Let $X$ be a smooth projective curve. Let $E_T$ be a family of vector bundles parameterised by scheme $T$ i.e a vector bundle over $X \times T$. Let for each $t \in T$ we have a surjection from $O_{X \times t}^p \to E_{X \times t}$.
Given a $t$ can I find an open set $V \subset T$ such that we have a surjection $O_{X \times V}^p \to E_{X \times V}$?
The situation is that we have quotient maps for each $t$, if I can at all produce a map (not necessarily surjection) on a neighborhood then I can proceed to show surjection (doing some stalk argument). But I am unable to glue the pointwise quotients to produce a local quotient on an open nbd.