By definition semistable sheaves are torsion-free. Any torsion-free $F$ includes into its double dual, $F\to F^{\ast\ast}$. The double dual is a reflexive sheaf, so any singularities occur in codimension 3. In the surface case, we conclude $F^{\ast\ast}$ is a line bundle with trivial determinant, so must be $\mathcal O_X$.

More generally, in "Vector Bundles on Complex Projective Spaces" (Okenek-Schneider-Spindler) it is shown that any reflexive sheaf on a smooth variety $X$ is necessarily a line bundle (Lemma 2.1.1.15). So even in higher dimensional cases, it follows that $F^{\ast\ast} = \mathcal O_X$.

(In fact, OSS defines the determinant of a torsion-free sheaf $F$ of rank $r$ by $$\det F = (\Lambda^r F)^{\ast\ast},$$ and notes that the determinant is always a line bundle by the cited lemma. In the rank one case, this is just the double dual, so we get a map $$F \to F^{\ast\ast} = \det F = \mathcal O_X.)$$

By definition semistable sheaves are torsion-free. Any torsion-free $F$ includes into its double dual, $F\to F^{\ast\ast}$. The double dual is a reflexive sheaf, so any singularities occur in codimension 3. In the surface case, we conclude $F^{\ast\ast}$ is a line bundle with trivial determinant, so must be $\mathcal O_X$.

More generally, in "Vector Bundles on Complex Projective Spaces" (Okenek-Schneider-Spindler) it is shown that any rank one reflexive sheaf on a smooth variety $X$ is necessarily a line bundle (Lemma 2.1.1.15). So even in higher dimensional cases, it follows that $F^{\ast\ast} = \mathcal O_X$.

(In fact, OSS defines the determinant of a torsion-free sheaf $F$ of rank $r$ by $$\det F = (\Lambda^r F)^{\ast\ast},$$ and notes that the determinant is always a line bundle by the cited lemma. In the rank one case, this is just the double dual, so we get a map $$F \to F^{\ast\ast} = \det F = \mathcal O_X.)$$

By definition semistable sheaves are torsion-free. Any torsion-free $F$ includes into its double dual, $F\to F^{\ast\ast}$. The double dual is a reflexive sheaf, so any singularities occur in codimension 3. In the surface case, we conclude $F^{\ast\ast}$ is a line bundle with trivial determinant, so must be $\mathcal O_X$.

More generally, in "Vector Bundles on Complex Projective Spaces" (Okenek-Schneider-Spindler) it is shown that any reflexive sheaf on a smooth variety $X$ is necessarily a line bundle (Lemma 2.1.1.15). So even in higher dimensional cases, it follows that $F^{\ast\ast} = \mathcal O_X$.

(In fact, OSS defines the determinant of a torsion-free sheaf $F$ of rank $r$ by $$\det F = (\Lambda^r F)^{\ast\ast},$$ and notes that the determinant is always a line bundle by the cited lemma. In the rank one case, this is just the double dual, so we get a map $$F \to \det F = \mathcal O_X.)$$

By definition semistable sheaves are torsion-free. Any torsion-free $F$ includes into its double dual, $F\to F^{\ast\ast}$. The double dual is a reflexive sheaf, so any singularities occur in codimension 3. In the surface case, we conclude $F^{\ast\ast}$ is a line bundle with trivial determinant, so must be $\mathcal O_X$.

More generally, in "Vector Bundles on Complex Projective Spaces" (Okenek-Schneider-Spindler) it is shown that any reflexive sheaf on a smooth variety $X$ is necessarily a line bundle (Lemma 2.1.1.15). So even in higher dimensional cases, it follows that $F^{\ast\ast} = \mathcal O_X$.

(In fact, OSS defines the determinant of a torsion-free sheaf $F$ of rank $r$ by $$\det F = (\Lambda^r F)^{\ast\ast},$$ and notes that the determinant is always a line bundle by the cited lemma. In the rank one case, this is just the double dual, so we get a map $$F \to F^{\ast\ast} = \det F = \mathcal O_X.)$$