Jumping line

In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in other words the structure of its restriction to the line "jumps". Jumping lines were introduced by R. L. E. Schwarzenberger (1961). The jumping lines of a vector bundle form a proper closed subset of the Grassmannian of all lines of projective space.

The Birkhoff–Grothendieck theorem classifies the n-dimensional vector bundles over a projective line as corresponding to unordered n-tupes of integers.

Example

Suppose that V is a 4-dimensional complex vector space with a non-degenerate skew-symmetric form. There is a rank 2 vector bundle over the 3-dimensional complex projective space associated to V, that assigns to each line L of V the 2-dimensional vector space L^⊥/L. Then a plane of V corresponds to a jumping line of this vector bundle if and only if it is isotropic for the skew-symmetric form.

References

• Mulase, Motohico (1979), "Poles of instantons and jumping lines of algebraic vector bundles on P³", Japan Academy. Proceedings. Series A. Mathematical Sciences 55 (5): 185–189, ISSN 0386-2194, MR 533544
• Schwarzenberger, R. L. E. (1961), "Vector bundles on algebraic surfaces", Proceedings of the London Mathematical Society. Third Series 11: 601–622, doi:10.1112/plms/s3-11.1.601, ISSN 0024-6115, MR 0137711
• Schwarzenberger, R. L. E. (1961), "Vector bundles on the projective plane", Proceedings of the London Mathematical Society. Third Series 11: 623–640, doi:10.1112/plms/s3-11.1.623, ISSN 0024-6115, MR 0137712