I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E\_{1,t}$ and $E\_{2,t}$, (I don't know what either $t$ runs through!) such that


(1), for any i={1，2}, the closed points of $X$ are the disjoint union of closed points of all $E\_{i,t}$.


(2), the intersection number of $E\_{1,t}$ and $E\_{2,t}$ is always 1.


Can we conclude that $X$ is an abelian surface?



Characterization of an Abelian surface
I think that the answer is yes, at least if you are working over an algebraically closed field of char $0$. Let me try to give an argument, I hope that there are no mistakes.


You have two flat families $E\_1 \to C\_1$ and $E\_2 \to C\_2$ of elliptic curves over some unknown bases $C\_1, C\_2$.


The first assumption says that the map $E\_1 \to X$ induces a bijection on closed points. If the field is of characteristic 0, one can use Zariski's main theorem to conclude that $E\_1 \to X$ is an isomorphism of varieties (see e.g. [Bijection implies isomorphism for algebraic varieties](https://mathoverflow.net/questions/264204/bijection-implies-isomorphism-for-algebraic-varieties)). The same holds for $E\_2 \to X$. Using these identifications $X \cong E\_1$ and $X \cong E\_2$, we get two flat projections $X \to C\_1$ and $X \to C\_2$, where the fibers are elliptic curves. In particular $C\_1$ and $C\_2$ must be smooth (by flat descent of smoothness).


I am interpreting the assumption that the intersection numbers are always 1 to mean that the fibers of the induced morphism $X \to C\_1 \times C\_2$ are singleton points, i.e. any two fibers of the families $E\_1$ and $E\_2$ respectively intersect in a single point (is this the precise condition you have in mind?). Since $C\_1 \times C\_2$ is smooth, this implies again that $X \to C\_1 \times C\_2$ is an isomorphism. But now by choosing a closed point $x\_1 \in C\_1$, we see that $C\_2 = x\_1 \times C\_2$ is isomorphic to the fiber of $E\_1 \to C\_1$ at $x\_1$. This means that $C\_1$ is an elliptic curve. The same holds for $C\_2$. Therefore $X \cong C\_1 \times C\_2$ is a product of elliptic curves.


