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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 n ] {\displaystyle {\tbinom {n+{\text{dim}}}{n}}} that, roughly, parametrizes n-th order Taylor expansions of sections of L.
Precisely, let I be the ideal sheaf defining the diagonal embedding X ↪ X × X {\displaystyle X\hookrightarrow X\times X} and p , q : V → X {\displaystyle p,q:V\to X} the restrictions of projections X × X → X {\displaystyle X\times X\to X} to V ⊂ X × X {\displaystyle V\subset X\times X}. Then the bundle of n-th order principal parts is
Then P 0 = L {\displaystyle P^{0}=L} and there is a natural exact sequence of vector bundles
where Ω X {\displaystyle \Omega _{X}} is the sheaf of differential one-forms on X.