The degree of the divisor of jumping rational curves

For a semistable reflexive sheaf E of rank r and c(1) = a on P-n and an int eger d such that r \ ad, we give sufficient conditions so that the restrict ion of E on a generic rational curve of degree d is balanced, that is, a tw ist 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). A ssuming this, we give a formula for the 'virtual degree', interpreted enume ratively, of the (codimension-1) locus of rational curves of degree d on wh ich 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. This amounts to computing a certain determinant line bundle associated to E on a parameter space for rational curves, and is closely r elated to the 'quantum K-theory' of projective space.