For each point $x$, choose a <global section> $s$ whose <germ> $s_x$ generates the rank-one free $\mathcal O_{X,x}$-module $\mathcal L_x$. The section $s\otimes s$ is global, and <tensor products> commute with taking <stalks>. Therefore
$$
(s\otimes s)_x=s_x\otimes s_x
$$
generates $(\mathcal L\otimes_{\mathcal O_X}\mathcal L)_x$. Thus \b[the tensor square is globally generated]. More generally, the <tensor product of globally generated line bundles> is globally generated: choose a generator section for each factor at the point and tensor them. This works on every scheme, including nonreduced schemes.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 113 / 3 / ii / Solution

The <tensor product> of two <globally generated line bundles> is globally generated. At a point choose generator germs arising from a global section of each factor. Their tensor is the germ of a global tensor-product section and generates the rank-one tensor-product stalk. Iteration gives the same property for every positive tensor power, on an arbitrary <scheme>.

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