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 [math]\displaystyle{ \tbinom{n+\text{dim}(X)}{n} }[/math] that, roughly, parametrizes n-th order Taylor expansions of sections of L. Precisely, let I be the ideal sheaf defining the diagonal embedding [math]\displaystyle{ X \hookrightarrow X \times X }[/math] and [math]\displaystyle{ p, q: V(I^{n+1}) \to X }[/math] the restrictions of projections [math]\displaystyle{ X \times X \to X }[/math] to [math]\displaystyle{ V(I^{n+1}) \subset X \times X }[/math]. Then the bundle of n-th order principal parts is[1]
- [math]\displaystyle{ P^n(L) = p_* q^* L. }[/math]
Then [math]\displaystyle{ P^0(L) = L }[/math] and there is a natural exact sequence of vector bundles[2]
- [math]\displaystyle{ 0 \to \mathrm{Sym}^n(\Omega_X) \otimes L \to P^n(L) \to P^{n-1}(L) \to 0. }[/math]
where [math]\displaystyle{ \Omega_X }[/math] 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 1998, Example 2.5.6.
- ↑ SGA 6 1971, Exp II, Appendix II 1.2.4.
- 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). Berlin; New York: Springer-Verlag. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8.
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