Low dimensional discriminant loci and scrolls

TL;DR

This paper classifies certain complex polarized varieties based on the codimension of their discriminant loci, revealing scrolls over curves and Grassmannian sections as key examples.

Contribution

It provides a classification of smooth complex polarized varieties with specific discriminant locus codimensions, identifying scrolls over curves and Grassmannian sections as unique cases.

Findings

Codimension cannot be $ ext{dim}(X)-4$ for the locus ${ ext{D}}(X,V)$.

Codimension $ ext{dim}(X)-3$ characterizes scrolls over curves.

For codimension $ ext{dim}(X)-5$, only Grassmannian embeddings or hyperplane sections appear.

Abstract

Smooth complex polarized varieties $(X,L)$ with a vector subspace $V \subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that the locus ${\Cal D}(X,V)$ of singular elements of $|V|$ has codimension equal to $\dim(X)-i$, $i=3,4,5$, the last case under the additional assumption that $X$ has Picard number one. In fact it is proven that this codimension cannot be $\dim(X)-4$ while it is $\dim(X)-3$ if and only if $(X,L)$ is a scroll over a smooth curve. When the codimension is $\dim(X)-5$ and the Picard number is one only the Pl\"ucker embedding of the Grassmannian of lines in $\Bbb P^4$ or one of its hyperplane sections appear. One of the main ingredients is the computation of the top Chern class of the first jet bundle of scrolls and hyperquadric fibrations. Further consequences of these computations are also provided.