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>:
$$
\boxed{\operatorname{Div}(X)\longrightarrow\operatorname{Pic}(X)\text{ is surjective}.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-118/3/e/solution