Rational conic fibrations of sectional genus two

TL;DR

This paper classifies rational conic fibrations of sectional genus two surfaces, analyzing their structure, ampleness conditions, and inflectional properties, especially focusing on non-minimal cases and the presence of lines.

Contribution

It provides a classification of polarized rational surfaces with conic fibrations of genus two, including conditions for ampleness and a detailed analysis of their geometric properties.

Findings

Non-minimal surfaces are blow-ups of _1 at points on distinct fibers.

Conditions for ampleness and very ampleness of the polarization are established.

Classification of conic fibrations with very ample line bundles containing a transverse line.

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.