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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 \rightarrow F \rightarrow 0$ where $F$ is a torsion free sheaf.

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

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 \rightarrow F \rightarrow 0 ] where $F$ is a torsion free sheaf.

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 \rightarrow F \rightarrow 0$ where $F$ is a torsion free sheaf.

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.

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 \rightarrow F \rightarrow 0 ] where $F$ is a torsion free sheaf.