Geometry of ${\mathbb{P}}^2$ blown up at seven points

Abstract

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 $(-2)$).

14 pages,Comments are welcome!