We give a proof for the sake of completeness.

Since $\mathcal F$ is weakly positive, $\widehat{S}^{2b}(\mathcal F)\otimes \mathcal H^{\otimes b}$ is generically generated by global sections for some positive integer $b$. By replacing $b$ with a multiple, we may assume that $\mathcal H^{\otimes b-1}$ is generated by global sections. Then $\widehat{S}^{2b}(\mathcal F\otimes \mathcal H)
\otimes \mathcal H^{\otimes -1}=
\widehat{S}^{2b}(\mathcal F)\otimes \mathcal H^{\otimes 2b-1}$ is generically generated by global sections. In particular, $\widehat {S}^{2b}(\mathcal F\otimes \mathcal H)
\otimes \mathcal H^{\otimes -1}$ is weakly positive. This implies that $\mathcal F\otimes \mathcal H$ is big by Lemma \[x-lem4.2\].

\[x-lem4.4\] Let $\mathcal F$ be a torsion-free coherent sheaf on a smooth quasi-projective variety $W$ and let $\tau:W'\to W$ be a finite surjective morphism from a smooth quasi-projective variety $W'$. Assume that $\tau^*\mathcal F$ is big. Then $\mathcal F$ is a big sheaf on $W$.

We include the proof for the benefit of the reader.

We take an ample invertible sheaf $\mathcal H$ on $W$. By replacing $W$ with $W\setminus \Sigma$ for some suitable closed subset $\Sigma$ of codimension $\geq 2$ (see, for example, [@fujino-iitaka Lemma 3.1.12 (i)]), we may assume that $\mathcal F$ is locally free. Since $\tau^*\mathcal F$ is big by assumption, there 