The inflectional loci of scrolls

This is a report on a recent result obtained with Raquel Mallavibarrena and Ragni Piene. Let X \subset P^N be a scroll over a smooth curve C and let L be its 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 their cohomology classes. In particular these formulas allow to prove that the only uninflected scrolls in projective spaces of appropriate dimensions are the balanced rational normal scrolls. This extends results due to Shifrin and Piene and Tai in the setting of surface scrolls. Some related results are also discussed in the talk.