The degree of the divisor of jumping rational curves

TL;DR

This paper provides conditions under which a semistable reflexive sheaf on projective space has balanced restriction on generic rational curves and derives a formula for the degree of the locus where this fails, generalizing classical results.

Contribution

It introduces new criteria for balanced restrictions of sheaves on rational curves and generalizes Barth's classical formula to higher ranks and degrees.

Findings

Conditions for balanced restriction on generic rational curves

A formula for the virtual degree of the jumping locus

Extension of classical results to higher rank sheaves

Abstract

For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $\P^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of the trivial bundle (for instance, if $E$ has balanced restriction on a generic line, or $r=2$ or $E$ is an exterior power of the tangent bundle). Assuming this, we give a formula for the 'virtual degree', interpreted enumeratively, of the locus of rational curves of degree $d$ on which the restriction of $E$ is not balanced, generalizing a classical formula due to Barth for the degree of the divisor of jumping lines of a semistable rank-2 bundle.