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Setting: $X$:projective surface over algebraically closed field $k$.

$T$:scheme over $k$.

$E$: Coherent sheaf on $X \times_k T$ , flat over $T$ and $\forall t \in T$, $E_t$ is rank 2 torsion-free sheaves.

Question We assume that $\exists t \in T$ s.t. $E_t$ has a non-degenerate global section. Then, is there $t \in U \subseteq T:$ open s.t. $\forall u \in U$, $E_u$ has a non-degenerate section ?

Originally, I was reading page 3, line 15 of this paper (https://arxiv.org/abs/alg-geom/9312011v1).

Any comment welcome! Thank you.

Edit:$E_t$ has a non degenerate global section means that there exists $s \in H^0(E_t)$ s.t by the $s$ we get a exact sequence

$ 0 \rightarrow \mathscr{O} \rightarrow E_t \rightarrow F \rightarrow 0$ where $F$ is a torsion free sheaf on X

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    $\begingroup$ What does non-degenerate global section mean to you? $\endgroup$
    – Mohan
    Commented May 25, 2020 at 15:58
  • $\begingroup$ Sorry.I added the definition of non-degenerate global section. $\endgroup$
    – Walter field
    Commented May 25, 2020 at 16:12
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    $\begingroup$ What if you take the family of $\mathscr L \oplus \mathscr L$ where $\mathscr L$ runs over all degree $0$ line bundles of a curve (or surface if you insist that $X$ is a surface)? Then the trivial line bundle has such a section, but the others don't have any nonzero global sections. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 25, 2020 at 16:14
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    $\begingroup$ By upper semicontinuity, a much more plausible question would be the opposite: if $E_t$ does not have a [insert adjective] section, then does the same hold in a neighbourhood? $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 25, 2020 at 16:23

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