Rational conic fibrations of sectional genus two

Abstract

Polarized rational surfaces $(X,\mathcal{L})$ of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of ${\mathbb{F}}_1$ at some points lying on distinct fibers. Ampleness and very ampleness of $\mathcal{L}$ are studied in terms of their location. When $\mathcal{L}$ is very ample and there is a line contained in $X$ and transverse to the fibers, the conic fibrations $(X,\mathcal{L})$ are classified and a related property concerned with the inflectional locus is discussed.

To appear in Advances in Geometry