  In this paper, we prove that $\mathbb{P}^2$ blown up at seven general points
admits a conic bundle structure over $\mathbb{P}^1$ and it can be embedded as
$(2,2)$ divisor in $\mathbb{P}^{1}\times\mathbb{P}^{2}$. Conversely, any smooth
surface in the complete linear system $\mid (2,2) \mid$ of
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ is obtained by blowing up $\mathbb{P}^2$
at seven points. We also show any smooth surface linearly equivalent to $(2,2)$
in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ has at most four $(-2)$ curves (the
curve which has self intersection $(-2)$).
