Rational conic fibrations of sectional genus two (Q2035035)

Let \(X\) be a smooth projective variety defined over the field of complex numbers and let \(L\) be an ample line bundle on \(X\). Then the pair \((X,L)\) is called a polarized manifold. In this paper, the authors considered the case that \(X\) is a rational surface fibered in conics over a smooth projective curve (we call this a rational conic fibration), and the sectional genus \(g(L)\) of \((X,L)\) is two. In particular, the authors treated the case where \(X\) is not minimal. If \(X\) is not minimal, then, by means of elementary transformations, \(X\) can be described as the blow-up of the Serre-Hirzebruch surface \(F_{1}\) at some points lying on distincts fibers. Under this situation, the authors studied ampleness, spannedness and very ampleness of \(L\). In particular, they characterized very ampleness of \(L\) when \((X,L)\) is a polarized rational conic fibrations with \(g(L)=2\). In connection with the above, the authors referred to Bese's result [\textit{E. M. Bese}, Math. Ann. 262, 225--238 (1983; Zbl 0491.14009)] for the spannedness of \(L\) but not for the very ampleness. They pointed out that some results of Bese specialized to the situation under consideration do not coincide with the results of this paper (see Remark 5 in this paper). Finally, the authors gave the complete list of polarized rational conic fibration \((X,L)\) with \(g(L)=2\) under the assumption that \(L\) is very ample and \((X,L)\) contains a line transverse to the fibers (see Theorem 12 in this paper).