Non zero sections of sheaves which vanish on all fibers

Question

I was stuck on the following question and I was wondering if someone more familiar with sheaf theory might be able to help me with it. $\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$

Suppose $(X,\mathcal{O}_X)$ is a locally ringed space and $\F$ is a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{F}_x$ denote the stalk at a point $x \in X$. We define the fiber of $\mathcal{F}$ at a point $x \in X$ to be the quotient: $$Fib(F)_x := \F_x / (\mathfrak{m}_x \mathcal{F}_x)$$ Given an open set $U$ containing $x$ there is a natural evaluation map $Ev_x \colon \F(U) \to Fib(F)_x$. Now here are my questions:

1. Can you give me an example of a sheaf with the property that there exists a section $s \in \F(U)$ for some open set $U$ such that for all $x \in U$ we have $Ev_x(s) = 0 $ but $s \neq 0$?

2. Are there any natural assumptions I can make to rule out this possibility? For the record, I would like something weaker than 'locally finitely generated and projective'.

Answer 1

I eventually found another example of 1 where the structure sheaf does not have the property.

$\newcommand{\O}{\mathcal{O}}$ Suppose $X$ is a smooth manifold and $\O_X$ is the sheaf of smooth functions on $X$. Let us fix a point $x_0$ and define $\mathcal{E}$ to be the submodule of $\O_X$ which consists of smooth functions which vanish to infinite order at $x_0$. That is, the elements of $\mathcal{E}$ vanish at $x_0$ and all derivatives vanish at $x_0$. This sheaf $\mathcal{E}$ does not yet have the desired property, but let us construct new sheaf.

Let $\mathcal{W}$ be the following sheaf. For an open $U \subset X$: $$ \mathcal{W}(U) = \begin{cases} 0 & x_0 \notin U \\ \mathcal{E}_{x_0} & x_0 \in U \end{cases} $$ Here $\mathcal{E}_{x_0}$ denotes the stalk of $\mathcal{E}$ at $x_0$. A fairly standard argument shows that $\mathcal{E}_{x_0}/ m_{x_0} \mathcal{E} = 0$. Therefore, any non-zero section of $\mathcal{W}$ has the desired property. So long as $dim (X) > 0$ the sheaf $\mathcal{W}$ is not trivial so we have a counter example.

This example clearly depends on properties of the ring of smooth functions. I am still curious if there exists an example where $\O_X$ is the sheaf of analytic functions on an analytic manifold.

Edit: I made a mistake in the definition of $\mathcal{W}$.