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 $\tbinom$ 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 $p, q: V(I^) \to X$ the restrictions of projections $X \times X \to X$ to $V(I^) \subset X \times X$. 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
:$0 \to \mathrm^n(\Omega_X) \otimes L \to P^n(L) \to P^(L) \to 0.$
where $\Omega_X$ 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

*
*Appendix II of Exp II of

Algebraic geometry

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