Inflectional loci of quadric fibrations (Q493748)

Continuing their previous work on inflection loci of scrolls over projective varieties [\textit{R. Mallavibarrena} et al., Indiana Univ. Math. J. 61, No. 2, 717--750 (2012; Zbl 1273.14015)], the authors scrutinize quadratic fibrations of dimension \(n\) in \(\mathbb{P}^N\) over a smooth curve. Once more, a careful investigating of the hyperplane bundle on the quadric fibration is crucial. Beside other results, the authors find the upper bound \(\min\{k(n+1), N+1\}\) for the highest dimension of the \(k\)-th osculating space and determine the cohomology class of a generic osculating space, provided the second inflectional locus attains its maximal dimension or is empty. In the later sections, they discuss special cases (conic fibrations, rational quadric fibrations) in more detail.