In algebraic geometry, given a line bundle L on a smooth variety X, the bundle of n-th order principal parts of L is a vector bundle of rank that, roughly, parametrizes n-th order Taylor expansions of sections of L.
Precisely, let I be the ideal sheaf defining the diagonal embedding and the restrictions of projections to . Then the bundle of n-th order principal parts is
Then and there is a natural exact sequence of vector bundles
where is the sheaf of differential one-forms on X.
See also Linear system of divisors (bundles of principal parts can be used to study the osculating (as in osculating plane) behaviors of a linear system.) Jet (mathematics) (a closely related notion)
References
Appendix II of Exp II of