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 𝔽1 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.
The authors are grateful to the referee for useful remarks. The first author is a member of G.N.S.A.G.A. of the Italian INdAM. He would like to thank the PRIN 2015 Geometry of Algebraic Varieties and the University of Milano for partial support. The second author wants to thank the Spanish Ministry of Science, Innovation and Universities (Project MTM 2015-65968-P “Geometría algebraica y analítica y aplicaciones”).
Communicated by: R. Cavalieri
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