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Let $k$ be an infinite field (not necessarily algebraically closed), $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. Is the quotient sheaf $F/F'$ locally free?
Let $k$ be an infinite field, $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. Is the quotient sheaf $F/F'$ locally free?
Let $k$ be an infinite field (not necessarily algebraically closed), $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. Is the quotient sheaf $F/F'$ locally free?
Is quotient by maximal destabilizing sheaf, torsion-free?
Let $k$ be an infinite field, $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. Is the quotient sheaf $F/F'$ locally free?
