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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