Inflectional loci of scrolls

Let X be a scroll over a smooth curve C, embedded in a projective space, and let L denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of L are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls