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

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

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

Then ${\displaystyle P^{0}(L)=L}$ and there is a natural exact sequence of vector bundles^[2]

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

where ${\displaystyle \Omega _{X}}$ 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 oscillating behaviors of a linear system.)
• Jet (mathematics) (a closely related notion)