[1][2] 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-tuples of integers.
This phenomenon cannot be generalized to higher dimensional projective spaces, namely, one cannot decompose an arbitrary bundle in terms of a Whitney sum of powers of the Tautological bundle, or in fact of line bundles in general.
, and it will decompose by the Birkhoff–Grothendieck theorem as a sum of powers of the Tautological bundle.
It can be shown that the unique tuple of integers specified by this splitting is the same for a 'generic' choice of line.
More technically, there is a non-empty, open sub-variety of the Grassmannian of lines in
Suppose that V is a 4-dimensional complex vector space with a non-degenerate skew-symmetric form.
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.