Inflectional loci of scrolls II

TL;DR

This paper extends the study of inflectional loci of scrolls to higher-dimensional bases, deriving formulas for osculating sheaves and inflectional locus classes under new dimension conditions.

Contribution

It generalizes previous work on scrolls over curves to higher-dimensional bases, providing explicit descriptions of osculating sheaves and inflectional locus classes.

Findings

Derived formulas for osculating sheaves of scrolls over higher-dimensional varieties.

Computed cohomology class and degree of inflectional loci for m ≥ 2.

Addressed increased complexity in formulas and dimension conditions.

Abstract

Let $X\subset \mathbb P^N$ be a scroll over a $m$-dimensional variety $Y$. We find the locally free sheaves on $X$ governing the osculating behavior of $X$, and, under certain dimension assumptions, we compute the cohomology class and the degree of the inflectional locus of $X$. The case $m=1$ was treated in \cite{LMP}. Here we treat the case $m\ge 2$, which is more complicated for at least two reasons: the expression for the osculating sheaves and the computations of the class of the inflectional locus become more complex, and the dimension requirements needed to ensure validity of the formulas are more severe.