The standard affine charts of $\mathbb P^3$ are copies of $\mathbb A^3$, and the product charts of $\mathbb P^1\times\mathbb P^1$ are copies of $\mathbb A^2$. Their local rings are localizations of <polynomial ring>[polynomial rings] over $k$ and hence are <regular local ring>[regular local rings]. Both schemes are therefore <regular scheme>[regular].

On a regular integral scheme every <Weil divisor> is <Cartier divisor>[Cartier], so the <divisor class group> is naturally the <Picard group>. Pullback of <line bundle>[line bundles] along the <Segre embedding> therefore defines
$$
\iota^*: \operatorname{Cl}(\mathbb P^3)\longrightarrow\operatorname{Cl}(\mathbb P^1\times\mathbb P^1).
$$
The two groups are
$$
\operatorname{Cl}(\mathbb P^3)=\mathbb Z[\mathcal O(1)],
\qquad
\operatorname{Cl}(\mathbb P^1\times\mathbb P^1)=\mathbb Z[\mathcal O(1,0)]\oplus\mathbb Z[\mathcal O(0,1)].
$$
The Segre coordinates are bihomogeneous of bidegree $(1,1)$, so $\iota^*\mathcal O_{\mathbb P^3}(1)=\mathcal O(1,1)$. In these bases the map is $m\mapsto(m,m)$ and
$$
\boxed{\operatorname{im}\iota^*=\mathbb Z(1,1)\subset\mathbb Z^2.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 2 / d / Solution

{wiki=Regular_scheme}

A locally Noetherian scheme is regular when every local ring $\mathcal O_{X,x}$ is a <regular local ring>.

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