Rational conic fibrations sectional genus two

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