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 $(\genfrac{}{}{0ex}{}{n+\text{dim}(X)}{n})$ 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

$0\to {\mathrm{S}\mathrm{y}\mathrm{m}}^n({\mathrm{\Omega }}_X)\otimes L\to P^n(L)\to P^{n-1}(L)\to 0.$

where ${\mathrm{\Omega }}_X$ 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)

References

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• Appendix II of Exp II of Template:Cite book