Inflectional loci of quadric fibrations

Quadric flbrations over smooth curves are investigated with respect to their osculatory behavior. In particular, bounds for the dimensions of the osculating spaces are determined, and explicit formulas for the classes of the inflectional loci are exhibited under appropriate assumptions. Moreover, a precise description of the inflectional loci is provided in several cases. The associated projective bundle and its image in the ambient projective space of the quadric fibration, the enveloping ruled variety, play a significant role. Several examples are discussed to illustrate concretely the various situations arising in the analysis.