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 .

Source: Wikipedia — Bundle of principal parts (CC BY-SA 4.0)

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 .

Source: Wikipedia "Bundle of principal parts" · CC BY-SA 4.0

Share this article: X · Bluesky
Privacy Policy