• Source: 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







(



n
+

dim

(
X
)

n


)






{\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



X
↪
X
×
X


{\displaystyle X\hookrightarrow X\times X}

and



p
,
q
:
V
(

I

n
+
1


)
→
X


{\displaystyle p,q:V(I^{n+1})\to X}

the restrictions of projections



X
×
X
→
X


{\displaystyle X\times X\to X}

to



V
(

I

n
+
1


)
⊂
X
×
X


{\displaystyle V(I^{n+1})\subset X\times X}

. Then the bundle of n-th order principal parts is





P

n


(
L
)
=

p

∗



q

∗


L
.


{\displaystyle P^{n}(L)=p_{*}q^{*}L.}


Then




P

0


(
L
)
=
L


{\displaystyle P^{0}(L)=L}

and there is a natural exact sequence of vector bundles




0
→


S
y
m


n


(

Ω

X


)
⊗
L
→

P

n


(
L
)
→

P

n
−
1


(
L
)
→
0.


{\displaystyle 0\to \mathrm {Sym} ^{n}(\Omega _{X})\otimes L\to P^{n}(L)\to P^{n-1}(L)\to 0.}


where




Ω

X




{\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.

Kata Kunci Pencarian:

• Bundle of principal parts
• Coherent sheaf
• Linear system of divisors
• Ehresmann connection
• Gauge theory (mathematics)
• Monopole
• Hopf fibration
• Principal–agent problem
• BRST quantization
• Nonabelian Hodge correspondence