$\widehat{S}^a(\mathcal F)\otimes \mathcal H^{\otimes -1}$ is weakly positive.

For the reader’s convenience, let us recall the following basic properties of big sheaves without proof.

\[x-lem4.2\] Let $\mathcal F$ be a nonzero torsion-free coherent sheaf on a smooth quasi-projective variety $W$. Then the following three conditions are equivalent.

-   There exist an ample invertible sheaf $\mathcal H$ on $W$, some positive integer $\nu$, and an inclusion $\bigoplus \mathcal H\hookrightarrow 
    \widehat{S}^\nu(\mathcal F)$, which is an isomorphism over a nonempty Zariski open set of $W$.

-   For every invertible sheaf $\mathcal M$ on $W$, there exists some positive integer $\gamma$ such that $\widehat {S}^\gamma (\mathcal F)\otimes \mathcal M^{\otimes -1}$ is weakly positive. In particular, $\mathcal F$ is a big sheaf.

-   There exist some positive integer $\gamma$ and an ample invertible sheaf $\mathcal M$ on $W$ such that $\widehat {S}^\gamma (\mathcal F)\otimes \mathcal M^{\otimes -1}$ is weakly positive.

We will use the following two easy lemmas on big sheaves in this paper. So we explicitly state them here for the reader’s convenience.

\[x-lem4.3\] Let $\mathcal F$ be a weakly positive sheaf and let $\mathcal H$ be an ample invertible sheaf on a smooth quasi-projective variety $W$. Then $\mathcal F\otimes \mathcal H$ is big.

