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 osculating (as in osculating plane) 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., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4 
• Appendix II of Exp II of Berthelot, Pierre, ed (1971) (in fr). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). 225. Berlin; New York: Springer-Verlag. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8.