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---
date: 2022-08-30 18:12
modification date: Tuesday 30th August 2022 18:12:24
title: "computations of picard groups"
aliases: [computations of picard groups]
---

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# computations of picard groups

## $\AA^n$

To show $\Pic(\AA^n) = 1$:

![](attachments/Pasted%20image%2020220830181227.png)

## $\PP^n$

Use smoothness and then toric geometry to compute the class group: $\Pic(\PP^n) = \CH^1(\PP^n) \cong \ZZ$.

## Smooth hypersurfaces in $\PP^n$

Identify $\Pic(X) = \CH^1(X)$ and use the Chow exact sequence to get $\Pic(X) \cong C_d$.