The twisting sheaves $\OO(n)$, and $\Pic(\PP^n) = \ZZ$

(Twisting sheaves)

$\OO_{\PP^n}(-1)$ is the tautological bundle: its fibre over $p \in \PP^n$ is the line in $\AA^{n+1}$ that $p$ is. Its dual is $\OO(1)$, and $\OO(n) = \OO(1)^{\tensor n}$; on $\Proj S$ these are the sheaves $\widetilde{S(n)}$ associated to the shifted graded modules.

$\Pic(\PP^n_k) = \ZZ$, generated by $\OO(1)$, and \[ H^0(\PP^n, \OO(d)) = S_d = \ts{\text{homogeneous polynomials of degree } d} , \] of dimension $\binom{n+d}{n}$ for $d \geq 0$ and zero for $d < 0$.

For $n \geq 1$, $\OO(-1)$ has no nonzero global sections, whereas the global sections of $\OO(1)$ are the linear forms.

By Serre’s theorem, every coherent sheaf on $\PP^n_k$ is a quotient of a finite sum of twists $\OO(-d_i)$. On a smooth projective curve, line bundles correspond to divisors modulo linear equivalence. On a curve of positive genus, two line bundles of the same degree can be non-isomorphic: on an elliptic curve $E$ with origin $O$ and a point $P\ne O$, $\OO_E(P-O)$ has degree $0$ and is not isomorphic to $\OO_E$.

What is the Serre twisting sheaf?

What is the tautological bundle on $\PP^n$?

What is $\Pic(\PP^n)$?

Dependencies

D-QNTZYQuasicoherent and coherent sheavesuses

Backlinks

T-MODSERRESerre's theorems A and Brelated-to
D-MODPICThe Picard grouprelated-to
T-MODRESCoherent sheaves on projective space have finite resolutions by twistsuses
T-MODEULERThe Euler sequence on projective spaceuses
T-IJW1KThe cohomology of $\OO_{\PP^n}(d)$uses
D-MODGRMODThe sheaf associated to a graded module, and $\Gamma_*$uses
FE-SHFPONETwisting sheaves and a skyscraper sequence on the projective lineuses
T-BIRKGROTHVector bundles on the projective line splituses

What links to this

Quasicoherence and twisting