar{k}$, such that the blow up $\tilde{S}$ along some point $x\in S$ can be written as $\tilde{S}=C_1\times C_2$ for two curves $C_i$?

Answer 1

No. If not, let $E$ be the exceptional curve in $C_1\times C_2$. Then I claim that the projections $E$ to both $C_i$ are onto. If one of them is not, say to the second factor, then $E=C_1\times p$ for $p\in C_2$. But, then, $E^2=0\neq -1$, proving the claim. So, both $C_i$ are projective lines and the the Picard group of the product is as described in the deleted answer and easy to see that it has no curves of negative self-intersection.