Bundle of principal parts explained

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{n + dim (X)}{n} that, roughly, parametrizes n-th order Taylor expansions of sections of L.X \hookrightarrow X x X and p, q: V(I ^{n+1} ) \to X the restrictions of projections X x X \to X to V(I ^{n+1} ) \subset X x X . Then the bundle of n-th order principal parts isP ^{n(L)} = p _{*} q ^{*} L. Then P ^{0(L)} = L and there is a natural exact sequence of vector bundles0 \to Sym ^{n(\Omega} _{X)} ⊗ L \to P ^{n(L)} \to P ^{n-1} (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 Book: Berthelot . Pierre . Pierre Berthelot (mathematician) . . . Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) . 1971 . . Berlin; New York . fr . xii+700 . true . 10.1007/BFb0066283 . 978-3-540-05647-8 . 0354655 . .