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 $\tbinom{n+\text{dim}(X)}{n}$ 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{{harvnb|Fulton|1998|loc=Example 2.5.6.}}

: $P^n(L) = p_* q^* L.$

Then $P^0(L) = L$ and there is a natural exact sequence of vector bundles{{harvnb|SGA 6|1971|loc=Exp II, Appendix II 1.2.4.}}

: $0 \to \mathrm{Sym}^n(\Omega_X) \otimes L \to P^n(L) \to P^{n-1}(L) \to 0.$

where $\Omega_X$ is the sheaf of differential one-forms on X.

References

{{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd |first=William |last=Fulton}}
Appendix II of Exp II of {{cite book

| editor-last = Berthelot

| editor-first = Pierre

| editor-link = Pierre Berthelot (mathematician)

| editor2=Alexandre Grothendieck

| editor3=Luc Illusie

| title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225)

| year = 1971

| volume = 225

| publisher = Springer-Verlag

| location = Berlin; New York

| language = fr

| pages = xii+700

| no-pp = true

|doi=10.1007/BFb0066283

|isbn= 978-3-540-05647-8

| mr = 0354655

| ref={{harvid|SGA 6|1971}}

}}

Category:Algebraic geometry

bundle of principal parts

See also

References