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

TL;DR

This paper characterizes the geometry of the projective plane blown up at seven points, showing it admits a conic bundle structure and can be embedded as a (2,2) divisor in a product of projective spaces, with specific properties of its curves.

Contribution

It establishes a correspondence between certain blown-up projective planes and (2,2) divisors in 12, detailing their geometric structures and curve configurations.

Findings

Blown-up 2 at seven points admits a conic bundle structure.

Such surfaces can be embedded as (2,2) divisors in 12.

Any smooth (2,2) surface in 12 has at most four (-2) curves.

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 $(-2)$).