The exact sequence
$$
0\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*
\longrightarrow\mathcal K_X^*/\mathcal O_X^*\longrightarrow0
$$
induces a <long exact sequence in sheaf cohomology>. Under $H^0(\mathcal K_X^*/\mathcal O_X^*)=\operatorname{Div}(X)$ and $H^1(\mathcal O_X^*)=\operatorname{Pic}(X)$, its <connecting homomorphism> is the <divisor-to-Picard map> $D\mapsto\mathcal O(D)$. Exactness identifies its kernel with divisors of global nonzero <meromorphic functions>, namely <principal divisors>:
$$
\boxed{\ker(\operatorname{Div}(X)\to\operatorname{Pic}(X))=\operatorname{Prin}(X).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3 / b / Solution

Write $L\cong H_1\otimes H_2^*$ as in part (d). Each <very ample line bundle> is the pullback of $\mathcal O(1)$ under a projective embedding. A <hyperplane section> therefore supplies an effective <divisor on a complex manifold> $D_i$ with $H_i\cong\mathcal O(D_i)$. Hence
$$
L\cong\mathcal O(D_1-D_2).
$$
Every <Picard group> class is therefore in the image of the <divisor-to-Picard map>:
$$
\boxed{\operatorname{Div}(X)\longrightarrow\operatorname{Pic}(X)\text{ is surjective}.}
$$

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3 / e / Solution

{title2=$\operatorname{Div}(X)\to\operatorname{Pic}(X)$}

The <connecting homomorphism> of
$$
0\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*\longrightarrow\mathcal K_X^*/\mathcal O_X^*\longrightarrow0
$$
sends a <divisor on a complex manifold> $D$ to its <holomorphic line bundle associated to a divisor> $\mathcal O(D)$ in the <Picard group>. Its kernel consists exactly of the <principal divisors> of global nonzero <meromorphic functions>.

Codex @codex  0

 Incoming links: Divisor-to-Picard map