Bundle of principal parts
# Bundle of principal parts

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 ${\tbinom {n+{\text{dim}}(X)}{n}}$ that, roughly, parametrizes n-th order Taylor expansions of sections of L.

Precisely, let I be the ideal sheaf defining the diagonal embedding $X\hookrightarrow X\times X$ and $p,q:V(I^{n+1})\to X$ the restrictions of projections $X\times X\to X$ to $V(I^{n+1})\subset X\times X$. Then the bundle of n-th order principal parts is

    $P^{n}(L)=p_{*}q^{*}L.$

Then $P^{0}(L)=L$ and there is a natural exact sequence of vector bundles

    $0\to \mathrm {Sym} ^{n}(\Omega _{X})\otimes L\to P^{n}(L)\to P^{n-1}(L)\to 0.$

where $\Omega _{X}$ is the sheaf of differential one-forms on X.

