Some properties of rational conic fibrations (Q2118357)

The authors study rational conic fibrations (RCFs) of sectional genus \(g=3\). In order to conduct their investigations, the authors study the interplay between rational conic fibrations and their enveloping bundles. Slightly more precisely, if \(X \subset \mathbb{P}^{N}\) is a rational conic fibration of sectional genus \(g\), then the enveloping projective bundle \(P\) is the rational \(\mathbb{P}^{2}\)-bundle giving rise to the ruled variety in \(\mathbb{P}^{N}\) swept out by the planes spanned by conics which are the fibres of \(X\). In Theorem 4 therein the authors provide possibilities for a RCF \((X, \mathcal{L})\) with \(\mathcal{L}\) being the hyperplane bundle of \(X\) of sectional genus \(g=3\) and its enveloping bundle. Moreover, the authors provide a complete description of rational conic fibrations containing a line or a conic, or a smooth twisted cubic transverse to the fibres.