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.