A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it together, hope that somebody can help me:

Let $X$ be a projective scheme, say a surface over $\mathbb{C}$, let $S=Spec(R)$ be a parameter scheme. Assume we have a familiy $\mathcal{F}$ of coherent sheaves on $X\times S$, flat over $S$, let $\pi: X\times S \rightarrow S$ be the projection.

Now assume $V=\pi_{*}\mathcal{F}$ is locally free on $S$ and commutes with base change. We denote the total space of the vector bundle associated to $V$ by $T$, so $T=Spec(Sym(V^{\vee}))$, and denote by $\alpha: T \rightarrow S$ the projection.

Define $\beta$ to be the map $(id,\alpha): X\times T \rightarrow X\times S$.

Question: Does $\beta^{*}\mathcal{F}$ have a canonical (or tautological) global section?

An element of $X\times T$ is a triple $(x,s,t)$ where $t\in H^0(X_s,\mathcal{F}_s)$. Now the fiber of $\beta^{*}\mathcal{F}$ over $(x,s,t)$ should be just the fiber of $\mathcal{F}$ over $(x,s)$. So i could look at the image $t((x,s))$ of $t_{(x,s)}$ in the fiber over $(x,s)$.

Is $\sigma: X\times T \rightarrow \beta^{*}\mathcal{F}$, $(x,s,t)\mapsto t((x,s))$ a well defined global section, that is: do we have $\sigma\in \Gamma(X\times T,\beta^{*}\mathcal{F})$? Is this a canonical or tautological section? I mean, there is no other reasonable way to define the map $\sigma$. Does this make sense at all?