# A torsion picard group.

Suppose $Y$ is a hypersurface in $\mathbb{P}_k^n$ corresponding to an irreducible degree $d$ polynomial.

Show that $Pic( \mathbb{P}_k^n - Y) \cong \mathbb{Z}/(d)$. This is related to the fact that the fundamental group $\pi_1(\mathbb{P}_k^n - Y) \cong \mathbb{Z}/(d)$.

The Picard group of a ringed space $X$, denoted by $Pic(X)$ is the group of isomorphism classes of invertible sheaves (or line bundles) on $X$, with the group operation being tensor product.

An invertible sheaf is a coherent sheaf $S$ on a ringed space $X$, for which there is an inverse $T$ with respect to the tensor product of $O_X$-modules. Invertible sheaves on a ringed space in algebraic geometry are topologically called line bundles.

These isomorphism classes form an abelian group, the Picard group under tensor product. $Pic$ is the Picard functor from ringed spaces to abelian groups.

They are probably related to the fundamental group in some way. But I haven’t figured that out yet.

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