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)

References

• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Appendix II of Exp II of Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.