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Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some intermediate modification.) Let $F_C$ be a torsion free sheaf on $C$. Pull it back: $F_{C'}:=\nu^*(F_C)/Torsion$.

Suppose $F_{C'}$ has global sections. Let $0\neq s\in H^0(F_{C'})$. When does $s$ descend to $C$, i.e. when is $s$ the pullback of a section of $F_C$?

Or, what is the condition that no global section of $F_{C'}$ comes from a global section of $F_C$? (Of course $h^0(F_{C'})=0$ is enough but probably is not necessary.)

Here is some local-to-global issue. Let $0\neq s\in H^0(F_{C'})$, such that for each point $pt\in C'$ this $s$ belongs to the preimage (not the pullback!) of the stalk of $F_C$, i.e.: $\nu^{-1}(F_{C,\nu(pt)})$. So, at each point of $C'$ the section $s$ "descends locally". Does this imply that $s$ descends globally?

What are the tools to treat such questions? Some sort of relative C'/C cohomology?

upd. According to Sándor Kovács' reply, it seems from $0\to F_C\to\nu_*(F_{C'})\to Skyscraper\to0$ that a global section of $F_{C'}$ descends to $F_C$ iff it descends locally at each point. The main question remains:

How to characterize the torsion free sheaves $F_{C'}$, pullbacks from $F_C$, whose global sections do not descend to $F_C$. (I'd like the answer in terms of $h^i(F_{C'})$, the genera of $C,C'$, the invariants of $\nu_* \mathcal{O}_{C'}/\mathcal{O}_C$, e.g. the lengths.)

Another related issue. If the conclusion is true, it seems some cohomology here vanishes (i.e. the local descent implies the global one). "Could it not vanish?" i.e. is there some similar situation (higher dimensional, etc) where a section descends locally at each point but not globally?

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# when a section descends?

Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some intermediate modification.) Let $F_C$ be a torsion free sheaf on $C$. Pull it back: $F_{C'}:=\nu^*(F_C)/Torsion$.

Suppose $F_{C'}$ has global sections. Let $0\neq s\in H^0(F_{C'})$. When does $s$ descend to $C$, i.e. when is $s$ the pullback of a section of $F_C$?

Or, what is the condition that no global section of $F_{C'}$ comes from a global section of $F_C$? (Of course $h^0(F_{C'})=0$ is enough but probably is not necessary.)

Here is some local-to-global issue. Let $0\neq s\in H^0(F_{C'})$, such that for each point $pt\in C'$ this $s$ belongs to the preimage (not the pullback!) of the stalk of $F_C$, i.e.: $\nu^{-1}(F_{C,\nu(pt)})$. So, at each point of $C'$ the section $s$ "descends locally". Does this imply that $s$ descends globally?

What are the tools to treat such questions? Some sort of relative C'/C cohomology?