# Extending Sections of Vector Bundles in Families

Let $f:X\rightarrow T$ be a smooth, projective morphism of finite type schemes over an algebraically closed field, with $T$ smooth, affine, and irreducible. Let $t_0$ be a closed point of $T$. Suppose given a vector bundle $\mathcal{W}$ on $X$ and a section $s$ of $\mathcal{W}|_{f^{-1}\{t_0\}}$. Can anything be said about locally extending $s$? E.g. is there an open (maybe only etale) neighborhood $U$ of $t_0$ and a section $\bar{s}$ of $\mathcal{W}|_{f^{-1}U}$ such that $\bar{s}$ restricts to $s$ on $f^{-1}\{t_0\}$? In my case, $X$ is a smooth relative curve over $T$, so feel free to assume that if it will help.

• Here's a promising related result in the world of smooth manifolds: math.stackexchange.com/questions/1213987/… – Cass Oct 25 '15 at 5:08
• Let $X=T\times C$ where $C$ is a smooth projective curve and $T=\mathrm{Pic}^0 C$ and $L$, the Poincare bundle. Then at $0\in T$, the fiber is is just $\mathcal{O}_C$, which has a section, but none of the nearby fibers have any sections (even at the etale level) – Mohan Oct 25 '15 at 22:53
• That's a fine answer if you care to submit it as one. – Cass Oct 26 '15 at 3:18