The Degree of the Divisor of Jumping Rational Curves

For a semistable reflexive sheaf E of rank r and c1 = 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 (codimension-1) 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. This amounts to computing a certain determinant line bundle associated to E on a parameter space for rational curves, and is closely related to the ’quantum K-theory’ of projective space.